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This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1] , the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O (h − 2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O (h − 1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O (h − 1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines. • This study extends the analysis of the Gibou method to irregular three-dimensional domains, demonstrating second-order convergence in irregular domains. • This section presents proof of the optimality of the MILU preconditioner, which reduces the condition number of the discretized system from O (h − 2) to O (h − 1). • It introduces and theoretically demonstrates the SMILU preconditioner with sectorized ordering, which improves parallel computing efficiency while maintaining optimal conditioning. • Comprehensive theoretical analyses and numerical experiments are conducted to confirm the theoretical analyses in both 2D and 3D scenarios. • This study contributes to the advancement of computational efficiency in solving large-scale Poisson problems in scientific and engineering domains. [ABSTRACT FROM AUTHOR]

Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC function of 2-level reconstruction based on BVD algorithm), is formulated as a two-stage spatial reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) functions with two-level steepness are used as the candidate reconstruction functions. The final reconstruction function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales. Highlights • An innovative fifth-order shock capturing scheme is proposed. • Low-dissipation linear scheme is retrieved for smooth solution over all wave numbers. • Discontinuous and smooth solution of wide-band scales are simultaneously solved with high accuracy. • The free-mode solutions are faithfully maintained in long term computation. [ABSTRACT FROM AUTHOR]

Highlights • Construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. • Treat the time derivative of the pressure as a source of the concentration equation. • Apply the algorithm on unstructured meshes. • In the flux limiter, use the second-order flux as the lower-order one. • Use L2-projection of the porosity and construct special limiters that suitable for multi-component fluid mixtures. Abstract In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the j th component, c j , should be between 0 and 1. There are three main difficulties. Firstly, c j does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all c j ′ s and enforce ∑ j c j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure d p / d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to c j. To construct the BP technique, we will not approximate c j directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control. • We devise an adaptive hyperbolic-cross-space mapped Jacobi method for solving unbounded-domain multidimensional spatiotemporal equations. • Our numerical analysis indicates that using the AHMJ method leads to effective error control. • Our AHMJ method is more efficient than previous adaptive Hermite methods for solving certain multidimensional spatiotemporal equations. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of high order semi-Lagrangian particle methods to handle high dimensional Vlasov–Poisson systems. It is based on recent developments in the numerical analysis of particle methods and the paper focuses on specific algorithmic features to handle large dimensions. The methods are tested with uniform particle distributions in particular against a recent multi-resolution wavelet based method on a 4D plasma instability case and a 6D gravitational case. Conservation properties, accuracy and computational costs are monitored. The excellent accuracy/cost trade-off shown by the method opens new perspective for accurate simulations of high dimensional kinetic equations by particle methods. [ABSTRACT FROM AUTHOR]

An accurate and predictive modelling of slug flows is required by many industries (e.g., oil and gas, nuclear engineering, chemical engineering) to prevent undesired events potentially leading to serious environmental accidents. For example, the hydrodynamic and terrain-induced slugging leads to unwanted unsteady flow conditions. This demands the development of fast and robust numerical techniques for predicting slug flows. The presented in this paper study proposes a multi-fluid model and its implementation method accounting for phase appearance and disappearance. The numerical modelling of phase appearance and disappearance presents a complex numerical challenge for all multi-component and multi-fluid models. Numerical challenges arise from the singular systems of equations when some phases are absent and from the solution discontinuity when some phases appear or disappear. This paper provides a flexible and robust solution to these issues. A fully implicit formulation described in this work enables to efficiently solve governing fluid flow equations. The proposed numerical method provides a modelling capability of phase appearance and disappearance processes, which is based on switching procedure between various sets of governing equations. These sets of equations are constructed using information about the number of phases present in the computational domain. The proposed scheme does not require an explicit truncation of solutions leading to a conservative scheme for mass and linear momentum. A transient two-fluid model is used to verify and validate the proposed algorithm for conditions of hydrodynamic and terrain-induced slug flow regimes. The developed modelling capabilities allow to predict all the major features of the experimental data, and are in a good quantitative agreement with them. [ABSTRACT FROM AUTHOR]

Solving acoustic equations in the time domain, possibly coupled with the description of flexible structure dynamics, remains attractive as compared to solving the same in the frequency domain: this allows for better consideration of local non-linearities (acoustics/structure), and the boundary integral formulation (also known as BEM) offers an exact description of the infinite acoustic field based on a simple surface mesh (no need for 3D-volume discretization). Some issues remain however: the required memory space and computation time continue to grow rapidly when the number of elements of the surface mesh increases. In the case of a structure with a regular non-slender shape, the computational cost, measured in terms of required memory space, varies by Helmholtz number to the power of 4. This paper illustrates how the accelerating method called NGTD helps overcome this difficulty. This paper shows the applicability of 2 level NGTD to acoustic and vibroacoustic problems described solely by the hypersingular formulation for surfaces. It goes into more detail on some important aspects of the interpolation process and on the memory saving obtained. Implementation within the MOT (“March-On-Time”) ASTRYD code shows the benefits of this method. The memory requirement shows an estimated trend lower than power 1.35 of the number of surface elements. [ABSTRACT FROM AUTHOR]

In this paper, we derive a family of source term quadrature formulas for preserving third-order accuracy of the node-centered edge-based discretization for conservation laws with source terms on arbitrary simplex grids. A three-parameter family of source term quadrature formulas is derived, and as a subset, a one-parameter family of economical formulas is identified that does not require second derivatives of the source term. Among the economical formulas, a unique formula is then derived that does not require gradients of the source term at neighbor nodes, thus leading to a significantly smaller discretization stencil for source terms. All the formulas derived in this paper do not require a boundary closure, and therefore can be directly applied at boundary nodes. Numerical results are presented to demonstrate third-order accuracy at interior and boundary nodes for one-dimensional grids and linear triangular/tetrahedral grids over straight and curved geometries. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast time adaptive numerical method for interface tracking. The method uses an explicit multiresolution description of the interface, which is represented by wavelet vectors that correspond to the details of the interface on different scale levels. The complexity of standard numerical methods for interface tracking, where the interface is described by N marker points, is O ( N / Δ t ) , when a time step Δ t is used. The methods that we propose in this paper have O ( tol − 1 / p log ⁡ N + N log ⁡ N ) computational cost, at least for uniformly smooth problems, where tol is some given tolerance and p is the order of the time stepping method that is used for time advection of the interface. The adaptive method is robust in the sense that it can handle problems with both smooth and piecewise smooth interfaces (e.g. interfaces with corners) while keeping a low computational cost. We show numerical examples that verify these properties. [ABSTRACT FROM AUTHOR]

The miscible displacement of one fluid by another in a porous medium has received considerable attention in subsurface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns – referred to as viscous fingering – may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods. [ABSTRACT FROM AUTHOR]

This paper presents preliminary developments of entropy-stable residual distribution methods for scalar problems. Controlling entropy generation is achieved by formulating an entropy conserved signals distribution coupled with an entropy-stable signals distribution. Numerical results of the entropy-stable residual distribution methods are accurate and comparable with the classic residual distribution methods for steady-state problems. High order accurate extensions for the new method on steady-state problems are also demonstrated. Moreover, the new method preserves second order accuracy on unsteady problems using an explicit time integration scheme. The idea of the multi-dimensional entropy-stable residual distribution method is generic enough to be extended to the system of hyperbolic equations, which will be presented in the sequel of this paper. [ABSTRACT FROM AUTHOR]

Using a simple one-dimensional shock problem as an example, the present paper investigates numerical properties of the original material point method (MPM), the generalized interpolation material point (GIMP) method, the convected particle domain interpolation (CPDI) method, and the dual domain material point (DDMP) method. For a weak isothermal shock of ideal gas, the MPM cannot be used with accuracy. With a small number of particles per cell, GIMP and CPDI produce reasonable results. However, as the number of particles increases the methods fail to converge and produce pressure spikes. The DDMP method behaves in an opposite way. With a small number of particles per cell, DDMP results are unsatisfactory. As the number of particles increases, the DDMP results converge to correct solutions, but the large number of particles needed for convergence makes the method very expensive to use in these types of shock wave problems in two- or three-dimensional cases. The cause for producing the unsatisfactory DDMP results is identified. A simple improvement to the method is introduced by using sub-points. With this improvement, the DDMP method produces high quality numerical solutions with a very small number of particles. Although in the present paper, the numerical examples are one-dimensional, all derivations are for multidimensional problems. With the technique of approximately tracking particle domains of CPDI, the extension of this sub-point method to multidimensional problems is straightforward. This new method preserves the conservation properties of the DDMP method, which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations. [ABSTRACT FROM AUTHOR]

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary can effectively reduce the dissipation in numerical flux. Differently from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, the BVD strategy presented in this paper switches between a high-order polynomial and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. This work provides a simple and accurate alternative of great practical significance to the current high-order Godunov paradigm which overly pursues the smoothness within mesh cells under the questionable premiss that discontinuities only appear at cell interfaces. [ABSTRACT FROM AUTHOR]

Compact difference schemes have been used extensively for solving the incompressible Navier–Stokes equations. However, the earlier formulations of the schemes are of central type (called central compact schemes, CCS), which are dispersive and susceptible to numerical instability. To enhance stability of CCS, the optimal upwind compact schemes (OUCS) are developed recently by adding high order dissipative terms to CCS. In this paper, it is found that OUCS are essentially not of the upwind type because they do not use upwind-biased but central type of stencils. Furthermore, OUCS are not the most optimal since orders of accuracy of OUCS are at least one order lower than the maximum achievable orders. New upwind compact schemes (called standard upwind compact schemes, SUCS) are developed in this paper. In contrast to OUCS, SUCS are constructed based completely on upwind-biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations. Furthermore, SUCS can achieve the maximum achievable orders of accuracy and hence be more compact than OUCS. More importantly, SUCS have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we explore flux vector splittings for the MHD system of equations. Our approach follows the strategy that was initially put forward in Toro and Vázquez-Cendón (2012) [55] . We split the flux vector into an advected sub-system and a pressure sub-system. The eigenvalues and eigenvectors of the split sub-systems are then studied for physical suitability. Not all flux vector splittings for MHD yield physically meaningful results. We find one that is completely useless, another that is only marginally useful and one that should work well in all regimes where the MHD equations are used. Unfortunately, this successful flux vector splitting turns out to be different from the Zha–Bilgen flux vector splitting. The eigenvalues and eigenvectors of this favorable FVS are explored in great detail in this paper. The pressure sub-system holds the key to finding a successful flux vector splitting. The eigenstructure of the successful flux vector splitting for MHD is thoroughly explored and orthonormalized left and right eigenvectors are explicitly catalogued. We present a novel approach to the solution of the Riemann problem formed by the pressure sub-system for the MHD equations. Once the pressure sub-system is solved, the advection sub-system follows naturally. Our method also works very well for the Euler system. Our FVS successfully captures isolated, stationary contact discontinuities in MHD. However, we explain why any FVS for MHD is not adept at capturing isolated, stationary Alfvenic discontinuities. Several stringent one-dimensional Riemann problems are presented to show that the method works successfully and can effectively capture the full panoply of wave structures that arise in MHD. This includes compound waves and switch-on and switch-off shocks that arise because of the non-convex nature of the MHD system. [ABSTRACT FROM AUTHOR]

In numerical dosimetry, the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate (SAR) characterizing the human exposure to electromagnetic waves. However, this procedure remains time-consuming and a single simulation can request several hours. As a consequence, the influence of uncertain input parameters on the SAR cannot be analyzed using crude Monte Carlo simulation. The solution presented here to perform such an analysis is surrogate modeling. This paper proposes a novel approach to build such a surrogate model from a design of experiments. Considering a sparse representation of the polynomial chaos expansions using least-angle regression as a selection algorithm to retain the most influential polynomials, this paper proposes to use the selected polynomials as regression functions for the universal Kriging model. The leave-one-out cross validation is used to select the optimal number of polynomials in the deterministic part of the Kriging model. The proposed approach, called LARS-Kriging-PC modeling, is applied to three benchmark examples and then to a full-scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device. The performances of the LARS-Kriging-PC are compared to an ordinary Kriging model and to a classical sparse polynomial chaos expansion. The LARS-Kriging-PC appears to have better performances than the two other approaches. A significant accuracy improvement is observed compared to the ordinary Kriging or to the sparse polynomial chaos depending on the studied case. This approach seems to be an optimal solution between the two other classical approaches. A global sensitivity analysis is finally performed on the LARS-Kriging-PC model of the fetus exposure problem. [ABSTRACT FROM AUTHOR]

High-fidelity computation of shock-associated noise places stringent requirements on the accuracy, linear and nonlinear spectral properties of shock-capturing scheme. In this paper, a novel weighted nonlinear compact scheme with seventh-order accuracy is developed for the purpose of improving the linear and nonlinear spectral properties of original scheme (Zhang et al., (2008) [1]). The numerical fluxes at cell centers are obtained using upwind-biased weighted nonlinear interpolation based on the new S-type smoothness indicator that be constant for sine and cosine functions (Wu et al., (2020) [2]). Through systematic spectral analysis and numerical experiments, it is demonstrated that the newly proposed scheme has very weak nonlinear effect and reduces the dissipation and dispersion at the medium and high wavenumbers. The resolution for the short waves and the fine-scale structures is improved. It has the potential to become a suitable candidate for the computation of shock-associated noise. • A novel seventh-order weighted compact scheme named WCOM7S is proposed. • WCOM7S reduces the dissipation and dispersion at medium and high wavenumbers. • WCOM7S has good nonlinear spectral properties and weak nonlinear effect. [ABSTRACT FROM AUTHOR]

Multidimensional Riemann solvers have been formulated recently by the author (Balsara, 2010, 2012) [6,7] . They operate at the vertices of a two-dimensional mesh, taking input from all the neighboring states and yielding the resolved state and fluxes as output. The multidimensional Riemann problem produces a self-similar strongly interacting state which is the result of several one-dimensional Riemann problems interacting with each other. The prior work was restricted to the use of one-dimensional HLLC Riemann solvers as building blocks. In this paper, we formulate the problem in similarity variables. As a result, any self-similar one-dimensional Riemann solver can be used as a building block for the multidimensional Riemann solver. This paper focuses on the structure of the strongly-interacting state. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course .) In this work the strongly-interacting state is expanded in a set of basis functions that depend on the similarity variables. Consequently, the resolved state and the fluxes can be endowed with considerably richer sub-structure compared to prior work. Unlike the multidimensional HLLC Riemann solver, the need to independently specify a direction for the evolution of the contact discontinuity is eliminated. The richer sub-structure in the strongly-interacting state naturally accommodates waves that may be moving in any direction relative to the mesh, thereby minimizing mesh-imprinting. Two formulations are presented. The first formulation does not linearize the problem around a favorable state. Its derivation takes a few cues from the derivation of the multidimensional HLL Riemann solver. The second formulation identifies such a state and carries out a linearization of the fluxes about that state. This paper is the very first time that a series solution of the multidimensional Riemann problem has been presented. Explicit formulae are presented for up to quartic variation in the self-similar variables. While linear variations are sufficient for numerical work, the higher order terms in the series solution could prove useful for analytical studies of the multidimensional Riemann problem. The formulation presented here is general enough to accommodate any hyperbolic conservation law. It can also accommodate any one-dimensional Riemann solver and yields a multidimensional version of the same. It has been incorporated in the author's RIEMANN code. As examples of the different types of hyperbolic conservation laws, we use Euler flow, Magnetohydrodynamics (MHD) and relativistic MHD. As examples of different types of Riemann solvers, we show multidimensional formulations of HLL, HLLC and HLLD Riemann solvers for MHD all working fluently within this formulation. Several stringent test problems are presented. [ABSTRACT FROM AUTHOR]

This paper presents a stable numerical algorithm for the Brinkman equations by using weak Galerkin (WG) finite element methods. The Brinkman equations can be viewed mathematically as a combination of the Stokes and Darcy equations which model fluid flow in a multi-physics environment, such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain. In such applications, the flow is dominated by Darcy in some regions and by Stokes in others. It is well known that the usual Stokes stable elements do not work well for Darcy flow and vice versa. The challenge of this study is on the design of numerical schemes which are stable for both the Stokes and the Darcy equations. This paper shows that the WG finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both Darcy and the Stokes dominated flows. Error estimates of optimal order are established for the corresponding WG finite element solutions. The paper also presents some numerical experiments that demonstrate the robustness, reliability, flexibility and accuracy of the WG method for the Brinkman equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error , where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to , where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small. [Copyright &y& Elsevier]

Abstract: In this paper, we propose energy-conserving numerical schemes for the Vlasov–Ampère (VA) systems. The VA system is a model used to describe the evolution of probability density function of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, including the total energy which is comprised of the kinetic and electric energy. Unlike the total particle number conservation, the total energy conservation is challenging to achieve. For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In this paper, we develop the first Eulerian solvers that can preserve fully discrete total energy conservation. The main components of our solvers include explicit or implicit energy-conserving temporal discretizations, an energy-conserving operator splitting for the VA equation and discontinuous Galerkin finite element methods for the spatial discretizations. We validate our schemes by rigorous derivations and benchmark numerical examples such as Landau damping, two-stream instability and bump-on-tail instability. [Copyright &y& Elsevier]

Abstract: An adaptive finite volume method for 2D steady Euler equations on unstructured grids is proposed. The framework of the finite volume method for the steady Euler equations follows the one in the paper [G.H. Hu, R. Li, and T. Tang, A robust WENO type finite volume solver for steady Euler equations on unstructured grids, Commun. Comput. Phys. 9 (2011) 627–648]. In this paper, we introduce the mesh adaptive methods to improve the above numerical method. The features of this work include: (i) different reconstruction stencils for WENO reconstruction are discussed in detail, including their performance on the convergence of steady state solutions and on the application of h-adaptive methods, and (ii) an effective indicator for generating quality nonuniform mesh is proposed, which is based on the entropy production. The improvement of the numerical methods is demonstrated by plenty of numerical examples. [Copyright &y& Elsevier]

Abstract: The use of moment closures for the prediction of continuum and moderately non-equilibrium flows offers modelling and numerical advantages over other methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however their are two issues that limit their practical implementation. Firstly, for closures that have a treatment for heat transfer, fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation. Secondly, for these same closures, there are physically possible moment states for which the entropy-maximization problem has no solution and the entire framework breaks down. This paper demonstrates that affordable closed-form moment closures that are inspired by the maximum-entropy framework can be proposed. It is known that closing fluxes in the maximum-entropy hierarchy approach a singularity as the region of non-solvability is approached. This paper shows that, far from a disadvantage, this singularity allows for smooth and accurate prediction of shock-wave structure, even for high Mach numbers. The presence of unphysical “sub-shocks” within shock-profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment-closure technique. The realization that smooth shock profiles are, in fact, possible for moment methods with a moderate number of moments greatly increases the method’s applicability to high-speed flows. In this paper, a 5-moment system for a simple one-dimensional gas and a 14-moment system for realistic gases are developed and examined. Numerical solution for shock-waves at a variety of incoming flow Mach numbers demonstrate both the robustness and the accuracy of the closures. [Copyright &y& Elsevier]

Abstract: This paper presents major and new results in the numerical treatment of conjugate heat transfer problems. The stability analysis of a 1D implicit model problem for heat transfer between a fluid and a solid is performed in the traditional finite-volume (fluid)/finite-element (solid) configuration. The interface stability study is carried out according to the Godunov–Ryabenkii theory normal-mode analysis. Two interface boundary conditions are studied. First, the commonly used Dirichlet–Robin algorithm is described in detail and, to the best of our knowledge, the exact expression of an optimal coupling coefficient is formulated for the first time. It is shown that this optimal coefficient is the best choice in terms of stability and convergence rate. The effect and impact on stability of various solid data are also discussed. In the last section of this paper the same stability analysis is carried out for a general Robin–Robin interface condition and an optimal relationship between the two coupling parameters is also provided. No stability restrictions are introduced by these optimal interface treatments and some noteworthy expressions are provided. [Copyright &y& Elsevier]

Abstract: When one solves PDEs modelling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. For instance, the underlying physical assumptions for the Euler equations are the positivity of both density and pressure variables. We consider in this paper an unstructured vertex-based tesselation in . Given a MUSCL finite volume scheme and given a reconstruction method (including a limiting process), the point is to determine whether the overall scheme ensures the positivity. The present work is issued from seminal papers from Perthame and Shu (On positivity preserving finite volume schemes for Euler equations, Numer. Math. 73 (1996) 119–130) and Berthon (Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2) (2006) 495–509). They proved in different frameworks that under assumptions on the corresponding one-dimensional numerical flux, a suitable CFL condition guarantees that density and pressure remain positive. We first analyse Berthon’s method by presenting the ins and outs. We then propose a more general approach adding non geometric degrees of freedom. This approach includes an optimization procedure in order to make the CFL condition explicit and as less restrictive as possible. The reconstruction method is handled independently by means of τ-limiters and of an additional damping parameter. An algorithm is provided in order to specify the adjustments to make in a preexisting code based on a certain numerical flux. Numerical simulations are carried out to prove the accuracy of the method and its ability to deal with low densities and pressures. [Copyright &y& Elsevier]

Abstract: In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result. [Copyright &y& Elsevier]

Abstract: This work is a continuation of the authors efforts to develop high-order numerical methods for solving elliptic problems with complex boundaries using a fictitious domain approach. In a previous paper, a new method was proposed, based on the use of smooth forcing functions with identical shapes, mutually disjoint supports inside the fictitious domain and whose amplitudes play the role of Lagrange multipliers in relation to a discrete set of boundary constraints. For one-dimensional elliptic problems, this method shows spectral accuracy but its implementation in two dimensions seems to be limited to a fourth-order algebraic convergence rate. In this paper, a spectrally accurate formulation is presented for multi-dimensional applications. Instead of being specified locally, the forcing function is defined as a convolution of a mollifier (smooth bump function) and a Lagrange multiplier function (the amplitude of the bump). The multiplier function is then approximated by Fourier series. Using a Fourier Galerkin approximation, the spectral accuracy is demonstrated on a two-dimensional Laplacian problem and on a Stokes flow around a periodic array of cylinders. In the latter, the numerical solution achieves the same high-order accuracy as a Stokes eigenfunction expansion and is much more accurate than the solution obtained with a classical third order finite element approximation using the same number of degrees of freedom. [Copyright &y& Elsevier]

Abstract: This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids. [Copyright &y& Elsevier]

Abstract: In this paper, the multidimensional limiter for the second order finite volume schemes on the unstructured grid, namely the Weighted Biased Average procedure developed in our previous paper is extended to high order finite volume schemes solving hyperbolic conservation laws. This extension relies on two key techniques: the secondary reconstruction and the successive limiting procedure. These techniques are discussed in detail in the present paper. Numerical experiments shows that this limiting procedure is very effective in removing numerical oscillations in the vicinity of discontinuities. And furthermore this procedure is efficient, robust and accuracy preserving. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce two numerical methods to get an electric source that gives a specified electromagnetic field some time after the source starts emitting. The first one is called the Time Reversal Method (TRM) and originates from acoustics and the second one is called Linear Combination of Configuration Fields (LCCF) and consists in constructing and solving a linear problem in order to find a possible required source. In this paper, we show on 1D and 2D examples that the last method gives lower relative errors when compared with the time reversal method. [Copyright &y& Elsevier]

Abstract: A systematic approach is presented for constructing higher-order immersed boundary and ghost fluid methods for CFD in general, and fluid–structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid–structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, immersed boundary and ghost fluid methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid–structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the immersed interfaces. In some cases, they are also provably inconsistent at these locations. A methodology is presented in this paper for addressing this issue. It is developed for inviscid flows and prescribed structural motions. For the sake of clarity, but without any loss of generality, this methodology is described in one and two dimensions. However, its extensions to flow-induced structural motions and three dimensions are straightforward. The proposed methodology leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the formal order of spatial accuracy of this scheme. It is illustrated in this paper by its application to various finite difference and finite volume methods. Its impact is also demonstrated by one- and two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces. [Copyright &y& Elsevier]

Abstract: In this follow up paper to our previous study in Bayona et al. (2011) , we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) , both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter. [Copyright &y& Elsevier]

Abstract: In the present work, we present a novel numerical algorithm to couple the Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann equation with a finite volume like method for the solution of the Euler equations. Recently we presented in different methodologies which permit to solve fluid dynamics problems with localized regions of departure from thermodynamical equilibrium. The methods rely on the introduction of buffer zones which realize a smooth transition between the kinetic and the fluid regions. In this paper we extend the idea of buffer zones and dynamic coupling to the case of the Monte Carlo methods. To facilitate the coupling and avoid the onset of spurious oscillations in the fluid regions which are consequences of the coupling with a stochastic numerical scheme, we use a new technique which permits to reduce the variance of the particle methods . In addition, the use of this method permits to obtain estimations of the breakdowns of the fluid models less affected by fluctuations and consequently to reduce the kinetic regions and optimize the coupling. In the last part of the paper several numerical examples are presented to validate the method and measure its computational performances. [Copyright &y& Elsevier]

Abstract: In the present paper a fourth/fifth order upwind biased limiting strategy is presented for the simulation of turbulent flows and combustion. Because high order numerical schemes usually suffer from stability problems and TVD approaches often prevent convergence to machine accuracy the multi-dimensional limiting process (MLP) is employed. MLP uses information from diagonal volumes of a discretization stencil. It interacts with the TVD limiter in such a way, that local extrema at the corner points of the volume are avoided. This stabilizes the numerical scheme and enables convergence in cases, where standard limiters fail to converge. Up to now MLP has been used for inviscid and laminar flows only. In the present paper this technique is applied to fully turbulent sub- and supersonic flows simulated with a low Reynolds-number turbulence closure. Additionally, combustion based on finite-rate chemistry is investigated. An improved MLP version (MLP ld , low diffusion) as well as an analysis of its capabilities and limitations are given. It is demonstrated, that the scheme offers high accuracy and robustness while keeping the computational cost low. Both steady and unsteady test cases are investigated. [Copyright &y& Elsevier]

Abstract: In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas–droplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N >2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the Eulerian Multi Size Moment (EMSM) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor–Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed. [Copyright &y& Elsevier]

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of Y. Efendiev et al. (2011) [22]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis. • We clarify and simplify the numerical analysis previously presented in [J. Comput. Phys. 230 (2011), 937-955]. • We design and analyze GMsFEM spaces constructed using local Neumann and Dirichlet eigenvalues problems. • We present an example with no optimal GMsFEM convergence rate when only Neumann eigenvalues are used. • A novel and general regularity numerical analysis tool for the high-contrast multiscale elliptic problems is presented. • We present numerical results with realistic high-contrast multiscale coefficients. [ABSTRACT FROM AUTHOR]

A novel dual-scale compact method for solving nonparaxial Helmholtz equations at high wavenumbers is proposed and analyzed. The approach is based on decomposing the axisymmetric transverse domain and governing equation according to interconnected micro and macro regions to maintain the smoothness of the underlying problem. Dual compact strategies are then implemented for acquiring highly accurate and efficient beam propagation computations. Aiming at the highly oscillatory solution features, the paper provides a rigorous analysis on the numerical stability. It is shown that the dual-scaled compact method is asymptotically stable. The analysis also reveals necessary constraints for the conventional stability. Computer experiments including self-focusing beam propagation simulations are conducted with various domain scaling factors to validate the theoretical results. • Dual-scale compact method for solving highly oscillatory Helmholtz problems. • Asymptotic stability analysis at high-wavenumbers for decomposed difference scheme. • Spectrum analysis follows to bound the 2-norm of amplification matrices. • Numerical self-focusing optical beam simulations reinforce reliability of method. • Flexibility in dual-domain scaling factor is favorable in optical beam applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for a dendritic solidification phase field model with melt convection in the liquid phase, which is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation, the heat equation, and the weighted Navier-Stokes equations together. We first reformulate the model into a form which is suitable for numerical approximations and establish the energy dissipative law. Then, we develop a linear, decoupled, and unconditionally energy stable numerical scheme by combining the modified projection scheme for the Navier-Stokes equations, the Invariant Energy Quadratization approach for the nonlinear anisotropic potential, and some subtle explicit-implicit treatments for nonlinear coupling terms. Stability analysis and various numerical simulations are presented. • We propose an efficient scheme for solving the anisotropic dendritic model with melt convection. • The model is formulated into a form which is suitable for energy dissipative law. • The scheme is decoupled, unconditionally energy stable, linear, and first-order accurate in time. • We present ample numerical tests that are consistent with the benchmarks. [ABSTRACT FROM AUTHOR]

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. • The Krylov implicit integration factor discontinuous Galerkin methods were first designed on sparse grids. • The new methods can simulate high spatial dimensional reaction-diffusion equations efficiently. • Both theoretical analysis and numerical experiments were performed to study the new methods. [ABSTRACT FROM AUTHOR]

Abstract Perfectly Matched Layer (PML) is a widely adopted non-reflecting boundary treatment for wave simulations. Reducing numerical reflections from a discretized PML has been a long lasting challenge. This paper presents a new discrete PML for the multi-dimensional scalar wave equation which produces no numerical reflection at all. The reflectionless discrete PML is discovered through a straightforward derivation using Discrete Complex Analysis. The resulting PML takes an easily-implementable finite difference form with compact stencil. In practice, the discrete waves are damped exponentially in the PML, and the error due to domain truncation is maintained at machine zero by a moderately thick PML. The numerical stability of the proposed PML is also demonstrated. Highlights • Perfectly Matched Layers can also be made reflectionless at the discrete level. • The measured reflection is maintained at machine zero at low cost. • Discrete complex analysis is introduced for deriving a reflectionless discrete PML. • Numerical evidence shows the stability of the proposed discrete PML. [ABSTRACT FROM AUTHOR]

Abstract A high-order numerical method for interface-preserving level-set reinitialization is presented in this paper. In the interface cells, the gradient of the level-set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved. In the non-interface cells, we compute the gradient of the level-set function by solving a Hamilton–Jacobi equation as a conservation law system using the discontinuous Galerkin method, following the work by Hu and Shu [SIAM J. Sci. Comput. 21 (1999) 660–690]; the missing constant is then recovered by the continuity of the level-set function while taking into account the characteristics. To handle highly distorted initial conditions, we develop a hybrid numerical flux that combines the Lax–Friedrichs flux and the penalty flux. Our method is stable for non-trivial test cases and handles singularities away from the interface very well. When derivative singularities are present on the interface, a second-derivative limiter is designed to suppress the oscillations. At least (N + 1) th order accuracy in the interface cells and N th order in the whole domain are observed for smooth solutions when N th degree polynomials are used. Two dimensional test cases are presented to demonstrate superior properties such as accuracy, long-term stability, interface-preserving capability, and easy treatment of contact lines. We also show some preliminary results on the pinch-off process of a pendant drop, where topological changes of the fluid interface are involved. Our method is readily extendable to three dimensions and adaptive meshes. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider Riemann problems for the shallow water equations with discontinuous topography whose initial conditions correspond to a wet–dry front: at time t = 0 there is vacuum on the right or on the left of the step. Besides the theoretical interest of this analysis, the results may be useful to design numerical methods and/or to produce reference solutions to compare different schemes. We show that, depending on the state at the wet side, 0, 1, or 2 self-similar solutions can be constructed by composing simple waves. In problems with 0 solutions, the step acts as an obstacle for the fluid and physically meaningful solutions can be constructed by interpreting the problem as a partial Riemann problems for the homogeneous shallow water system. Some numerical results are shown where different numerical methods are compared. In particular, it is shown that, in the non-uniqueness cases, the numerical solutions can converge to one or to the other solution, what is the reason that explains the huge differences observed when different numerical methods are applied to the shallow water system with abrupt changes in the bottom. Moreover, problems with zero solutions will be reinterpreted as Partial Riemann problems for the homogeneous system what will allow us to build a physically solution. When one side of the step is wet and the other one is dry. We will specify the regions where we can find zero, one or two solutions, giving the form of the solution when it is possible and giving an alternative when it is not possible. Highlights • Wet–dry Riemann problems for the shallow water system are considered. • 0, 1, or 2 solutions are found depending on the wet state. • Solutions based on a reinterpretation are proposed in the case of 0 solutions. • This analysis is useful to better understand the behavior of the numerical methods. • The correct simulation of wet–dry fronts is crucial in applications. [ABSTRACT FROM AUTHOR]

Abstract Numerical simulations of multi-scale flow problems such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics and various other flow problems with complex physics require high-order methods with high spectral resolutions. For instance, the receptivity mechanisms in the hypersonic boundary layer are the resonant interactions between forcing waves and boundary-layer waves, and the complex wave interactions are difficult to be accurately predicted by conventional low-order numerical methods. High-order methods, which are robust and accurate in resolving a wide range of time and length scales, are required. Currently, the high-order finite difference methods for simulations of hypersonic flows are usually upwind schemes or compact schemes with fifth-order accuracy or lower [1]. The objective of this paper is to develop and analyze a new very high-order numerical scheme with the spectral-like resolution for flow simulations on structured grids, with focus on smooth flow problems involving multiple scales. Specifically, a new upwind multi-layer compact (MLC) scheme with spectral-like resolution up to seventh order is derived in a finite difference framework. By using the 'multi-layer' idea, which introduces first derivatives into the MLC schemes and approximates the second derivatives, the resolution of the MLC schemes can be significantly improved within a compact grid stencil. The auxiliary equations are required and they are the only nontrivial equations, which contributes to good computational efficiency. In addition, the upwind MLC schemes are derived based on the idea of constructing upwind schemes on centered stencils with adjustable parameters to control the dissipation. Fourier analysis is performed to show that the MLC schemes have small dissipation and dispersion in a very wide range of wavenumbers in both one- and two-dimensional cases, and the anisotropic error is much smaller than conventional finite difference methods in the two-dimensional case. Furthermore, the stability analysis with matrix method shows that high-order boundary closure schemes are stable because of compactness of the stencils. The accuracies and rates of convergence of the new schemes are validated by numerical experiments of the linear advection equation, the nonlinear Euler equations, and the Navier–Stokes equations. The numerical results show that good computational efficiency, very high-order accuracies, and high spectral resolutions especially on coarse meshes can be attained with the MLC schemes. Overall, the MLC scheme has the properties of simple formulations, high-order accuracies, spectral-like resolutions, and compact stencils, and it is suitable for accurate simulation of smooth multi-scale flows with complex physics. Highlights • Multi-layer idea to build high-order and high-resolution schemes on short stencils. • Upwind schemes on centered stencil with an adjustable dissipation parameter. • Explicit compact finite difference formulas derived on structured grids. • Linear scheme to maintain low computational costs and spectral-like resolution. [ABSTRACT FROM AUTHOR]

Abstract This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain "shifted equivalent sources" which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme (2014) [4]). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications. Highlights • First high-order accelerated solver for periodic scattering including Wood anomalies. • The methodology greatly reduces the number of shifted Green function evaluations. • Additional acceleration is obtained by means of a dual spatial/spectral approach. • Efficient solution of highly challenging practical 2D scattering problems. • A three-dimensional version of this approach has been found equally effective. [ABSTRACT FROM AUTHOR]

Abstract: The present paper aims to provide an efficient and flexible method for the initialization of a zonal RANS/LES type calculation when the resolution of the near wall region is treated in RANS mode. Indeed, when part of the boundary layer must be resolved in LES mode, one generally experiences a very long transient state, which makes this approach inapplicable to industrial applications. The skillful combination of a Zonal Detached Eddy Simulation method (ZDES), a Synthetic Eddy Method (SEM) and a self-adaptative dynamic forcing approach enables this. The two former being taken as framework, while the latter, based on innovative considerations, is the purpose of the paper. The main strength of the dynamic forcing method comes from its local nature, enabling to treat geometrically complex applications. A new definition of the dynamic forcing method, based on error, is derived from the original one. It dramatically increases the efficiency of the inflow generation. Indeed, the dynamic forcing method allows to reduce the transition distance up to 76%, compared to the SEM inflow by itself, when a RANS/LES type resolution is employed. Thus the use of a synthetic turbulence generation method is now affordable for industrial applications both technically and economically. A particular attention is brought to the behavior and the parametrization of such an approach, with regards to the other simulation parameters. The authors will try to give all the information required to successfully apply the present strategy on a particular case. [Copyright &y& Elsevier]