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We propose general non-accelerated [The results for non-accelerated methods first appeared in December 2020 in the preprint (A. Agafonov, D. Kamzolov, P. Dvurechensky, and A. Gasnikov, Inexact tensor methods and their application to stochastic convex optimization, preprint 2020. arXiv:2012.15636)] and accelerated tensor methods under inexact information on the derivatives of the objective, analyse their convergence rate. Further, we provide conditions for the inexactness in each derivative that is sufficient for each algorithm to achieve a desired accuracy. As a corollary, we propose stochastic tensor methods for convex optimization and obtain sufficient mini-batch sizes for each derivative. [ABSTRACT FROM AUTHOR]

Unsteady counterflows are employed to understand and model the effect of turbulence on flames. We present a novel numerical approach for the simulation of one-dimensional unsteady counterflow flames with fourth order spatial discretization and up to fourth order time discretization. The approach couples a three-stage Lobatto IIIa formula for boundary value problems and variable-order, variable time step size Backward Differentiation Formulas for time integration. The framework is explained in detail, its computational performance is analysed, and its use is demonstrated for the case of stochastic forcing of premixed counterflow flames, whereby the imposed rate of strain is a multi-scale lognormal discrete random process with exponential autocorrelation. High-order spatial and temporal discretization make the approach well-suited for the accurate and computationally efficient simulation of the effect of turbulence on flames, which are characterised by large amplitude stochastic fluctuations of the local rate of strain. [ABSTRACT FROM AUTHOR]

Highly accurate simulation of plasma transport is needed to drive the successful design and operation of magnetically confined fusion reactors. Unfortunately, the extreme anisotropy present in magnetized plasmas results in thin boundary layers that are expensive to resolve. This work investigates how various mesh refinement strategies might reduce that expense to allow for more efficient simulation by comparing standard variable refinement approaches that use a field quantity to an adaptive approach that uses an error estimator. It is first verified that higher order discretization only realizes the proper rate of convergence once the mesh resolves the thin boundary layer, therefore motivating the focusing of refinement on the boundary layer. For three two-dimensional test cases that contain characteristic features of tokamak magnetic fields, an exponential refinement strategy based on the magnetic flux function, which is the standard refinement approach in the field, is compared to an adaptive strategy utilizing the established Zienwiekicz and Zhu error estimator. The adaptive mesh refinement strategy consistently achieves the same accuracy using orders of magnitude less degrees of freedom than either exponential or uniform refinement. This result makes the adaptive refinement strategy more efficient than the exponential refinement strategy while also being more generalizable to problems with complex magnetic geometries. Scaling laws are derived that quantify the improvement in cost of the adaptive refinement strategy over other refinement approaches in 2D and 3D. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a novel collocation method to solve nonlinear transient heat transfer problems in typical intelligent materials. The new method is derived from the weighted residual method in which the weight of Jacobian is proven to be the best choice. In the paper, the physical variables and their fluxes are respectively approximated by two series of Lagrange interpolation polynomials at Chebyshev-Gauss-Lobatto nodes. In the proposed method, the mass matrices of the final linear system of equations are diagonal and the explicit time discretization method can be employed. Using the proposed method, we solved the heat transfer in intelligent materials and it is proven to be quite efficient and accurate, which benefits from that no large sparse matrix is generated and solved. By the proposed method, one can easily solve the problems with complex geometries. [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin method and related flavors of high-order spectral element methods provide many well-known benefits for the spatial discretization of partial differential equations such as the Navier–Stokes equations. However, practical problems of engineering relevance such as large-eddy simulation of turbulent flows over complex geometries are computationally intractable by standard explicit time integration methods, necessitating the use of implicit methods. The efficient solution of the nonlinear algebraic systems arising from implicit time integration methods applied to DG discretizations of nonlinear PDEs is challenging; standard linearization methods result in very stiff block-sparse systems with prohibitive computational and memory requirements. This paper presents a low-memory, computationally efficient, implicit solution method that combines a framework of nonlinear polynomial multigrid, adaptive explicit Runge–Kutta smoothers, implicit Jacobian-free coarse level smoothers, nonlinear Krylov subspace acceleration and adaptive time stepping using feedback control of the nonlinear solver convergence rate. [ABSTRACT FROM AUTHOR]

The solution of conservation laws with parametrised shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop reduced order models for compressible flows. The optimal transport theory allows us to compute high-order r-adaptive meshes from a starting reference mesh by solving the Monge–Ampère equation. A high-order discretization of the conservation laws enables high-order solutions to be computed on the resulting r-adaptive meshes. Furthermore, the Monge–Ampère solutions contain mappings that are used to reduce the spatial locality of the resulting solutions and make them more amenable to model reduction. We use a non-intrusive model reduction method to construct reduced order models of both the mesh and the solution. The procedure is demonstrated on three supersonic and hypersonic test cases, with the hybridisable discontinuous Galerkin method being used as the full order model. [ABSTRACT FROM AUTHOR]

This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most O (log ⁡ (σ 0 − 1 ϵ − 2)) outer iterations to generate an ϵ-approximate KKT point, where σ 0 is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of O (ϵ − 4) , when σ 0 = ϵ − 2 and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of O (ϵ − (p + 1) / p) is established when appropriate p-order methods ( p ≥ 2) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper studies the efficient methods for option pricing under multivariate rough volatility models. The characteristic functions of the asset log-price, which play important role in the option pricing under the multivariate rough volatility models, are determined by a system of parametric nonlinear fractional Riccati equations. This paper obtains the results on the existence, uniqueness and regularity of the solutions to the parametric nonlinear fractional Riccati equations, proposes a high-order scheme to solve the system and proves the high-order convergence. The option pricing problem is solved by the Fourier-cosine formula with the fast approximation of the characteristic functions. Numerical examples are carried out to confirm the theoretical results and show efficiency of the methods. [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre–Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing. [ABSTRACT FROM AUTHOR]

The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in some previous papers for explicit schemes based on the application of a well-balanced reconstruction operator is applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws. [ABSTRACT FROM AUTHOR]

We propose high-order schemes for nonlinear fractional initial value problems. We split the fractional integral into a history term and a local term. We take advantage of the sum of exponentials (SOE) scheme in order to approximate the history term. We also use a low-order quadrature scheme to approximate the fractional integral appearing in the local term and then apply a spectral deferred correction (SDC) method for the approximation of the local term. The resulting one-step time-stepping methods have high orders of convergence, which make adaptive implementation and accuracy control relatively simple. We prove the convergence and stability of the proposed schemes. Finally, we provide numerical examples to demonstrate the high-order convergence and adaptive implementation. [ABSTRACT FROM AUTHOR]

The following work concerns the construction of an entropy dissipative finite volume solver based on the convex combination of an entropy conservative and an entropy dissipative flux. We aim to construct a semidiscrete scheme that is entropy stable in the sense of the entropy criterion of Dafermos as well as in the classical sense entropy dissipative. The proposed semidiscrete scheme shows nice properties like 2p order accuracy in smooth regions as well as a non-oscillatory behavior around shocks. [ABSTRACT FROM AUTHOR]

The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network. [ABSTRACT FROM AUTHOR]

In this paper, we study a fully composite formulation of convex optimization problems, which includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems with simple nondifferentiable components. We treat all these formulations in a unified way, highlighting the existence of very natural optimization schemes of different order p ≥ 1. As the result, we obtain new high-order (p ≥ 2) optimization methods for composite formulation. We prove the global convergence rates for them under the most general conditions. Assuming that the upper-level component of our objective function is subhomogeneous, we develop efficient modification of the basic fully composite first-order and second-order methods and propose their accelerated variants. [ABSTRACT FROM AUTHOR]

We present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients. • Developed and analyzed novel sharp immersed method for PDEs with moving boundaries. • The approach is compatible with a wide range of high-order time integration schemes. • The moving boundary treatment achieves high order spatial/temporal convergence rates. • High order convergence maintained for 3D simulations with moving nonconvex bodies. • Surface quantities remain accurate and non-oscillatory even at low resolutions. [ABSTRACT FROM AUTHOR]

We present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations. • A new stabilised spectral element method based on continuous solutions and face upwinding. • Using a special set of local nodes, the scheme is stable for arbitrary polynomial degrees. • For the special case of constant coefficient 1D problems, the scheme can be written as a Spectral Difference method. • In general, the method can be understood as a nodally-integrated Petrov-Galerkin scheme. • Some benefits over the popular Discontinuous Galerkin method include fewer degrees of freedom, sparser Jacobian matrices, and better CFL condition. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the Basic Tensor Methods, which use approximate solutions of the auxiliary problems. The quality of this solution is described by the residual in the function value, which must be proportional to ϵ p + 1 p , where p ≥ 1 is the order of the method and ϵ is the desired accuracy in the main optimization problem. We analyse in details the auxiliary schemes for the third- and second-order tensor methods. The auxiliary problems for the third-order scheme can be solved very efficiently by a linearly convergent gradient-type method with a preconditioner. The most expensive operation in this process is a preliminary factorization of the Hessian of the objective function. For solving the auxiliary problem for the second order scheme, we suggest two variants of the Fast Gradient Methods with restart, which converge as O (1 k 6 ) , where k is the iteration counter. Finally, we present the results of the preliminary computational experiments. [ABSTRACT FROM AUTHOR]

We consider time-harmonic Maxwell’s equations set in a heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in L 2 , we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order Nédélec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell’s equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the right-hand side only exhibits L 2 regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmholtz equation and showing the interest of high-order methods. [ABSTRACT FROM AUTHOR]

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding ε-approximate stationary points of convex functions that are p-times differentiable with ν-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O ϵ − 1 / (p + ν − 1) iterations to reduce the norm of the gradient of the objective below given ϵ ∈ (0 , 1). For accelerated tensor schemes, we establish improved complexity bounds of O ϵ − (p + ν) / [ (p + ν − 1) (p + ν + 1) ] and O | log ⁡ (ϵ) | ϵ − 1 / (p + ν) , when the Hölder parameter ν ∈ [ 0 , 1 ] is known. For the case in which ν is unknown, we obtain a bound of O ϵ − (p + 1) / [ (p + ν − 1) (p + 2) ] for a universal accelerated scheme. Finally, we also obtain a lower complexity bound of O ϵ − 2 / [ 3 (p + ν) − 2 ] for finding ε-approximate stationary points using p-order tensor methods. [ABSTRACT FROM AUTHOR]

An adaptive 2D nonhydrostatic dynamical core is proposed by using the multi-moment constrained finite-volume (MCV) scheme and the Berger-Oliger adaptive mesh refinement (AMR) algorithm. The MCV scheme takes several point-wise values within each computational cell as the predicted variables to build high-order schemes based on single-cell reconstruction. Two types of moments, such as the volume-integrated average (VIA) and point value (PV), are defined as constraint conditions to derive the updating formulations of the unknowns, and the constraint condition on VIA guarantees the rigorous conservation of the proposed model. In this study, the MCV scheme is implemented on a height-based, terrain-following grid with variable resolution to solve the nonhydrostatic governing equations of atmospheric dynamics. The AMR grid of Berger-Oliger consists of several groups of blocks with different resolutions, where the MCV model developed on a fixed structured mesh can be used directly. Numerical formulations are designed to implement the coarse-fine interpolation and the flux correction for properly exchanging the solution information among different blocks. Widely used benchmark tests are carried out to evaluate the proposed model. The numerical experiments on uniform and AMR grids indicate that the adaptive model has promising potential for improving computational efficiency without losing accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized framework for the derivation of compact difference schemes for the approximation of an arbitrary order derivative. We formulate a methodology for calculating the approximation coefficients involving a large number of mesh points leading to very high-order formulas, e.g., 20th or 30th order. The main emphasis is on the first and second order derivative occurring in the Navier-Stokes equations governing fluid flow problems. Contrary to most studies on compact schemes in which the computational stencils employed in the discretization are narrow and consist of only a few mesh points K , in the presented paper we consider wide stencils that in practice are limited only by the overall number of the nodes in a computational grid. We analyze the features of such wide stencil based schemes in physical and spectral space. It is shown that the solutions obtained applying the schemes of the same order but involving different number of the mesh points characterize significantly different accuracy. This is presented by direct comparisons of the analytically derived formula for the leading terms of the truncation errors as well as in practical simulations. The test cases include a second order ordinary differential equation with variable coefficients, 1D Burgers equation and three typical benchmark problems used in Computational Fluid Dynamics (CFD), i.e., inviscid vortex advection, convected Taylor-Green vortices and a double jet flow. The obtained results are compared with available analytical solutions and the solutions obtained with the help of a pseudo-spectral method. We found that very high-order compact schemes reach the spectral accuracy characterized by the error decrease proportional to O (h K). In some cases, they show even higher accuracy, and moreover, reaching the limit of the machine precision they are substantially more resistant to the round-off errors. [ABSTRACT FROM AUTHOR]

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods. [ABSTRACT FROM AUTHOR]

A diffused-interface approach based on the Allen-Cahn phase field equation is developed within a high-order discontinuous Galerkin framework. The interface capturing technique is based on the balance between explicit diffusion and sharpening terms in the phase field equation, where the former term involves the computation of the local interface normal vectors. Due to the well-known Gibbs phenomenon encountered in high-order discretisations, the accurate evaluation of the normal vectors requires special consideration. To this end, a non-linear preconditioning strategy is proposed where an additional smooth level-set function, advected by the velocity field, is used for the evaluation of the normal vectors. It is shown that for appropriate choices of numerical fluxes and parameters of the model, the phase field remains bounded without any need for explicit regularisation. In order to preserve isolated interfaces, a quasi-conservative discretisation of the five-equation model is employed. A series of numerical experiments of increasing complexity are performed in order to assess the accuracy and robustness of the developed methodology, including two-phase flows involving viscous effects, gravitational forces, and surface tension. [ABSTRACT FROM AUTHOR]

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. In this work, we extend implicit shock tracking to time-dependent problems using a slab-based space-time approach. This is achieved by reformulating a time-dependent conservation law as a steady conservation law in one higher dimension and applying existing implicit shock tracking techniques. To avoid computations over the entire time domain and unstructured mesh generation in higher dimensions, we introduce a general procedure to generate conforming, simplex-only meshes of space-time slabs in such a way that preserves features (e.g., curved elements, refinement regions) from previous time slabs. The use of space-time slabs also simplifies the shock tracking problem by reducing temporal complexity. Several practical adaptations of the implicit shock tracking solvers are developed for the space-time setting including 1) a self-adjusting temporal boundary, 2) nondimensionalization of a space-time slab, 3) adaptive mesh refinement, and 4) shock boundary conditions, which lead to accurate solutions on coarse space-time grids, even for problems with complex flow features such as curved shocks, shock formation, shock-shock and shock-boundary interaction, and triple points. [ABSTRACT FROM AUTHOR]

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-order Runge–Kutta method from 17 to 16. Unlike the vast majority of published Runge–Kutta methods, the method presented in this paper was not constructed via symbolic analysis or the use of simplifying assumptions, but instead by directly applying numerical optimization and root-finding algorithms to the order conditions. A naïve implementation of these algorithms would be made computationally infeasible by the considerable size and complex structure of the Butcher equations. However, we present a collection of software optimizations that greatly accelerate the evaluation of the Butcher equations and their derivatives while mitigating the effects of destructive cancellation, pushing these techniques within the realm of feasibility. While we do not have a formal proof of order, we present the results of numerical experiments to demonstrate that our method satisfies the Butcher equations to an accuracy of over 3000 decimal digits. [ABSTRACT FROM AUTHOR]

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation. [ABSTRACT FROM AUTHOR]

In this note we consider the continuous Galerkin time‐stepping method of arbitrary order as a possible discretization scheme of nonlinear initial value problems. In addition, we develop and generalize a well‐known existing result for the discrete solution by applying a general linearizing procedure to the nonlinear discrete scheme including also the simplified Newton solution procedure. In particular, the presented existence results are implied by choosing sufficient small time steps locally. Furthermore, the established existence results are independent of the local approximation order. Moreover, we will see that the proposed solution scheme is able to significantly reduce the number of iterations. Finally, based on existing and well‐known a priori error estimates for the discrete solution, we present some numerical experiments that highlight the proposed results of this note. [ABSTRACT FROM AUTHOR]

Linear theory continues to play a critical role in extracting the underlying primary instability dynamics of boundary layer transition. However, such methods have difficulty in isolating the dynamics of secondary instabilities, which dominate the later stages of transition. A Floquet analysis can bridge this gap; however, typical matrix-based approaches can be cumbersome to implement in general compressible scenarios. We address this difficulty by developing a matrix-free perturbation technique to extract the three-dimensional linearized secondary dynamics of compressible inhomogeneous flowfields. The method can be implemented into traditional Navier-Stokes codes in a straightforward manner. It is then used to examine the transition mechanism in a hypersonic boundary layer saturated by a second-mode instability. The linear response of the time-periodic basic-state is shown to precisely identify the non-parallel amplification and breakdown of oblique modes that disintegrate the boundary layer. It also correctly predicts the fundamental resonance mechanism that leads to transition. [ABSTRACT FROM AUTHOR]

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. In this article it is shown that for the standard space of polynomials used with triangular elements, denoted T(p), where p is the degree of the space, there is no diagonal approximate mass matrix that permits accurate solutions where accuracy is defined as giving an exact projection of functions in T(p - 1). In light of this, a lower-triangular pseudo-mass matrix method is introduced that requires only local operations, and the method's accuracy is demonstrated for the space T(3). The pseudo-mass matrix and accompanying high-order basis allow for computationally efficient time-stepping techniques without sacrificing the accuracy of the spatial approximation for unstructured triangular meshes. [ABSTRACT FROM AUTHOR]

The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators has received significant attention in the literature in the context of Poisson‐type problems, but preconditioners and solvers for inverting more general operators are not very well‐developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high‐order H1 finite element problems on the same mesh, efficient H(curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor‐product basis functions on an unstructured mesh of quadrilaterals. Our approach uses a sparsified H1 solver constructed on a low‐order mesh of the nodal points of the underlying high‐order space, and we show that the resulting H(curl) preconditioner is effective at very high polynomial orders for two‐dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O(pd + 1) floating point operation count and optimal O(pd) memory transfer requirements, outperforming existing Maxwell preconditioners in the high‐order regime. [ABSTRACT FROM AUTHOR]

In this paper, we design a truly exact perfect absorbing layer (PAL) for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This technique is based on a complex compression coordinate transformation in polar coordinates and a judicious substitution of the unknown field in the artificial layer. Compared with the widely used perfectly matched layer (PML) methods, the distinctive features of PAL lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain reduced by the truncation layer; (ii) with the substitution, the PAL-equation is free of singular coefficients and the substituted unknown field is essentially nonoscillatory in the layer; and (iii) the construction is valid for general star-shaped domain truncation. By formulating the variational formulation in Cartesian coordinates, the implementation of this technique using standard spectralelement or finite-element methods can be made easy as a usual coding practice. We provide ample numerical examples to demonstrate that this method is highly accurate and robust for very high wavenumbers and thin layers. It outperforms the classical PML and the recently advocated PML using unbounded absorbing functions. Moreover, it can fix some flaws of the PML approach. [ABSTRACT FROM AUTHOR]

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the auxiliary problems that appear in p-order tensor methods for unconstrained minimization of convex functions with ν-Hölder continuous pth derivatives. This type of auxiliary problems corresponds to the minimization of a (p + ν) -order regularization of the pth-order Taylor approximation of the objective. For the case p = 3, we consider the use of Gradient Methods with Bregman distance. When the regularization parameter is sufficiently large, we prove that the referred methods take at most O (log ⁡ (ϵ − 1)) iterations to find either a suitable approximate stationary point of the tensor model or an ε-approximate stationary point of the original objective function. [ABSTRACT FROM AUTHOR]

MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM's goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. [ABSTRACT FROM AUTHOR]

High-order implicit shock tracking (fitting) is a class of high-order, optimization-based numerical methods to approximate solutions of conservation laws with non-smooth features by aligning elements of the computational mesh with non-smooth features. This ensures the non-smooth features are perfectly represented by inter-element jumps and high-order basis functions approximate smooth regions of the solution without nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. In this work, we introduce a robust implicit shock tracking framework specialized for problems with parameter-dependent lead shocks (i.e., shocks separating a farfield condition from the downstream flow), which commonly arise in high-speed aerodynamics and astrophysics applications. After a shock-aligned mesh is produced at one parameter configuration, all elements upstream of the lead shock are removed and the nodes on the lead shock are positioned for new parameter configurations using the implicit shock tracking solver. The proposed framework can be used for most many-query applications involving parametrized lead shocks such as optimization, uncertainty quantification, parameter sweeps, "what-if" scenarios, or parameter-based continuation. We demonstrate the robustness and flexibility of the framework using a one-dimensional space-time Riemann problem, and two- and three-dimensional supersonic and hypersonic benchmark problems. • Robust many-query analysis of flow problems with parametrized lead shock. • Parameter-dependent lead shock treated as moving boundary. • Moving shock boundary determined using implicit shock tracking. • Elements upstream of lead shock removed from computational domain. • High-order resolution of 2D and 3D high-speed flows on coarse meshes. [ABSTRACT FROM AUTHOR]

Dye experimentation is a widely used method in experimental fluid mechanics for flow analysis or for the study of the transport of particles within a fluid. This technique is particularly useful in biomedical diagnostic applications ranging from hemodynamic analysis of cardiovascular systems to ocular circulation. However, simulating dyes governed by convection-diffusion partial differential equations (PDEs) can also be a useful post-processing analysis approach for computational fluid dynamics (CFD) applications. Such simulations can be used to identify the relative significance of different spatial subregions in particular time intervals of interest in an unsteady flow field. Additionally, dye evolution is closely related to non-discrete particle residence time (PRT) calculations that are governed by similar PDEs. This contribution introduces a pseudo-spectral method based on Fourier continuation (FC) for conducting dye simulations and non-discrete particle residence time calculations without numerical diffusion errors. Convergence and error analyses are performed with both manufactured and analytical solutions. The methodology is applied to three distinct physical/physiological cases: 1) flow over a two-dimensional (2D) cavity; 2) pulsatile flow in a simplified partially-grafted aortic dissection model; and 3) non-Newtonian blood flow in a Fontan graft. Although velocity data is provided in this work by numerical simulation, the proposed approach can also be applied to velocity data collected through experimental techniques such as from particle image velocimetry. • Fast FC-based solver for 3D advection-diffusion systems. • Mild CFL, long-term stability, high-order convergence and limited numerical dispersion/diffusion. • Treatment of hyperbolic-parabolic PDEs with mixed Dirichlet and Neumann boundaries. • Velocity fields can be provided by any suitable data (experimental or numerical). [ABSTRACT FROM AUTHOR]

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems. • High-order conservative projection of limited quantities (CPL) is developed. • CPL ensures consistency of high-order quantities with multi-material closure laws. • CPL avoids the need for excessive limiting, thereby ensuring high-order accuracy. • Since CPL relies on solution basis only, it is a local (element-based) procedure. • CPL can be used with any slope limiter and can be extended to other PDE systems. [ABSTRACT FROM AUTHOR]

In this paper, we study p-order methods for unconstrained minimization of convex functions that are p-times differentiable (p ≥ 2) with ν-Hölder continuous pth derivatives. We propose tensor schemes with and without acceleration. For the schemes without acceleration, we establish iteration complexity bounds of O(ε-1/(p+ν-1)) for reducing the functional residual below a given ε ∈ (0, 1). Assuming that ν is known, we obtain an improved complexity bound of O(ε-1/(p+ν)) for the corresponding accelerated scheme. For the case in which ν is unknown, we present a universal accelerated tensor scheme with iteration complexity of O(ε-p/[(p+1)(p+ν-1)]). A lower complexity bound of O(ε-2/[3(p+ν)-2]) is also obtained for this problem class. [ABSTRACT FROM AUTHOR]

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds $\mathbb{M} \subset \mathbb{R}^3$ of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and surface advection-diffusion. Finally, we demonstrate the applicability of our formulation to advection-diffusion-reaction equations on manifolds described purely as point clouds. Our surface discretizations use the recently developed RBF-least orthogonal interpolation method and, with the addition of hyperviscosity, are now empirically high-order accurate, stable, and free of stagnation errors. [ABSTRACT FROM AUTHOR]

We present several essential improvements to the powerful scalar auxiliary variable (SAV) approach. Firstly, by using the introduced scalar variable to control both the nonlinear and the explicit linear terms, we are able to reduce the number of linear equations with constant coefficients to be solved at each time step from two to one, so the computational cost of the new SAV approach is essentially half of the original SAV approach while keeping all its other advantages. This technique is also extended to the multiple SAV approach. Secondly, instead of discretizing the dynamical equation for the auxiliary variable, we use a first-order approximation of the energy balance equation, which allows us to construct high-order unconditionally energy-stable SAV schemes with uniform and, more importantly, variable time step sizes, enabling us to construct, for the first time, high-order unconditionally stable adaptive time-stepping backward differentiation formula schemes. Representative numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, operator-compensation based high order methods are presented to solve the Gross-Pitaevskii equation modeling Bose-Einstein condensation (BEC). We begin with the high-order approximation to the Laplacian by the central finite difference method combined with operator-compensation technique, and the Crank-Nicolson and time-splitting methods are then proposed for the time discretization. We show that the Crank-Nicolson operator-compensation method keeps the mass and energy conservation, while the time-splitting operator-compensation method only conserves the mass. Then by extending the high-order operator-compensation methods for the first order derivative, the time-splitting finite difference method is developed to solve the Gross-Pitaevskii equation with an angular momentum rotation term. Numerical results are given to test accuracy and verify the conservation properties. [ABSTRACT FROM AUTHOR]

In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear L 2 -stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions. [ABSTRACT FROM AUTHOR]

In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form (− Δ) p u = f , p ≥ 1. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in suitable norms. [ABSTRACT FROM AUTHOR]

We analyse the convergence of finite element discretizations of time-harmonic wave propagation problems. We propose a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber |$k$|⁠. This methodology is formally based on an expansion of the solution in powers of |$k$|⁠ , which permits to split the solution into a regular, but oscillating part, and another component that is rough, but behaves nicely when the wavenumber increases. The method is developed in its full generality and is illustrated by three particular cases: the elastodynamic system, the convected Helmholtz equation and the acoustic Helmholtz equation in homogeneous and heterogeneous media. Numerical experiments are provided, which confirm that the stability conditions and error estimates are sharp. [ABSTRACT FROM AUTHOR]

This paper presents a novel method to obtain the solution to the initial orbit determination problem for optical observations as a continuum of orbits—namely the orbit set—that fits the set of acquired observations within a prescribed accuracy. Differential algebra is exploited to analytically link the uncertainty in the observations to the state of the orbiting body with truncated power series, thus allowing for a compact analytical description of the orbit set. The automatic domain splitting tool controls the truncation error of the polynomial approximation by patching the uncertainty domain with different polynomial expansions, effectively creating a mesh. The algorithm is tested for different observing strategies to understand the working boundaries, thus defining the region for which the admissible region is necessary to extract meaningful information from observations and highlight where the new method can achieve a smaller uncertainty region, effectively showing that for some observing strategies it is possible to extract more information from a tracklet than the attributable. Consequently, the method enables comparison of orbit sets avoiding sampling when looking for correlation of different observations. Linear regression is also implemented to improve the uncertainty estimation and study the influence of the confidence level on the orbit set size. This is shown both for simulated and real observations obtained from the TFRM observatory. [ABSTRACT FROM AUTHOR]