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In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretized on the dual mesh at the aid of an explicit local ADER finite volume scheme, while the implicit pressure terms are discretized at the aid of a continuous $\mathbb{P}^{1}$ finite element method on the nodes of the primal mesh. In the zero Mach number limit, the new scheme automatically reduces to the hybrid FV/FE approach forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As such, the method is asymptotically consistent with the incompressible limit of the governing equations and can therefore be applied to flows at all Mach numbers. Due to the chosen semi-implicit discretization, the CFL restriction on the time step is only based on the magnitude of the flow velocity and not on the sound speed, hence the method is computationally efficient at low Mach numbers. In the chosen discretization, the only unknown is the scalar pressure field at the new time step. Furthermore, the resulting pressure system is symmetric and positive definite and can therefore be very efficiently solved with a matrix-free conjugate gradient method. In order to assess the capabilities of the new scheme, we show computational results for a large set of benchmark problems that range from the quasi incompressible low Mach number regime to compressible flows with shock waves.

Use of curvilinear coordinates is sometimes indicated by the inherent geometry of a fluid dynamics problem, but this introduces fictitious forces into the momentum equations that spoil strict conservative form. If one is willing to work in three dimensions, these fictitious forces can be eliminated by solving for rectangular (Cartesian) momentum components on a curvilinear mesh. A thoroughly geometric approach to fluid dynamics on spacetime demonstrates this transparently, while also giving insight into a greater unity of the relativistic and nonrelativistic cases than is usually appreciated., Comment: 11 pages

In this paper, we apply a weak Galerkin method for solving one dimensional coupled Burgers’ equations. Based on a conservation form for nonlinear term and some of the technical derivational. Theorticly, we drive the optimal order error in $$L^2$$ and $$H^1$$ norm for both continuous and discrete time weak Galerkin finite element schemes, also the stability of continuous time weak Galerkin finite element method is proved. Numerically, the accuracy and effectiveness of the weak Galerkin finite element method are illustrated by using Numerical examples with the lower order Raviart–Thomas element $$RT_k$$ for discrete weak derivative space.

We review the progress on the theory and applications of the vortex-surface field (VSF). The VSF provides a systematic Lagrangian-based framework for the identification, characterization, and modeling of flow structures. From the theoretical perspective, the vorticity is simplified to a scalar field as the VSF. The VSF isosurface is a vortex surface consisting of vortex lines. By introducing the virtual circulation-preserving velocity, the Helmholtz theorem and the Ertel theorem are extended to some non-ideal flows, so that the VSF evolution equation can be expressed in a Lagrangian conservation form. Thus the VSF isosurfaces of the same threshold at different times have strong coherence, facilitating the tracking of vortex surfaces. As a general flow diagnostic tool, the numerical VSF solution can be constructed in arbitrary flow fields by solving a pseudo-transport equation driven by the frozen, instantaneous vorticity. In addition, the local optimization method and the boundary-constraint method can further improve the smoothness and convergence of VSF solutions. Then the two-time method is developed for calculating the temporal evolution of VSFs. From post-processing of large-scale database of numerical simulations, the VSF elucidates mechanisms in the flows with essential vortex dynamics, such as turbulence and transition. For example, the VSF reveals the complex network of tangling vortex tubes in isotropic turbulence, consistent with the vorticity equation and dynamics. In addition, Eulerian vortex-identification criteria cannot identify complete vortex tubes, so the visual “breakdown” of worm-like structures were often reported in the literature. The universal framework of the VSF evolution illustrates an ordered evolution process of coherence structures in transitional flows, in which it is no need to ad hoc change the structure-identification method and isocontour threshold. Moreover, the quantitative VSF study elucidates the interaction between the vortex surface and the shock wave, flame, or electromagnetic field in multi-physics coupled flows. Based on the characterization of VSFs, we seek robust statistical features, and then determine and model correlations between the features and critical quantities to be predicted in engineering applications. For example, we develop a predictive model of the skin-friction coefficient in boundary-layer transition based on the small-scale inclination angle in experimental images. Some open problems in the VSF study are discussed, including the regularization for the breakdown of virtual conservation theorems at vorticity nulls and the speedup of the numerical construction and evolution of VSFs.

In this paper, an accurate and efficient numerical scheme is developed to solve the Navier-Stokes equations for two-phase flow systems with the nonlinear inertial term in conservation form. The major difficulty of discretizing the nonlinear inertial term in conservation form in two-phase flows is rooted in the discontinuous nature of momentum across interfaces. To overcome this difficulty, we developed the mixed Upwind/Central WENO scheme so that no grid deformation or mapping, or explicit interface reconstruction is necessary. Our mixed Upwind/Central WENO scheme also removes the dependence on the interface tracking/capturing method so it is compatible with any popular interface tracking/capturing methods. To decouple the overall equations of the two-phase system, an efficient semi-implicit projection scheme is constructed, which shares the efficiency of pressure stabilization method while enforces divergence-free condition to machine precision, to benefit interface advection, reduction consistency as well as large density ratio. The accuracy and reduction consistency of this projection scheme are validated by numerical experiments. Several benchmark cases are performed to show the capability of the overall scheme handling realistic two-phase problems with multiple physical effects, complex interface evolution and/or large density ratio of the order of ( ρ water / ρ air ). Both quantitative and qualitative comparisons have been made and very good agreements with analytical solutions, published numerical and/or experimental results are achieved.

New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration., 37 pages, 15 figures

In this study, an alternative way, a so called caisson based type of oscillating water column (OWC) wave energy converting system was proposed to capture and convert wave energy. Since the caisson structure is constructed to protect the coastal line or ports, it is important to know if a built-in associated OWC system will be a burden to affect the safety of the structure or it is safe enough to work appropriately. In this study, three steps of structural analysis were performed: firstly, the analysis for the structural safety of the whole caisson structure, secondly, performing the mechanic analysis for the chamber of the associated OWC system, and finally, performing the analysis for the wave induced air-pressure in the chamber under the design conditions of a local location during the wave-converting operation. For the structural safety analysis, a typical structural model associated with caisson breakwater was built and analyzed while the shape of the structure, material applied to the construction, and associated boundary conditions were all set-up according to the wave and structures. The motion and the strain distribution of the caisson structure subjected to designated waves of 50-year return period were evaluated and compared to the safety requirement by the code. For the analysis of the energy converting performance, a numerical method by using a theorem of unsteady Navier&ndash, Stokes equations in conservation form was used to analyze the proposed OWC model when the structure subjected to an incident wave of a 10-year return period.

The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task for complicated physical or engineering problems because the associated Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them.

In this study, numerical simulation is performed with a focus on the application of new modified preconditioned all-speed Roe scheme to simulate natural convection flows. The new modified preconditioned all-speed Roe Scheme primarily utilizes the local flow parameters for calculation of the coefficients of numerator in dissipation term of the scheme instead of using the global cut-off Mach number strategy (which is advantageous for natural convection flows) and has been shown to have better accuracy than preconditioned Roe Scheme for Low Mach number flows. For the present simulation, the compressible governing equation in conservation form, new modified preconditioned all-speed Roe scheme and dual time stepping are employed. The validation of numerical algorithm is divided into two investigations a) natural convection flow within differentially heated enclosed square cavity and b) open-ended vertical channel asymmetrically heated for a wide range of Rayleigh number with air (Pr 0.72) as working fluid. Visualization of fluid flow dynamics conducted for both classes of geometries for all range of Rayleigh number show similar phenomena in accordance with previous literatures and compared data also show very good agreement with previous literatures. All results indicate that the new modified preconditioned all-speed Roe scheme is very much competent and accurate for simulation of buoyancy-induced compressible convection flows without relying on the correct prediction of global cut-off Mach number.

Shen et al. [1] , [2] proposed an upwind space-time conservation element and solution element (CESE) scheme for 1D and 2D hydrodynamics (HD) in rectangular coordinates, which combined the advantages of CESE and upwind scheme, namely, guaranteed strictly the space–time conservation law as well as captured discontinuities very efficiently. All kinds of upwind schemes can be combined very flexibly for different problems to achieve the perfect combination of CESE and finite volume method (FVM). However, in many physical applications, we need to consider geometries that are more sophisticated. Hence, the main objective of this paper is to extend the upwind CESE scheme to multidimensional magneto-hydrodynamics (MHD) in general curvilinear coordinates by transforming the MHD equations from the physical domain (general curvilinear coordinates) to the computational domain (rectangular coordinates) and the new equations in the computational domain can be still written in the conservation form. For the 3D case, the derivations of some formulas are much more abstract and complex in a 4D Euclidean hyperspace, and some technical problems need to be solved in the debugging process. Unlike in HD, keeping the magnetic field divergence-free for MHD problems is also a challenge especially in general curvilinear coordinates. These are the main obstacles we have overcome in this study. The test results of benchmarks demonstrate that we have successfully extended the upwind CESE scheme to general curvilinear coordinates for both 2D and 3D MHD problems.

Two-phase flow through a porous medium with hysteresis effects is considered. The model consists of a system of two coupled nonlinear equations: a transport equation for the water saturation and an evolution equation for the hysteresis variable. The latter is not in conservation form and contains discontinuous functions of the two unknown variables as coefficients. Some qualitative properties of piecewise smooth solutions of the system are proved. In particular, the hysteresis variable is shown to satisfy a maximum principle, and that its total variation is bounded by the total variation of its initial value. The traveling waves are investigated under the assumption that the convective term is convex. Riemann solvers for the inviscid system are constructed. Non-uniqueness due to hysteresis loops is finally discussed; several solutions are discarded by the maximum principle for the hysteresis variable.

A nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented. This system constitutes a full nonlinear wave model that has been formulated in the conservation form. Initially, this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the Lax–Wendroff theorem, and the scheme can match the solutions of the Westervelt and Burgers equations numerically. Here, high-order numerical descriptions of strongly nonlinear wave propagations become of great interest. For that matter we consider finite difference formulations of the weighted essentially non-oscillatory (WENO) schemes associated with explicit strong stability preserving Runge–Kutta (SSP-RK) time integration methods. Although this strategy is known to be computationally demanding, it is found to be effective when implemented to be solved in graphical processing units (GPUs). As we consider wave propagations in unbounded domains, perfectly matching layers (PML) have been also considered in this work. The proposed system model is validated and illustrated by using one- and two-dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media.

A solver for the Euler equations with optional diffusion based on the Discontinuous Galerkin (DG) method is presented. To be usable for simulation problems in the atmosphere from the global scale down to the meso-scale, the horizontally explicit, vertically implicit (HEVI) approach is applied to the DG discretization, to avoid tiny time-steps by thin grid cells. To consider orography in the correct approximation order, terrain-following coordinates are used, and to nevertheless keep the local conservation properties of DG, the equations are formulated in the strong conservation form. IMEX-Runge-Kutta time integration enables at least third order approximation in time. In a similar manner, diffusion both as a physical process (for turbulence parameterization) and as a physically motivated stabilization mechanism is treated by the Bassi, Rebay (1997) approach in terrain-following coordinates and in a conserving manner, too. Several test cases relevant for the atmosphere demonstrate the validity of the approach. Although most of the presented theory is applicable both in two and three dimensions (2D and 3D), these test cases are purely 2D. They show, in particular, that the proposed scheme seems to tolerate very steep terrain.

This paper presents numerical and experimental investigations of the combined effect on pressure transients of air pockets and homogenous water–air bubble mixtures. An air pocket can accumulate at a high point of a pipeline along the control section located at the transition between pipes with sub- and supercritical slope, forcing open channel flow conditions underneath the pocket that ends in a hydraulic jump at the downward sloping pipe. The turbulence action at the jump generates small air bubbles that are entrained and transported along the pipe producing a two-component bubbly flow within the continuous liquid phase. A numerical model is developed, combining the explicit–implicit scheme proposed by McGuire and Morris and the method of characteristics for solving the quasi-linear hyperbolic partial differential equations for transient two-phase flow expressed in conservation form. To verify the proposed model, an experimental apparatus made of PVC was used to carry out hydraulic transient experiments. Tests were conducted in a tank–pipe–valve system and a valve with a pneumatic actuator at the downstream end generated transients. Numerical results at the test section pipe compares favorably with experimental data. The results show that pressure transients are significantly reduced with increasing air-pocket volumes and bubbly flow air content.

Purpose The purpose of this paper is to present a fully conservative numerical algorithm for solving the coupled shallow water hydro-sediment-morphodynamic equations governing fluvial processes, and also to clarify the performance of a conventional algorithm, which redistributes the variable water-sediment mixture density to the source terms of the governing equations and accordingly the hyperbolic operator is rendered similar to that of the conventional shallow water equations for clear water flows. Design/methodology/approach The coupled shallow water hydro-sediment-morphodynamic equations governing fluvial processes are arranged in full conservation form, and solved by a well-balanced weighted surface depth-gradient method along with a slope-limited centred scheme. The present algorithm is verified for a spectrum of test cases, which involve complex flows with shock waves and sediment transport processes with contact discontinuities over irregular topographies. The computational results of the conventional algorithm are compared with those of the present algorithm and evaluated by available referenced data. Findings The fully conservative numerical algorithm performs satisfactorily over the spectrum of test cases, and the conventional algorithm is confirmed to work similarly well. Originality/value A fully conservative numerical algorithm, without redistributing the water-sediment mixture density, is proposed for solving the coupled shallow water hydro-sediment-morphodynamic equations. It is clarified that the conventional algorithm, involving redistribution of the water-sediment mixture density, performs similarly well. Both algorithms are equally applicable to problems encountered in computational river modelling.

We propose a numerical method to study both the nonlinear dynamics and the stability of equilibrium solutions of one-dimensional problems modeled by systems of differential-algebraic equations (DAE) written in conservation form. The method relies on a quadrature rule (finite volume approach) for the spatial discretization, while an implicit scheme is used in time to compute the transient dynamics. The stability of equilibrium solutions is analyzed by solving the generalized eigenvalue problem obtained after a standard linearisation. The Jacobian matrices needed for the computations - which are the same independently of whether one is interested in computing the equilibrium shapes, the nonlinear transient evolution or the linear stability analysis–have been systematically and efficiently computed with the help of a symbolic toolbox. To illustrate the method, we have applied it to several problems in the one-dimensional dynamics of capillary jets. For this purpose, and apparently for the first time, we have rewritten all the equations governing the jet kinematics and dynamics in conservation form and have shown the numerical advantages of this formulation, together with the numerical method proposed, when applied to the global stability of jets stretched by gravity, to the jetting to dripping transition and to the recoil of the filament remaining after the jet breakup. For the latter problem, we show that the results obtained with the one dimensional equations are in reasonable good agreement with the those previously reported in the literature and obtained with complicated potential three dimensional codes.

A Chebyshev collocation spectral method (CCSM) is developed to solve the radiative transfer equation (RTE) in an infinitely long, cylindrically symmetric, homogeneous medium. Both the spatial and angular computational domains of the RTE are discretized by the Chebyshev collocation points. For the CCSM, taking the conservation form of RTE, which is commonly used in the discrete ordinates method (DOM), will produce poor accuracy, whereas taking the non-conservation form can yield good prediction. To test the applicability of different collocation point schemes in the radial direction, three solvers are developed. SOLVER1 and SOLVER2 use the Chebyshev-Gauss-Radau (CGR) points and the Chebyshev-Gauss-Lobatto (CGL) points on the radius, respectively. SOLVER 3 uses the CGL points on the diameter, and an even number of nodes is taken to exclude the origin. The results show that SOLVER1 and SOLVER2 suffer from the “singularity effect”. This effect can be reduced by increasing the grid number in radial direction. Whereas SOLVER3 can avoid the “singularity effect”. We also show that the mapping (the Kosloff–Tal-Ezer transformation) actually cannot improve the accuracy. Besides, the numerical accuracy is declined by using cosine of the polar angle instead of the polar angle itself as the independent variable. Our results demonstrate that the CCSM performs much better than the DOM and produces the exponential convergence in both the spatial and angular domains. The CCSM is a superior method to achieve high accuracy for thermal radiation problem of homogeneous medium with cylindrical symmetry.

A lattice Boltzmann model is proposed with a newly modified equilibrium distribution function for solving the conservation form of the energy equation to treat the interphase conjugate heat transfer problems under both steady state and unsteady state. The temperature and heat flux continuity conditions at the interface can be inherently satisfied without needing any additional treatments, such as iterative computation, correcting procedure for the incoming distribution function, and the complicated calculation procedure for the source term, to account for the interphase conjugate heat transfer. The implementation of the present LB model, therefore, is more straightforward and more efficient than those in most previous models, especially for problems with complex interfaces. The applicability and accuracy of the proposed LB model were evaluated by some benchmark problems including both simple flat interface and complex interface geometry. The results show excellent agreements with analytical soluti...

The study of viscous flow in a lid-driven cavity is carried out using alternating direction implicit method. The conservation form of incompressible Navier–Stokes equation in stream function–vorticity form is solved using second-order accurate central difference scheme in a uniform finite-difference grid mesh. The numerical solution is obtained up to highest Reynolds number 32,500 from the lowest 0.00001 using the grid sizes 129 × 129, 257 × 257 and 513 × 513. Good agreement of the result is found with Erturk et al. (Int J Numer Methods Fluids 48:747–774, 2005) [1]. The study of flow properties in the form of velocity profiles, stream function and vorticity contour plots and location of primary and secondary eddies are carried out. The novelty of this study is that the magnitude of the vorticity value does not cross the theoretical limit −1.8859 (Burggraf 24:113–151, 1966 [2]).

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers. The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form with the standard and the split-form variants of the DGSEM. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system. The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

Cosserat rods [1] are the prefered choice for modeling large spatial deformations of slender flexible structures at small local strains. Discrete Cosserat rod models [6, 7] based on geometric finite differences preserve essential properties of the continuum theory. In previous work [9] kinetic aspects of discrete quaternionic Cosserat rods defined on a staggered grid were investigated. In particular it was shown that equilibrium configurations obtained by energy minimization correspond to solutions of finite difference type discrete balance equations for the sectional forces and moments in conservation form. The present contribution complements the numerical studies shown in [9] by considering localized helix configurations [2] of discrete Cosserat rods as more complex benchmark examples.

A modified enthalpy porosity formulation is introduced to capture melting and solidification of pure substances. When melting and solidification of pure substances are addressed by the fixed grid based volume averaging technique, it is possible to obtain two equivalent and interchangeable mathematical formulations of the energy conservation equation if the governing equation is expressed in terms of temperature as the primary dependent variable. Between these two formulations, only one form is shown to provide physically consistent numerical solutions when very large difference in specific heats for liquid and solid phases is involved. A modified enthalpy updating scheme is proposed to predict the solid/liquid fraction during melting and solidification process of pure substances having large difference in phase specific heats. The results from the proposed scheme are validated with the existing results from literature involving numerical prediction of freezing of water. The physical consistency of the simulation results obtained by solving two interchangeable forms of energy conservation equation is tested and compared considering a case study involving melting of ice. While one of the conservation forms fails to predict the melting process, the other conservation form successfully predicts physically consistent result. The proposed formulation is capable of predicting melting and solidification of all pure substances including those with large difference in phase specific heats such as water and paraffin wax.

Accepted: 2016-03-24, 資料番号: SA1160148000

A well-balanced explicit/semi-implicit finite element scheme is proposed for the simulation of dam break flows over complex domains involving wetting and drying. The numerical model is based on the nonlinear shallow water equations in the hyperbolic conservation form. The governing equations are discretized by a fractional finite element method using a characteristic-Galerkin procedure. Firstly, the intermediate increment of a conserved variable is obtained explicitly neglecting the pressure gradient term. And then, the increment is corrected for the effects of pressure once the pressure increment is obtained from the Poisson equation. In order to maintain the “well-balanced” property, the pressure gradient term and bed slope terms are incorporated into the Poisson equation. Moreover, a local bed slope modification technique is employed in drying-wetting interface treatments. The new model is validated against several benchmark tests and laboratory experimental datas related to dam-break flood wave propagation and promising results are obtained.

We describe a scheme for the numerical approximation of uniformly or locally weakly hyperbolic systems of equations in conservation and non-conservation forms. The strategy focuses on generalizations of the Roe scheme for non-conservation systems that can be considered as upstream differencing discretizations. The present work explicitly treats an advection matrix that is not diagonalizable by extending previous ideas that introduce a perturbation into the weakly hyperbolic problem to convert it to a nearby strongly hyperbolic problem. The perturbative expansion allows one to identify and remove the terms that would otherwise result in an apparent singularity and isolate the regular part of the numerical method. The performance of the method is showcased for uniformly and locally weakly hyperbolic systems in one and two dimensions. Generalization to higher dimensional systems can be performed by following the usual practice of dimension-by-dimension splitting.

We present a three-dimensional (3D) numerical solver of the linearized compressible Euler equations (Global Acoustic Linearized Euler), used to model acoustic oscillations throughout the solar interior. The governing equations are solved in conservation form on a fully global spherical mesh (0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π, 0 ≤ r ≤ R ⊙) over a background state generated by the standard solar model S. We implement an efficient pseudospectral computational method to calculate the contribution of the compressible material derivative dyad to internal velocity perturbations, computing oscillations over arbitrary 3D background velocity fields. This model offers a foundation for a “forward-modeling” approach, using helioseismology techniques to explore various regimes of internal mass flows. We demonstrate the efficacy of the numerical method presented in this paper by reproducing observed solar power spectra, showing rotational splitting due to differential rotation, and applying local helioseismology techniques to measure travel times created by a simple model of single-cell meridional circulation.

Inviscid, compressible flows are described by a first-order system of hyperbolic partial differential equations in conservation form, and are solved numerically with simple schemes. These schemes show the ability to reproduce well-known solutions and benchmarks for the two-dimensional Euler equations. Finally, the schemes are used to explore the anomaly of non-unique solutions in the transonic regime. Furthermore, numerical solutions for the isentropic Euler equations with γ = 2, which are analogous to the shallow water equations, also exhibit non-uniqueness. In the appendix, results computed with OVERFLOW also exhibit non-unique solutions for the same geometries considered in the text.

This paper aims to introduce a new class of high order conservative schemes to solve systems of conservation laws. The idea is to couple the conservation form of the system with, possibly simpler, ...

We introduce a second-order numerical scheme for compressible atmospheric motions at small to planetary scales. The collocated finite volume method treats the advection of mass, momentum, and mass-weighted potential temperature in conservation form while relying on Exner pressure for the pressure gradient term. It discretises the rotating compressible equations by evolving full variables rather than perturbations around a background state, and operates with time steps constrained by the advection speed only. Perturbation variables are only used as auxiliary quantities in the formulation of the elliptic problem. Borrowing ideas on forward-in-time differencing, the algorithm reframes the authors' previously proposed schemes into a sequence of implicit midpoint, advection, and implicit trapezoidal steps that allows for a time integration unconstrained by the internal gravity wave speed. Compared with existing approaches, results on a range of benchmarks of nonhydrostatic- and hydrostatic-scale dynamics are competitive. The test suite includes a new planetary-scale inertia-gravity wave test highlighting the properties of the scheme and its large time step capabilities. In the hydrostatic-scale cases the model is run in pseudo-incompressible and hydrostatic mode with simple switching within a uniform discretization framework. The differences with the compressible runs return expected relative magnitudes. By providing seamless access to soundproof and hydrostatic dynamics, the developments represent a necessary step towards an all-scale blended multimodel solver.