Multidimensional Systems Of Conservation Laws Research Articles

Nonlinear Hyperbolic Problems: Modeling, Analysis, and Numerics

The workshop gathered together leading international experts, as well as most promising young researchers, working on the modelling, the mathematical analysis, and the numerical methods for nonlinear hyperbolic partial differential equations (PDEs). The meeting focussed on addressing outstanding issues and identifying promising new directions in all three fields, i.e. modelling, analysis, and numerical discretization. Key questions settled around the lack of well-posedness theories for multidimensional systems of conservation laws and the use of hyperbolic modelling beyond the classical topic of gas dynamics. A focal point in numerics has been the discretization of random evolutions and uncertainty quantification. Equally important, new multi-scale methods and schemes for asymptotic regimes have been considered.

Nonuniqueness of Admissible Weak Solution to the Riemann Problem for the Full Euler System in Two Dimensions

The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a second space dimension. It was shown that there exist infinitely many bounded entropy admissible weak solutions to such a 2D Riemann problem for isentropic Euler equations, if the initial data give rise to a 1D self-similar solution containing a shock. In this work we study such a 2D Riemann problem for the full Euler system in two space dimensions and prove the existence of infinitely many bounded entropy admissible weak solutions in the case that the Riemann initial data give rise to the 1D self-similar solution consisting of two shocks and possibly a contact discontinuity.

Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) [39]) and symmetric hyperbolic systems (Hou and Liu (2007) [36]), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) [5] and Gassner (2013) [19]. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss–Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection–diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

AbstractIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions witha posterioridetection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.

Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We propose a simple modification of standard WENO finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

Systems of conservation laws with rotation and Galilean symmetry

We show that the structure of multidimensional systems of conservation laws, equipped with a single, convex entropy, and with both rotation and Galilean symmetries, is perhaps surprisingly limited. The following are established.The simplest examples of such systems, with only one vector field, are at most simple extensions of the familiar Euler systems.In such models of magnetohydrodynamic flow, including a solenoidal, Galilean invariant magnetic field, the property of finite signal propagation speeds is often lost.The only such system describing adiabatic multi-fluid flow, with the mass of each species conserved separately, corresponds to independent flow of each species. Whatever interaction between the different species occurs in an energy equation or through some other dependent variable, and not simply through expressions for the pressures in terms of the speciesʼ densities. A similar conclusion holds for incompressible flow.

TESS: A RELATIVISTIC HYDRODYNAMICS CODE ON A MOVING VORONOI MESH

We have generalized a method for the numerical solution of hyperbolic systems of equations using a dynamic Voronoi tessellation of the computational domain. The Voronoi tessellation is used to generate moving computational meshes for the solution of multi-dimensional systems of conservation laws in finite-volume form. The mesh generating points are free to move with arbitrary velocity, with the choice of zero velocity resulting in an Eulerian formulation. Moving the points at the local fluid velocity makes the formulation effectively Lagrangian. We have written the TESS code to solve the equations of compressible hydrodynamics and magnetohydrodynamics for both relativistic and non-relativistic fluids on a dynamic Voronoi mesh. When run in Lagrangian mode, TESS is significantly less diffusive than fixed mesh codes and thus preserves contact discontinuities to high precision while also accurately capturing strong shock waves. TESS is written for Cartesian, spherical and cylindrical coordinates and is modular so that auxilliary physics solvers are readily integrated into the TESS framework and so that the TESS framework can be readily adapted to solve general systems of equations. We present results from a series of test problems to demonstrate the performance of TESS and to highlight some of the advantages of the dynamic tessellation method for solving challenging problems in astrophysical fluid dynamics.

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t → ∞ in L loc 1 of the space of directions ζ = x / t . That is, the solution z ( t , x , ξ ) of the perturbed Cauchy problem for the corresponding BGK system satisfies ∫ X z ( t , t ζ , ξ ) d μ ( ξ ) → R ( ζ ) as t → ∞ , in L loc 1 ( R n ) , where R ( ζ ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in Lloc1 of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in Lloc1(ℝn), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

In this paper, we present some interesting connections between a number of Riemann-solver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach . Furthermore, Ostkamp's equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach . At the same time, we clarify some important differences between these two approaches.

ASYMPTOTIC STABILITY OF NON-PLANAR RIEMANN SOLUTIONS FOR MULTI-D SYSTEMS OF CONSERVATION LAWS WITH SYMMETRIC NONLINEARITIES

We study the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form [Formula: see text], where the gα are real smooth functions defined in [0,+∞), and when the initial data are perturbations of two-state nonplanar Riemann data. Specifically, if R0(x) is such Riemann data and ψ∈L∞(ℝd)n satisfies ψ(Tx)→0 in [Formula: see text], as T→∞, then an entropy solution, u(x,t), of the Cauchy problem with u(x,0)=R0(x)+ψ(x) satisfies u(ξt,t)→R(ξ) in [Formula: see text], as t→∞, where R(x/t) turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.

The MoT-ICE: A New High-Resolution Wave-Propagation Algorithm for Multidimensional Systems of Conservation Laws Based on Fey's Method of Transport

Fey's Method of Transport (MoT) is a multidimensional flux-vector-splitting scheme for systems of conservation laws. Similarly to its one-dimensional forerunner, the Steger–Warming scheme, and several other upwind finite-difference schemes, the MoT suffers from an inconsistency at sonic points when used with piecewise-constant reconstructions. This inconsistency is due to a cell-centered evolution scheme, which we call MoT-CCE, that is used to propagate the waves resulting from the flux-vector-splitting step. Here we derive new first-order- and second-order-consistent characteristic schemes based on interface-centered evolution, which we call MoT-ICE. We prove consistency at all points, including the sonic points. Moreover, we simplify Fey's wave decomposition by distinguishing clearly between a linearization and a decomposition step. Numerical experiments confirm the stability and accuracy of the new schemes. Owing to the simplicity of the two new ingredients of the MoT-ICE, its second-order version is several times faster than that of the MoT-CCE.

Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

Nonoscillatory High Order Accurate Self-Similar Maximum Principle Satisfying Shock Capturing Schemes I

This is the first paper in a series in which a class of nonoscillatory high order accurate self-similar local maximum principle satisfying (in scalar conservation law) shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed. In this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented, using scalar one-dimensional initial value problems as a model. For this model, the schemes are made to satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modifications, imposed as needed. The first enforces a local maximum principle; the second guarantees that no new extrema develop. The schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [Math. Comp., 49 (1987), pp. 105–121, Math. Comp...

A Variant of Van Leer's Method for Multidimensional Systems of Conservation Laws

We present a new variant of Van Leer's construction of upwind finite volume schemes for hyperbolic systems of conservation laws. Fluxes are computed with second-order accuracy using an interpolation rather than a slope reconstruction. A correction of the interpolated values is necessary and performed globally on each cell by a conservation argument. It can be used on a rectangular or triangles based dual grid to obtain a genuinely multidimensional scheme. One of our main concerns in this construction, is to prove that the second-order reconstruction, combined with a Boltzmann solver, gives nonnegative values of the pressure and density for gas dynamics, even on an unstructured mesh. This allows us to derive a rigorous CFL condition. Thus our approach is very robust.