Systems Of Conservation Laws Research Articles

δ − shocks and vacuum states in the Riemann problem for isothermal van der Waals dusty gas under the flux approximation

In this paper, we analyze the Riemann problem for concentration and cavitation phenomena to the isothermal Euler equations for van der Waals gas with dust particles in the presence of flux approximation. By the Riemann problem, we mean that it is an initial value problem for the system of conservation laws supplemented by constant discontinuous initial data involving one breaking. The existence of δ−shock and vacuum state in the Riemann problem for the system under consideration is examined. Further, it is shown that as the flux approximation and pressure vanish, the Riemann solution containing two shock waves to the isothermal Euler equation for van der Waals gas with dust particles converges to δ−shock solution and the Riemann solution containing two rarefaction waves tends to the vacuum state solution of the transport equations. Numerical simulations presenting the formation of δ−shocks and vacuum states for different flux approximations are also shown.

A MOOD-like compact high order finite volume scheme with adaptive mesh refinement

In this paper, a novel Finite Volume (FV) scheme for obtaining high order approximations of solutions of multi-dimensional hyperbolic systems of conservation laws within an Adaptive Mesh Refinement framework is proposed. It is based on a point-wise polynomial reconstruction that avoids the recalculation of reconstruction stencils and matrices whenever a mesh is refined or coarsened. It also couples both the limiting of the FV scheme and the refinement procedure, taking advantage of the Multi-dimensional Optimal Order Detection (MOOD) detection criteria. The resulting computational procedure is employed to simulate test cases of increasing difficulty using two models of Partial Differential Equations: the Euler system and the radiative M1 model, thus demonstrating its efficiency.

Preface: Hyperbolic System of Conservation Laws and Related Topics

Preface: Hyperbolic System of Conservation Laws and Related Topics

System of kinematical conservation laws

System of kinematical conservation laws

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

Here, we study the Riemann problem for a strictly hyperbolic system of conservation laws, which occurs in gas dynamics and nonlinear elasticity. We establish the existence and uniqueness of the solution of Riemann problem containing delta shock wave by employing self‐similar vanishing viscosity approach. We prove that delta shock is stable under self‐similar viscosity perturbation, which ensures that delta shock wave is a unique entropy solution.

Delta shock wave as limits of vanishing viscosity for zero‐pressure gas dynamics with energy conservation law

AbstractThis paper studies the system of conservation laws of mass, momentum, and energy in zero‐pressure gas dynamics. By the vanishing viscosity method, the stability of the solutions involving delta shock wave with Dirac delta functions developing in both state variables is established.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded.

Central random choice methods for hyperbolic conservation laws

In this article, we briefly review the random choice method (RCM) and ADER methods for solving one and two-dimensional hyperbolic conservation laws. The main advantage of RCM is that it computes discontinuities with infinite resolution. In this method, the original problem is reduced to a set of local Riemann problems (RPs). The exact solutions of these RPs are used to form the solution of the original problem. However, RCM has the following disadvantages: (1) one should solve the RP exactly, however, the exact solutions are usually complex and unavailable for many problems. (2) The accuracy of the smooth region of the flow is poor. ADER methods are explicit, one-step schemes with a very high order of accuracy in time and space. They depend on the solution of the generalized RP (GRP) exactly. In Zahran (J Math Anal Appl 346:120–140, 2008), an improved version of ADER methods (central ADER) was introduced where the RPs were solved numerically and used central fluxes, instead of upwind fluxes. The improved central ADER schemes are more accurate, faster, simple to implement, RP solver free, and need less computer memory. To fade the drawbacks of the above schemes and keep their advantages, we propose, in this paper, an improved version of the RCM. We merge the central ADER technique with the RCM. The resulting scheme is called Central RCM (CRCM). The improvements are listed as follows: we use the WENO reconstruction for the initial data instead of constant reconstruction in RCM, we solve the RPs numerically by using central finite difference schemes and use random sampling to update the solution, as the original RCM. Here we use the staggered and non-staggered RCM. To enhance the accuracy of the new methods, we use a third-order TVD flux (Zahran in Bull Belg Math Soc Simon Stevin 14:259–275, 2007), instead of a first-order flux. Compared with the original RCM and the central ADER, the new methods combine the advantages of RCM, ADER, and central finite difference methods as follows: more accurate, very simple to implement, need less computer memory, and RP solver free. Moreover, the new methods capture the discontinuities with infinite resolution and improve the accuracy of the smooth parts. The new methods have less CPU time than the central ADER methods, this is due to less flux evaluation in CRCM. An extension of the schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented. We present several numerical examples for one and two-dimensional problems. The results confirm that the presented schemes are superior to the original RCM, ADER, and central ADER schemes.

Residual Viscosity Stabilized RBF-FD Methods for Solving Nonlinear Conservation Laws

In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and the oversampled RBF-FD method. Computational tests confirm that the stabilized methods are reliable and accurate in solving scalar conservation laws and conservation law systems such as compressible Euler equations.

Erratum to: On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws

Erratum to: On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws

Variational principles for nonlinear PDE systems via duality

A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-Jacobi equations.

Dispersive shock waves in lattices: A dimension reduction approach

Dispersive shock waves (DSWs), which connect states of different amplitude via a modulated wave train, form generically in nonlinear dispersive media subjected to abrupt changes in state. The primary tool for the analytical study of DSWs is Whitham’s modulation theory. While this framework has been successfully employed in many space-continuous settings to describe DSWs, the Whitham modulation equations are cumbersome in most spatially discrete systems. In this article, we illustrate the relevance of the reduction of the DSW dynamics to a planar ODE in a broad class of lattice examples. Solutions of this low-dimensional ODE accurately describe the orbits of the DSW in self-similar coordinates and the local averages in a manner consistent with the modulation equations. We use data-driven and quasi-continuum approaches within the context of a discrete system of conservation laws to demonstrate how the underlying low dimensional structure of DSWs can be identified and analyzed. The connection of these results to Whitham modulation theory is also discussed.

Uniqueness and Weak-BV Stability for 2times 2 Conservation Laws

Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small BV functions which are global solutions of this equation. For any small BV initial data, such global solutions are known to exist. Moreover, they are known to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even BV) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-BV uniqueness result, where only one of the solutions is supposed to be small BV, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with 2 unknowns. The method is L^2 based, and builds up from the theory of a-contraction with shifts, where suitable weight functions a are generated via the front tracking method.

On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws

It was shown in Ferapontov et al. (Lett Math Phys 108(6):1525–1550, 2018) that the classification of n-component systems of conservation laws possessing a third-order Hamiltonian structure reduces to the following algebraic problem: classify n-planes H in \(\wedge ^2(V_{n+2})\) such that the induced map \(Sym^2H\longrightarrow \wedge ^4V_{n+2}\) has 1-dimensional kernel generated by a non-degenerate quadratic form on \(H^*\). This problem is trivial for \(n=2, 3\) and apparently wild for \(n\ge 5\). In this paper we address the most interesting borderline case \(n=4\). We prove that the variety \({\mathcal {V}}\) parametrizing those 4-planes H is an irreducible 38-dimensional \(PGL(V_6)\)-invariant subvariety of the Grassmannian \(G(4, \wedge ^2V_6)\). With every \(H\in {\mathcal {V}}\) we associate a characteristic cubic surface \(S_H\subset \mathbb {P}H\), the locus of rank 4 two-forms in H. We demonstrate that the induced characteristic map \(\sigma : {\mathcal {V}}/ PGL(V_6) \dashrightarrow {\mathcal {M}}_{c},\) where \({\mathcal {M}}_{c}\) denotes the moduli space of cubic surfaces in \(\mathbb {P}^3\), is dominant, hence generically finite. Based on Manivel and Mezzetti (Manuscr Math 117:319–331, 2005), a complete classification of 4-planes \(H\in {\mathcal {V}}\) with the reducible characteristic surface \(S_H\) is given.

Riemann problems for a hyperbolic system of nonlinear conservation laws from the Liou–Steffen pressure system

This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations.

Multiresolution-Based Mesh Adaptation and Error Control for Lattice Boltzmann Methods with Applications to Hyperbolic Conservation Laws

Lattice Boltzmann methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this work, we design a class of fully adaptive LBM methods with dynamic mesh adaptation and error control relying on multiresolution analysis. This wavelet-based approach allows us to adapt the mesh based on the regularity of the solution and leads to a very efficient compression of the solution without loosing its quality and with the preservation of the properties of the original LBM method on the finest grid. This yields a general approach for a large spectrum of schemes and allows precise error bounds, without the need for deep modifications on the reference scheme. An error analysis is proposed. For the purpose of validating this error analysis, we conduct a series of test cases for various schemes and scalar and systems of conservation laws, where solutions with shocks are to be found and local mesh adaptation is especially relevant. Theoretical estimates are retrieved while a reduced memory footprint is observed. It paves the way to an implementation in a multidimensional framework and high computational efficiency of the method for both parabolic and hyperbolic equations, which is the subject of a companion paper.

A High-Order Velocity-Based Discontinuous Galerkin Scheme for the Shallow Water Equations: Local Conservation, Entropy Stability, Well-Balanced Property, and Positivity Preservation

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas. In this work, we present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor (in: Tezduyar, Hughes T (eds) Proceedings of the winter annual meeting of the American Society of Mechanical Engineering 1986) back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions. The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge–Kutta (SSP–RK) time integrators (Gottlieb et al. in SIAM Rev. 43, 89–112, 2001) to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter (Cockburn and Shu in J Comput Phys 141:199–224, 1998) on the conservative variables using the Fu-Shu troubled cell indicator (Fu and Shu in J Comput Phys 347:305–327, 2017) in each inner stage of the Runge–Kutta time stepping to suppress numerical oscillations. This fully discrete can be readily applied to various SWEs simulations without dry areas where the water height is close to zero. The case with dry areas need further special attention, where the velocity approximation can be unphysically large near cells with a small water height, which may eventually crashes the simulation if no special treatment is used near these cells. Here we propose a simple wetting/drying treatment for the velocity update without violating the local conservation property to enhance the robustness of the overall scheme. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

Entropy stable and positivity preserving Godunov-type schemes for multidimensional hyperbolic systems on unstructured grid

This paper describes a novel subface flux-based Finite Volume (FV) method for discretizing multi-dimensional hyperbolic systems of conservation laws of general unstructured grids. The subface flux numerical approximation relies on the notion of simple Eulerian Riemann solver introduced in the seminal work [Gallice (2003) [18]]. The Eulerian Riemann solver is constructed from its Lagrangian counterpart by means of the Lagrange-to-Euler mapping. This systematic procedure ensures the transfer of good properties such as positivity preservation and entropy stability. In this framework, the conservativity and the entropy stability are no more locally face-based but result respectively from a node-based vectorial equation and a scalar inequation. The corresponding multi-dimensional FV scheme is characterized by an explicit time step condition ensuring positivity preservation and entropy stability. The application to gas dynamics provides an original multi-dimensional conservative and entropy-stable FV scheme wherein the numerical fluxes are computed through a nodal solver which is similar to the one designed for Lagrangian hydrodynamics. The robustness and the accuracy of this novel FV scheme are assessed through various numerical tests. We observe its insensitivity to the numerical pathologies that plague classical face-based contact discontinuity preserving FV formulations.

Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws

Let u(t, x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H1 because by Sobolev embedding theorem H1 functions are Hölder continuous. However, the author notes that from any point (t, x), he can draw a generalized characteristic downward which meets the initial axis at y = α(t, x). If he regards u as a function of (t, y), it indeed belongs to H1 as a function of y if the initial data belongs to H1. He may call this generalized persistence (of high regularity) of the entropy weak solutions. The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2 × 2 Temple system of hyperbolic conservation laws in one space dimension.

On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any ntimes n system of conservation laws has a solution. The solution concept is an extension of the notion of singular delta -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of delta -distributions as a part of solutions to systems of conservation laws.