On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws

On a nonlocal regularization of a non-strictly hyperbolic system of conservation laws

In this paper, we consider the Cauchy problem for a generic hyperbolic system of conservation laws and assume it is provided by a standard Riemann semigroup of solutions, for summable initial data with small total variation. Then, we introduce an integral functional involving the solutions of the Cauchy problem and investigate when such a functional has a (nontrivial) minimum in case the initial data vary in suitable admissible classes of functions.

Abstract A system of hyperbolic conservation laws <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtable columnalign="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msubsup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> as well as its viscous regularisation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mtable columnalign="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> </mml:mrow> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>diag</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> , the solution also contains undercompressive shocks and Jouguet waves.

This paper extends the deterministic Lyapunov-based stabilisation framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite-volume discretisation method from [M. Banda and M. Herty, Math. Control Relat. Fields, 3 (2013), pp. 121–142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilisation of the linear wave equations and linearised shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearised Saint-Venant system with source terms.

Feedback stabilization for entropy solutions of a 2 × 2 hyperbolic system of conservation laws at a junction

ABSTRACT The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter‐cells, and time‐integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation‐dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge‐Kutta impacts deeply on the quality of the solution. We analyze one‐dimensional dissipation‐dispersion to select the best combination of the space‐time discretization for high Courant numbers. Then, we apply our findings to the integration of one‐dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time‐steps based solely on accuracy requirements. High‐order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first‐order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.

ABSTRACT In this paper, we present the high‐order positivity‐preserving finite volume (PPFV) schemes for radiation hydrodynamics equations (RHE) in the equilibrium‐diffusion limit. The equations are composed of a hyperbolic system of conservation laws and a radiation diffusion term. First, the weighted essentially non‐oscillatory (WENO) reconstruction is applied to the spatial discretization of the hyperbolic system to achieve the high‐order numerical approximation. Subsequently, the adjacent cells are utilized to reconstruct high‐order polynomials for approximating the diffusion flux. Furthermore, the third‐order strong‐stability‐preserving (SSP) Runge‐Kutta scheme is employed for time discretization. Additionally, the positivity‐preserving limiter is implemented to prevent non‐physical numerical results. The positivity‐preserving property of the schemes is also proven. Finally, several numerical examples are presented to show the good performance of the schemes.

Abstract Systems of the first‐order partial differential equations with singular solutions appear in many multiphysics problems and the weak formulation of the solution involves, in many cases, the product of distributions. In this paper, we study such a system derived from Eulerian droplet model for air particle flow. This is a non‐strictly hyperbolic system of conservation laws with linear damping. We first study a regularized viscous system with variable viscosity term, obtain a weak asymptotic solution with general initial data and also get the solution in Colombeau algebra. We study the vanishing viscosity limit and show that this limit is a distributional solution. Further, we study the large‐time asymptotic behavior of the viscous system. This important system is not very well studied due to complexities in the analysis. As far as we know, the only work done on this system is for Riemann type of initial data. The significance of this paper is that we work on the system having general initial data and not just initial data of the Riemann type.

ABSTRACT This article addresses the interactions of delta shock waves with elementary waves in a strictly hyperbolic system of conservation laws derived from a traffic flow model. On utilizing the technique of splitting delta function, the global structures of solutions are constructed and analyzed one by one. Moreover, we prove that the solutions of the Riemann problems are stable under the small perturbations of the Riemann data.

Concerning on the consensus problem of a networked hyperbolic system of two linear conservation laws, this paper proposes a boundary consensus protocol and establishes a theoretical framework for consensus analysis under both fixed and switching communication topologies. Firstly, we present a consensus analysis under fixed topologies by utilizing the Lyapunov approach, where both undirected graph and digraph cases are considered. Under the assumption that the undirected graph is connected or the digraph is balanced and rooted, we derive sufficient conditions w.r.t. the boundary control matrices and Laplacian eigenvalues for ensuring the asymptotic consensus. Secondly, we further consider the consensus problem under switching topologies, in which the undirected graph (or balanced digraph) is relaxed to be jointly connected (or jointly rooted). By using the Lyapunov approach combined with the related space decomposition technique, we derive sufficient conditions w.r.t. the boundary control gain based on a priori knowledge of possible Laplacian matrices for ensuring the asymptotic consensus. Finally, we provide numerical examples and an application to the consensus control of a multi-lane road traffic flow system described by the Aw-Rascle Equations, to demonstrate the effectiveness of our theoretical results.

Abstract The study explores the Riemann problem and wave interactions for a two-lane two velocities traffic flow model with ideal isentropic pressure governing 3 × 3 system of hyperbolic conservation laws. Utilizing the method of characteristics, we construct the elementary waves, namely shock waves, rarefaction waves, and contact discontinuities of the Riemann problem in one-parameter family of curves. A condition for the existence of a unique solution is proved on arbitrary initial data which is not necessarily closed. Also, we establish a necessary and sufficient condition to determine whether a shock wave or a rarefaction wave exist of the solution in one-family and three-family of characteristic fields. Further, we analyze the combination of two shocks originating from the same family. Moreover, we solve the Riemann problem in a qualitative manner by considering the projections of the elementary waves in phase plane. At the end, the conditions on initial data are demonstrated to the waves structure of the Riemann problem along with the vacuum state.

We derive a depth-averaged model consistent with the $\mu (I)$ rheology for an incompressible granular flow down an inclined plane. The first two variables of the model are the depth and the depth-averaged velocity. The shear is also taken into account via a third variable called enstrophy. The obtained system is a hyperbolic system of conservation laws, with an additional equation for the energy. The system is derived from an asymptotic expansion of the flow variables in powers of the shallow-water parameter. This method ensures that the model is fully consistent with the rheology. The velocity profile is a Bagnold profile at leading order and the first-order correction to this profile can be calculated for flows that are not steady uniform. The first-order correction to the classical granular friction law is also consistently written. As a consequence, the instability threshold of the steady uniform flow is the same for the depth-averaged model and for the governing equations. In addition, a higher-order version that contains diffusive terms is also presented. The spatial growth rate, the phase velocity and the cutoff frequency of the version with diffusion are in good agreement with the experimental data and with the theoretical predictions for the rheology. The mathematical structure of the equations enables us to use well-known and stable numerical solvers. Numerical simulations of granular roll waves are presented. The model has the same limitations as the $\mu (I)$ rheology, in particular for the solid/ liquid and liquid/gas transitions, and needs therefore a regularisation for these transitions.

We present a rigorous approach and related techniques to construct global solutions of the two-dimensional (2-D) Riemann problem with four-shock interactions for the Euler equations of potential flow. With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions, and the sonic and detachment angles—whose existence and uniqueness follow from our rigorous proof of the strict monotonicity of the steady detachment and sonic angles for 2-D steady potential flow with respect to the upstream Mach number, we classify all configurations of the Riemann solutions for the interaction of two forward and two backward shocks, including the subsonic-subsonic reflection configuration that has not emerged in the previous results. To achieve this, we recast the 2-D Riemann problem as a shock reflection-diffraction problem with respect to a symmetry line, which is further reformulated as a free boundary problem for a second-order quasilinear equation of mixed elliptic-hyperbolic type. The difficulties arise from the degenerate ellipticity of the nonlinear equation near the sonic boundaries, the nonlinearity of the free boundary condition, the singularity of the solution near the corners of the domain, and the geometric properties of the free boundary. To solve the problem, we analyze the solutions of a quasilinear degenerate elliptic equation by using the maximum principle for the mixed boundary value problem, the theory of oblique derivative boundary value problems, uniform a priori estimates, and a sophisticated iteration method. To the best of our knowledge, this is the first rigorous result for the 2-D Riemann problem with four-shock interactions for the Euler equations. The approach and techniques developed for the Riemann problem with four-shock interactions should be useful in solving other 2-D Riemann problems for more general Euler equations and related nonlinear hyperbolic systems of conservation laws.

Abstract We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.

Delta-shock for a class of non-strictly hyperbolic systems of conservation laws as self-similar viscosity limits

In this paper, we establish the existence and nonlinear stability of a hyperbolic system of conservation laws derived from a repulsive singular chemotaxis model. By the phase plane analysis alongside Poincaré-Bendixson theorem, we first prove that this hyperbolic system admits three different types of traveling wave profiles, which are explicitly illustrated with numerical simulations. Then using a unified weighted energy estimates and technique of taking anti-derivatives, we prove that all types of traveling wave profiles, including non-monotone pulsating wave profiles, are nonlinearly and asymptotically stable if the initial data are small perturbations with zero mass from the spatially shifted traveling wave profiles.

We present a graph-based numerical method for solving hyperbolic systems of conservation laws using discontinuous finite elements. This work fills important gaps in the theory as well as practice of graph-based schemes. In particular, four building blocks required for the implementation of flux-limited graph-based methods are developed and tested: a first-order method with mathematical guarantees of robustness; a high-order method based on the entropy viscosity technique; a procedure to compute local bounds; and a convex limiting scheme. Two important features of the current work are the fact that (i) boundary conditions are incorporated into the mathematical theory as well as the implementation of the scheme. For instance, the first-order version of the scheme satisfies pointwise entropy inequalities including boundary effects for any boundary data that is admissible; (ii) sub-cell limiting is built into the convex limiting framework. This is in contrast to the majority of the existing methodologies that consider a single limiter per cell providing no sub-cell limiting capabilities. From a practical point of view, the implementation of graph-based methods is algebraic, meaning that they operate directly on the stencil of the spatial discretization. In principle, these methods do not need to use or invoke loops on cells or faces of the mesh. Finally, we verify convergence rates on various well-known test problems with differing regularity. We propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

On the Gelfand problem and viscosity matrices for two-dimensional hyperbolic systems of conservation laws

We consider the approximation of entropy solutions of nonlinear hyperbolic conservation laws using neural networks. We provide explicit computations that highlight why classical PINNs will not work for discontinuous solutions to nonlinear hyperbolic conservation laws and show that weak (dual) norms of the PDE residual should be used in the loss functional. This approach has been termed “weak PINNs” recently. We suggest some modifications to weak PINNs that make their training easier, which leads to smaller errors with less training, as shown by numerical experiments. Additionally, we extend wPINNs to scalar conservation laws with weak boundary data and to systems of hyperbolic conservation laws. We perform numerical experiments in order to assess the accuracy and efficiency of the extended method.