A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.

In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2-norm. Therefore, we first prove that the L2-norm of the α-th order (1≤ α≤ k+1) divided difference of the DG error with upwind fluxes is of order k+(3-α)/2, provided that the flux Jacobian matrix, f'(u), is symmetric positive definite. Furthermore, using the duality argument, we are able to derive superconvergence estimates of order 2k+(3-α)/2 for the negative-order norm, indicating that some particular compact kernels can be used to extract at least (3k/2+1)-th order superconvergence for nonlinear systems of conservation laws. Numerical experiments are shown to demonstrate the theoretical results.

In this course evolution equations defining non-linear hyperbolic conservation laws, some general theory of non-linear systems of conservation laws and solution methods will be presented. The notions of a weak solution and entropy will be introduced. This will lead into an investigation of solutions of the so called Riemann problem. For scalar conservation laws, analytical solutions will be derived using characteristics methods. In general numerical methods are used to solve or simulate such problems. Therefore, ideas guiding the design of numerical schemes for such equations will be discussed. Some numerical schemes for the numerical integration of such initial boundary value problems related to systems of conservation laws will be analyzed. A collection of case studies from application areas like gas dynamics, and networked flow will be used to demonstrate how non-linear hyperbolic conservation laws are used to model, understand and predict the dynamics governing real-world problems.KeywordsWeak SolutionRiemann ProblemBurger EquationEntropy SolutionEntropy ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A class of hyperbolic systems of conservation laws satisfying weaker conditions than genuine nonlinearity

Two-dimensional Riemann problem involving three J’s for a hyperbolic system of nonlinear conservation laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.

We consider weakly coupled systems of nonlinear hyperbolic conservation laws on moving surfaces. As in the Euclidean space, see, for example, (Levy, Commun Partial Differ Equ 17(3–4):657–698, 1992, [9], Rohde, Weakly coupled systems of hyperbolic conservation laws. PhD thesis, Mathematische Fakultat der Albert-Ludwigs-Universitat Freiburg 1996, [16]), the coupling is realized by a source term, which only depends on (x, t) and the unknown function u(x, t) but not on its derivatives. Scalar conservation laws on moving surfaces were considered in Dziuk et al., Interfaces Free Boundaries 15:202–236, 2013, [4]), Lengeler and Muller (J Differ Equ 254(4):1705–1727, 2013, [10]). The velocity of the surface is given by a smooth function, and we assume the surface to be compact. We prove the existence for an entropy solution. First, we consider the regularized parabolic problem with viscosity parameter \(\varepsilon \) and show that there exists a weak solution by decoupling and linearizing the problem. Then, we prove the boundedness of this solution in \(L^\infty (G_T)\), use standard regularity results to prove that this solution is a solution in the classical sense, and show uniform boundedness in \(L^\infty (G_T)\) and \(W^{1,1}(G_T)\) with respect to \(\varepsilon \).

Central schemes offer a simple and versatile approach for computing approximate solutions of non-linear systems of hyperbolic conservation laws and related PDEs. The solution of such problems often involves the spontaneous evolution of steep gradients. The multiscale aspect of these gradients poses a main computational challenge for their numerical solution. Central schemes utilize a minimal amount of information on the propagation speeds associated with the problems, in order to accurately detect these steep gradients. This information is then coupled with high–order, non-oscillatory reconstruction of the approximate solution in ‘the direction of smoothness’: that is, information of smoothness does not cross regions of steep gradients. The use of central stencils enables us to realize the reconstructed solutions through simple quadratures. In this manner, central schemes avoid the intricate and time-consuming details of the eigen-structure of the underlying PDEs, and in particular, the use of (approximate) Riemann solvers, dimensional splitting, etc. The resulting family of central schemes offers relatively simple, black-box solvers for a wide variety of problems governed by multi-dimensional systems of non-linear hyperbolic conservation laws and related convection-diffusion problems.

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition.

Multidimensional Riemann problem with self-similar internal structure. Part II – Application to hyperbolic conservation laws on unstructured meshes

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the LP-norm for all p __> 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in L p (p >= 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.

Multiresolution-based adaptive central high resolution schemes for modeling of nonlinear propagating fronts

Hyperbolic Conservation Laws

The problem of well-posedness for a system of nonstrictly hyperbolic conservation laws is studied. A finite difference scheme is used to prove the existence of an entropy solution with bounded variation. It is proved that the entropy solution of the system is unique, and that the solution depends continuously on its initial data in a proper topology. The analysis is based on a smoothness property of one of the Riemann invariants of the system.