Discrete Shock Profiles Research Articles

Stability Analysis of Discontinuous Galerkin Discrete Shock Profiles for Steady Scalar Conservation Laws

We present a stability analysis of stationary discrete shock profiles for a discontinuous Galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux. We consider the Godunov method to evaluate the numerical flux and use either an explicit third-order Runge--Kutta scheme or an implicit backward Euler scheme for the time integration. Applying a linear stability analysis, we show that the steady solutions may become unstable when the numerical flux is not differentiable. In particular, the situation of a shock at an interface of the mesh corresponds to an unstable solution for a space discretization accuracy higher than second-order, whatever the time integration method and the physical flux. Spectra of the linearized operator indicate that the fourth-order numerical scheme with the explicit time integration is also unstable for a continuous range of shock positions around an interface.

Discrete shock profiles for scalar conservation laws with discontinuous fluxes

The paper considers a scalar conservation law with a discontinuous flux F of the form F(x,u)=H(x)g(u)+(1−H(x))h(u) where H(x) is the Heaviside function. Herein, the fluxes g and h are supposed to have one minimum and no maximum and at most one crossing in the interior of the domain of definition. The aim is to verify a weak solution of such a problem in the following way: We are looking for discrete shock profiles for its continuously differentiable perturbation with a parameter ε and Godunov's scheme for conservation laws with spatially varying flux functions. The obtained discrete shock profile satisfies a discrete entropy condition of Kruzkhov type and after letting ε→0, approaches an entropy weak solution of the original equation.

Stationary Discrete Shock Profiles for Scalar Conservation Laws with a Discontinuous Galerkin Method

We present an analysis of stationary discrete shock profiles for a discontinuous Galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux. Using the Godunov method for the numerical flux, we characterize the steady-state solutions for arbitrary approximation orders and show that they are oscillatory only in one mesh cell and are parametrized by the shock strength and its relative position in the cell. In the particular case of the inviscid Burgers equation, we derive analytical solutions of the numerical scheme and predict their oscillations up to fourth-order accuracy. Moreover, a linear stability analysis shows that these profiles may become unstable at points where the Godunov flux is not differentiable. Theoretical and numerical investigations show that these results can be extended to other numerical fluxes. In particular, shock profiles are found to vanish exponentially fast from the shock position for some class of monotone numerical fluxes and the oscillatory and unstable characters of their solutions present strong similarities with that of the Godunov method.

On two deficiencies of the AUFS scheme for Euler flows and possible fixes

The AUFS scheme by Sun and Takayama is a flux splitting scheme without breakdown of discrete shock profiles, usually called carbuncle, but still with a good resolution of entropy waves. Unfortunately, numerical tests with this scheme yield that the viscosity on entropy waves is too small while the viscosity on shear waves is too large. In this paper, we prove that both deficiencies are inherent to the construction of the scheme and provide fixes to overcome them.

Modified Suliciu relaxation system and exact resolution of isolated shock waves

We present a new approximate Riemann solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general nonlinear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is positive, Lipschitz-continuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunov-type method, or a Glimm-type (random sampling-based Godunov-type) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches.

A note on the carbuncle phenomenon in shallow water simulations

AbstractA classic problem in gas dynamics simulation is the occurrence of the carbuncle phenomenon, a breakdown of discrete shock profiles. We show that for high Froude number, this also occurs in shallow water simulations. Numerical evidence is given that commonly accepted cures developed for the numerics of gas dynamics should also work for shallow water flows.

An investigation of the internal structure of shock profiles for shock capturing schemes

The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.

Existence and asymptotic stability of relaxation discrete shock profiles

kIn this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in L 2 , 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 relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

Green's function pointwise estimates for the modified Lax–Friedrichs scheme

The aim of this paper is to find estimates of the Green's function of stationary discrete shock profiles and discrete boundary layers of the modified Lax–Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [CITE] in the continuous viscous setting.

Existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws

When shock speed s times Δt/Δx is rational, the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved. A sufficient condition under which the above solutions can be extended to - ∞ < j < ∞, implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with C 1 numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with Lax-Friedrichs' splitting, and the Godunov scheme for scalar conservation laws are found to satisfy this condition. Thus, the existence of discrete shock profiles for these schemes is established when sΔt/Δx is rational.

Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws

Numerical simulations often provide strong evidences for the existence and stability of discrete shocks for certain finite difference schemes approximating conservation laws. This paper presents a framework for converting such numerical observations to mathematical proofs. The framework is applicable to conservative schemes approximating stationary shocks of one-dimensional scalar conservation laws. The numerical flux function of the scheme is assumed to be twice differentiable but the scheme can be nonlinear and of any order of accuracy. To prove existence and stability, we show that it would suffice to verify some simple inequalities, which can usually be done using computers. As examples, we use the framework to give an unified proof of the existence of continuous discrete shock profiles for a modified first-order Lax--Friedrichs scheme and the second-order Lax--Wendroff scheme. We also show the existence and stability of discrete shocks for a third-order weighted essentially nonoscillatory (ENO) scheme.

Decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws

This paper is to study the decay rate for perturbations of stationary discrete shocks for the Lax-Friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method. If the summation of the initial perturbation over (-∞,j) is small and decays at the algebraic rate as |j|→ ∞, then the solution approaches the stationary discrete shock profiles at the corresponding rate as n - ∞. This rate seems to be almost optimal compared with the rate in the continuous counterpart. Proofs are given by applying the weighted energy integration method to the integrated scheme of the original one. The selection of the time-dependent discrete weight function plays a crucial role in this procedure.

Discrete shock profiles for systems of conservation laws

AbstractThe existence of discrete shock profiles for difference schemes approximating a system of conservation laws is the major topic studied in this paper. The basic theorem established here applies to first‐order accurate difference schemes; for weak shocks, this theorem provides necessary and sufficient conditions involving the truncation error of the linearized scheme which guarantee entropy satisfying or entropy violating discrete shock profiles. Several explicit difference schemes are used as examples illustrating the interplay between the entropy condition, monotonicity, and linearized stability. Entropy violating stationary shocks for second‐order accurate Lax‐Wendroff schemes approximating systems are also constructed. The only tools used in the proofs are local analysis and the center manifold theorem.