Asymptotic-preserving Godunov-type numerical schemes for hyperbolic systems with stiff and non-stiff relaxation terms (original) (raw)

Methods for hyperbolic systems with stiff relaxation

International Journal for Numerical Methods in Fluids, 2002

Three methods are analyzed for solving a linear hyperbolic system that contains stiff relaxation. We show that the semi-discrete discontinuous Galerkin method, with a linear basis, is accurate when the relaxation time is unresolved (asymptotic preserving -AP). The two other methods are shown to be non-AP. To discriminate between AP and non-AP methods, we argue that in the limit of small relaxation time, one should fix the dimensionless parameters that characterize the near-equilibrium limit.

Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations

Mathematics of Computation, 2012

We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff source term. On one hand, we determine the relevant asymptotic expansion, derive a reduced system in the limit, and discuss the role of an entropy condition to establish the diffusive nature of the reduced system. On the other hand, we propose a new numerical scheme of finite volume type, which allows us to recover the correct asymptotic regime. The associated discrete form of the diffusion system is achieved via a suitable discretization compatible with the stiff source term. Our theoretical results are illustrated with several models from continuum physics and numerical experiments demonstrating the relevance of the proposed theory and numerical strategy.

Implicit-Explicit methods for hyperbolic systems with hyperbolic and parabolic relaxation

In this talk we discuss the problem of constructing effective high order methods for the numerical solution of hyperbolic systems of balance laws, in presence of stiff source. Because of the stiffness, the use of implicit integrators is advisable, so that no restrictions on the time step due to small relaxation time will appear. Two different relaxation systems will be considered, namely hyperbolic and parabolic relaxation. Because of the different nature of the problems, the two cases will be considered separately. A common denominator of both treatments is the choice of space discretization. Most schemes for conservation or balance laws are discretized by finite volume (FV), conservative finite difference (FD), or discontinuous Galerkin (DG). Here we choose conservative finite difference since it is probably the simplest general approach for the construction of high order shock capturing schemes for such problems.

Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation

Siam Journal on Numerical Analysis, 1997

We develop high resolution shock capturing numericalschemes for hyperbolicsystems with relaxation. In such systems the relaxation time may vary from order one to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly sti , and underresolved numerical schemes may produce spurious results. Usually one can not decouple the problem into separate regimes and handle di erent regimes with di erent methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a second order scheme that works e ectively, with a xed spatial and temporal discretization, for all range of mean free path. Formal uniform consistency proof for a rst order scheme, and numerical convergence proof for the second order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.

Numerical methods for nonconservative hyperbolic systems: A theoretical framework

SIAM journal on numerical analysis, 2007

The goal of this paper is to provide a theoretical framework allowing to extend some general concepts related to the numerical approximation of 1d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and the analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of Approximate Riemann Solvers and we give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe and Relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.

A Godunov scheme for solving hyperbolic systems in a nonconservative form

Journal de Physique IV (Proceedings), 2005

In this paper, we developed a Godunov scheme for solving nonconservative systems. The main idea of this method is a new type of projection which illustrated the essential role of the numerical viscosity to determine the solution with shocks for system in a nonconservative form. We apply our study to a system modeling elasticity and we observe a complete agreement between the theory and the numerical results.

A Unified IMEX Runge--Kutta Approach for Hyperbolic Systems with Multiscale Relaxation

SIAM Journal on Numerical Analysis

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis.

Higher-order relaxation schemes for hyperbolic systems of conservation laws

Journal of Numerical Mathematics, 2000

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in . The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge-Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

Provably non-stiff implementation of weak coupling conditions for hyperbolic problems

Numerische Mathematik, 2022

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant–Friedrichs–Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superco...