Results 1  10
of
21
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
Abstract

Cited by 27 (5 self)
 Add to MetaCart
(Show Context)
We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
Entropy stability theory for difference approximations of nonlinear conservation laws and related timedependent problems
 ACTA NUMERICA
, 2003
"... ..."
Explicit Diffusive Kinetic Schemes For Nonlinear Degenerate Parabolic Systems
 Parabolic Systems, Math. Comp
, 2000
"... We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the sourceterm depend singularly on the relaxation parameter. General stability conditions are derived, a ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the sourceterm depend singularly on the relaxation parameter. General stability conditions are derived, and convergence is proved to the entropy solutions for scalar equations.
Numerical approximations of the 10moment Gaussian closure
"... Abstract. We propose a numerical scheme to approximate the weak solutions of the 10moment Gaussian closure. The moment Gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensori ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We propose a numerical scheme to approximate the weak solutions of the 10moment Gaussian closure. The moment Gaussian closure for gas dynamics is governed by a conservative hyperbolic system supplemented by entropy inequalities whose solutions satisfy positiveness of density and tensorial pressure. We consider a Suliciu type relaxation numerical scheme to approximate the solutions. These methods are proved to satisfy all the expected positiveness properties and all the discrete entropy inequalities. The scheme is illustrated by several numerical experiments. 1.
A Reduced Stability Condition for Nonlinear Relaxation to Conservation Laws
 J. Hyperbolic Differ. Equ
, 2003
"... We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore, Liu in Comm. Pure Appl. Math. 47, 787830, namely the existence of an entr ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore, Liu in Comm. Pure Appl. Math. 47, 787830, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the ChapmanEnskog expansion. This reduced stability condition has the advantage to involve only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition.
A MULTIWAVE APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD BASED ON RELAXATION II NUMERICAL IMPLEMENTATION WITH 3 AND 5 WAVES
"... Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In the first part of this work ([5]), we introduced an approximate Riemann solver for onedimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of [5]. We present a 3wave solver that well resolves fast waves and material contacts, and a 5wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7wave solver, which is highly accurate on all types of waves, will be described in a followup paper. We test the solvers on onedimensional shock tube data and smooth shear waves. (1.1) (1.2)
A relaxation scheme for the approximation of the pressureless Euler equations, Num
 Method Partial Di. Eq
"... In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δshocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one sp ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δshocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed firstorder scheme but also a classical MUSCLtype secondorder extension confirm the reliability and robustness of the relaxation approach. The paper extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established and twodimensional
Entropysatisfying relaxation method with large timesteps for Euler IBVPs
, 2007
"... This paper could have been given the title: “How to positively and implicitly solve Euler equations using only linear scalar advections. ” The new relaxation method we propose is able to solve Eulerlike systems —as well as initial and boundary value problems — with real state laws at very low cost, ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
This paper could have been given the title: “How to positively and implicitly solve Euler equations using only linear scalar advections. ” The new relaxation method we propose is able to solve Eulerlike systems —as well as initial and boundary value problems — with real state laws at very low cost, using a hybrid explicitimplicit time integration associated with the Arbitrary LagrangianEulerian formalism. Furthermore, it enjoys many desirable properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of minmax principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The design of this optimal timestep, which takes into account data not only from the inner domain but also from the boundary conditions, is the main novel feature we emphasize on.
Finite difference schemes with cross derivatives correctors for multidimensional parabolic systems
 Journal of Hyperbolic Differential Equations
"... Abstract. We propose finite difference schemes for multidimensional quasilinear parabolic systems whose main feature is the introduction of correctors which control the secondorder terms with mixed derivatives. We show that with these correctors the schemes inherit physically relevant properties p ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We propose finite difference schemes for multidimensional quasilinear parabolic systems whose main feature is the introduction of correctors which control the secondorder terms with mixed derivatives. We show that with these correctors the schemes inherit physically relevant properties present at the continuous level, such as the existence of invariant domains and/or the nonincrease of the total amount of entropy. The analysis is performed with some general tools that could be used also in the analysis of finite volume methods of flux vector splitting type for firstorder hyperbolic problems on unstructured meshes. Applications to the compressible NavierStokes system are given. 1.
NUMERICAL COMPARISON OF RIEMANN SOLVERS FOR ASTROPHYSICAL HYDRODYNAMICS
"... Abstract. The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with stateof theart algorithms for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPMcode, the Prometheus code, and also made a ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The idea of this work is to compare a new positive and entropy stable approximate Riemann solver by Francois Bouchut with stateof theart algorithms for astrophysical fluid dynamics. We implemented the new Riemann solver into an astrophysical PPMcode, the Prometheus code, and also made a version with a different, more theoretically grounded higher order algorithm than PPM. We present shock tube tests, twodimensional instability tests and forced turbulence simulations in three dimensions. We find subtle differences between the codes in the shock tube tests, and in the statistics of the turbulence simulations. The new Riemann solver increases the computational speed without significant loss of accuracy. 1.