Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes

Liu, Hailiang; Wang, Jinghua; Yang, Tong

doi:10.1007/bf03167326

Cited by 6 publications

(15 citation statements)

References 14 publications

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Section: Hailiang Liumentioning

confidence: 92%

“…The existence of such discrete solutions and further properties, see Proposition 2.1, were proved by Liu, Wang and Yang [17]. Moreover, by using the energy method the authors in [17] were able to show that these stationary discrete travelling waves are nonlinearly stable with respect to initial perturbations, provided the total mass of the perturbation is zero. The main goal of this paper is to improve the nonlinear stability results in [17] by establishing the time convergence rates to a stationary discrete travelling wave (U j , V j ) j∈Z .…”

Section: Hailiang Liumentioning

confidence: 92%

“…Moreover, by using the energy method the authors in [17] were able to show that these stationary discrete travelling waves are nonlinearly stable with respect to initial perturbations, provided the total mass of the perturbation is zero. The main goal of this paper is to improve the nonlinear stability results in [17] by establishing the time convergence rates to a stationary discrete travelling wave (U j , V j ) j∈Z . To the author's knowledge, this seems the first time-asymptotic convergence rate for a difference scheme applied to a relaxation system of conservation laws.…”

Section: Hailiang Liumentioning

confidence: 99%

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Convergence rates to the discrete travelling wave for relaxation schemes

Liu

1999

Math. Comp.

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This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time, provided the sums of the initial perturbations for the u-component equal zero. A polynomially weighted l 2 norm on the perturbation of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.

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Section: Hailiang Liumentioning

confidence: 92%

Section: Hailiang Liumentioning

confidence: 92%

Section: Hailiang Liumentioning

confidence: 99%

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Convergence rates to the discrete travelling wave for relaxation schemes

Liu

1999

Math. Comp.

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“…There are several authors who have studied the stability of shock profiles; see [2,3,5,6,25,26]. In the study of discrete relaxation conservation laws, most works have concentrated on the scalar case, such as Liu, Wang and Yang [13,14,15]. In this paper, we will concentrate on the case of systems.…”

Section: Mao Yementioning

confidence: 99%

Existence and asymptotic stability of relaxation discrete shock profiles

2004

Math. Comp.

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Abstract. In 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.

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“…Jennings [13] studied the approximation of scalar equations by monotone conservative finite difference schemes and proved the existence and stability of traveling discrete shocks. Discrete profiles for scalar equations were also studied in [14,21,26]. The analysis was then extended to systems in [24,25].…”

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confidence: 99%

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

Renac¹

2015

SIAM J. Numer. Anal.

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Abstract. 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 of 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.Key words. discontinuous Galerkin method, discrete shock profile, scalar conservation laws, convex flux, inviscid Burgers equation, linear stability, spectral viscosity AMS subject classifications. 65N30, 65N121. Introduction. Discontinuous Galerkin (DG) methods are high-order finite element discretizations which were introduced in the early 1970s for the numerical simulation of the first-order hyperbolic neutron transport equation [20,30]. In recent years, these methods have become very popular for the solution of nonlinear convection dominated flow problems [7,8,17]. These methods allow high-order of accuracy and locality, which make them well suited to parallel computing, hp-refinement, hpmultigrid, unstructured meshes, application of boundary conditions, etc.However, the DG method suffers from spurious oscillations in the vicinity of discontinuities that develop in solutions of hyperbolic systems of conservation laws. These oscillations are due to the Gibbs phenomenon [9] and may cause the solution to become locally nonphysical leading to robustness issues for the computation. Quadrature rules are usually used to compute integrals in the discretization of the equations. Properties of the DG method therefore depend on the local structure of the numerical solution at faces and within elements of the mesh. The control of such oscillations at a reasonable cost while keeping accuracy, robustness and stability is essential for the efficiency of the DG method and remains a challenge. Strategies have been proposed such as limiters [7,8], non-oscillatory reconstructions [1,29], hp-adaptation [12], shock capturing techniques [27,10], etc. The latter methods aim at adding artificial viscosity to spread the structure of the discontinuity so that it can be resolved at the discrete level. For a DG method with polynomials of degree p > 0 and cells of size h, the resolution scale is h/p thus meaning that the m...

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Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes

Cited by 6 publications

References 14 publications

Convergence rates to the discrete travelling wave for relaxation schemes

Convergence rates to the discrete travelling wave for relaxation schemes

Existence and asymptotic stability of relaxation discrete shock profiles

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

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