Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws

Zhang, Qiang; Shu, Chi‐Wang

doi:10.1137/s0036142902404182

Cited by 182 publications

(199 citation statements)

References 28 publications

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“…As such, they provide an attractive alternative to the method of characteristics since the present methods are more easily extendible to higher order. For DG methods, the two above results for RK2 schemes have been obtained by Zhang and Shu in the more general context of nonlinear scalar conservation laws [27] and symmetrizable systems of nonlinear conservation laws [28]. Our result for the RK3 scheme is, to the best of our knowledge, new.…”

Section: Introduction Let ω Be An Open Bounded Lipschitz Domain In Rsupporting

confidence: 62%

Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

Burman¹,

Ern²,

Fernández³

2010

SIAM J. Numer. Anal.

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Abstract. We analyze explicit Runge-Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit second-and third-order Runge-Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish L 2 -norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge-Kutta schemes and any polynomial degree in space and for second-order RungeKutta schemes and first-order polynomials in space. For second-order Runge-Kutta schemes and higher polynomial degrees in space, a tightened 4/3-CFL condition is required. Numerical results are presented for the advection and wave equations.

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Section: Introduction Let ω Be An Open Bounded Lipschitz Domain In Rsupporting

confidence: 62%

Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

Burman¹,

Ern²,

Fernández³

2010

SIAM J. Numer. Anal.

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“…For notations of different constants we will follow [29] and [26]. By C, we refer to a positive constant that is independent of the mesh size h, but it may depend on other parameters of the problem.…”

Section: Notations For Different Constantsmentioning

confidence: 99%

“…A quantity related to the numerical flux. In [29], Zhang and Shu introduced an important quantity to measure the difference between a monotone numerical flux and the physical flux.…”

Section: A Appendix: Collections Of Lemmas and Proofsmentioning

confidence: 99%

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

2007

Self Cite

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Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

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“…Ainsi, on comprend bien d'où provient la condition de stabilité ∆t ≤ ∆x α avec α > 1 qui se rencontre parfois dans les simulations numériques où domine la convection [9]. Et une simple analyse de stabilité de von Neumann permet d'en rendre compte,à condition de considérer un accroissement de l'erreur en C∆tà chaque pas de temps.…”

Section: Resultsunclassified

Stabilité sous condition CFL non linéaire

Deriaz

Kolomenskiy

2012

ESAIM: Proc.

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Abstract. We present a basic althought little known numerical stability condition: for convection equations, the von Neumann stability constraint un+1 L 2 ≤ (1 + C ∆t) un L 2 drives to the stability condition ∆t ≤ C∆x α with α = p(2q−1) q(2p−1) where p is an integer linked to the stability domain of the time scheme and q ≥ p an integer linked to the upwind property of the space discretization (in the centered case we have q = +∞ and α = 2p 2p−1 ).Résumé. Nous présentons uneétude de stabilité aux conclusions originales : pour leséquations de convection, la contrainte un+1 L 2 ≤ (1 + C ∆t) un L 2 fait apparaître la condition de stabilité ∆t ≤ C∆x α avec α = p(2q−1) q(2p−1) où p est un entier lié au domaine de stabilité du schéma en temps et q ≥ p un autre entier lié au caractère upwind de la discrétisation en espace (dans le cas centré, q = +∞). Des exemples numériques montrent la pertinence de cetteétude.

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Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws

Cited by 182 publications

References 28 publications

Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

Stabilité sous condition CFL non linéaire

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