Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws

Jiang, Guang-Shan; Yu, Shih‐Hsien

doi:10.1137/s0036142996307090

Cited by 16 publications

(9 citation statements)

References 8 publications

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“…As has been shown in Section 4.2, the ESWENO scheme can be written in the semi-discrete conservative form (13) with the flux f ES ¼ f W þê, where f W is the conventional WENO flux, andê is the additional dissipation flux term given by Eq. (48).…”

Section: Conservationmentioning

confidence: 99%

Third-order Energy Stable WENO scheme

Yamaleev

Carpenter

2009

Journal of Computational Physics

116

102

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Section: Conservationmentioning

confidence: 99%

Third-order Energy Stable WENO scheme

Yamaleev

Carpenter

2009

Journal of Computational Physics

116

102

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“…In the study of discrete conservation laws, Jennings [10] showed the stability of discrete shock profiles of general scalar first-order monotone schemes. The existence of discrete shock profiles of high order scalar difference scheme was established in [4,8] recently. For the study of the system of conservation laws, the existence of discrete shock profiles of finite-difference methods which are accurate to first order for systems of conservation laws was established by Majda and Ralson [21] using the center manifold construction and by Michelson in [22].…”

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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“…This kind of investigation has been proved important in understanding the convergence behavior of numerical shock computations and is essential for the error analysis around shocks. For scalar conservative schemes many stability results have been obtained by various authors, see e.g., Jennings [5], Ying and Zhou [24], Jiang and Yu [6], Fan [1], Liu and Wang [7,8,9], and Smyrlis [21]. The existence of discrete shocks for system began with Majda and Ralston [14]; see also Michelson [13].…”

Section: Introductionmentioning

confidence: 99%

The $l^1$ global decay to discrete shocks for scalar monotone schemes

Liu¹

2001

Math. Comp.

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Abstract. Given a family of discrete shocks φ of a monotone scheme, we prove that the discrete shock profile with rational shock speed η is asymptotically stable with respect to the l 1 topology · 1 : if u 0 − φ ∈ l 1 , then u n − φ ·−nη 1 → 0 as n → ∞ under no restriction conditions of the initial data to the interval [inf φ, sup φ]. The asymptotic wave profile is uniquely identified from the above family by a mass parameter.

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Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws

Cited by 16 publications

References 8 publications

Third-order Energy Stable WENO scheme

Third-order Energy Stable WENO scheme

Existence and asymptotic stability of relaxation discrete shock profiles

The $l^1$ global decay to discrete shocks for scalar monotone schemes

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