Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations

Zhang, Qian; Xu, Yan; Shu, Chi‐Wang

doi:10.4208/cicp.oa-2020-0027

Cited by 4 publications

(3 citation statements)

References 37 publications

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“…then there holds ‖v‖ 2 L 2 (I×J) ≤ Ch 2k , for k ≥ 2. This lemma can be proven in a similar way to Reference [37]. By the above preparations, the proof of Theorem 3.4 can be provided.…”

Section: • the Error Equationmentioning

confidence: 65%

“…The authors proposed the energy stable DG scheme for the Camassa‐Holm (CH) equation in Reference [30] to approximate different solutions. Moreover, the numerical schemes for the Benjamin‐Bona‐Mahony [19] and Fornberg‐Whitham equations [37] offer inspiration for our DG scheme. In Reference [36], the operator

{\partial}^{-1}

was handled by constructing an integration DG scheme, which will be a critical step to simplify the calculation for the CH‐KP type equations.…”

Section: Introductionmentioning

confidence: 99%

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A discontinuous Galerkin method for the Camassa‐Holm‐Kadomtsev‐Petviashvili type equations

Zhang

Liu

2023

Numerical Methods Partial

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This paper develops a high-order discontinuous Galerkin (DG) method for the Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) type equations on Cartesian meshes. The significant part of the simulation for the CH-KP type equations lies in the treatment for the integration operator 𝜕 −1 . Our proposed DG method deals with it element by element, which is efficient and applicable to most solutions. Using the instinctive energy of the original PDE as a guiding principle, the DG scheme can be proved as an energy stable numerical scheme. In addition, the semi-discrete error estimates results for the nonlinear case are derived without any priori assumption. Several numerical experiments demonstrate the capability of our schemes for various types of solutions.

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Section: • the Error Equationmentioning

confidence: 65%

{\partial}^{-1}

was handled by constructing an integration DG scheme, which will be a critical step to simplify the calculation for the CH‐KP type equations.…”

Section: Introductionmentioning

confidence: 99%

A discontinuous Galerkin method for the Camassa‐Holm‐Kadomtsev‐Petviashvili type equations

Zhang

Liu

2023

Numerical Methods Partial

Self Cite

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show abstract

“…In [19], the authors constructed high-order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. Then they gave the proof for the dissipation or conservation of related conservative quantities.…”

Section: Introductionmentioning

confidence: 99%

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

Gao¹,

Sun²

2023

NMTMA

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A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the secondorder accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

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A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

Li,

Guan,

Shi

et al. 2025

Applied Numerical Mathematics

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Dissipative and Conservative Local Discontinuous Galerkin Methods for the Fornberg-Whitham Type Equations

Cited by 4 publications

References 37 publications

A discontinuous Galerkin method for the Camassa‐Holm‐Kadomtsev‐Petviashvili type equations

A discontinuous Galerkin method for the Camassa‐Holm‐Kadomtsev‐Petviashvili type equations

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation

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