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Numerical preservation of multiple local conservation laws

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  • Frasca-Caccia, Gianluca
  • Hydon, Peter E.

Abstract

There are several well-established approaches to constructing finite difference schemes that preserve global invariants of a given partial differential equation. However, few of these methods preserve more than one conservation law locally. A recently-introduced strategy uses symbolic algebra to construct finite difference schemes that preserve several local conservation laws of a given scalar PDE in Kovalevskaya form. In this paper, we adapt the new strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs. The Benjamin–Bona–Mahony equation and a system equivalent to the nonlinear Schrödinger equation are used as benchmarks, showing that the strategy yields conservative schemes which are robust and highly accurate compared to others in the literature.

Suggested Citation

  • Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    • Handle: RePEc:eee:apmaco:v:403:y:2021:i:c:s0096300321002939
    DOI: 10.1016/j.amc.2021.126203
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    References listed on IDEAS

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    1. Islas, A.L. & Schober, C.M., 2005. "Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(3), pages 290-303.
    2. Barletti, L. & Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2018. "Energy-conserving methods for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 3-18.
    3. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
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    Cited by:

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    2. Sun, Zhengjie, 2022. "A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach," Applied Mathematics and Computation, Elsevier, vol. 423(C).

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