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High resolution compact implicit numerical scheme for conservation laws

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  • Frolkovič, Peter
  • Žeravý, Michal

Abstract

We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.

Suggested Citation

  • Frolkovič, Peter & Žeravý, Michal, 2023. "High resolution compact implicit numerical scheme for conservation laws," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    • Handle: RePEc:eee:apmaco:v:442:y:2023:i:c:s0096300322007883
    DOI: 10.1016/j.amc.2022.127720
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    References listed on IDEAS

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    1. Frolkovič, Peter & Mikula, Karol, 2018. "Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 129-142.
    2. Michel-Dansac, Victor & Thomann, Andrea, 2022. "TVD-MOOD schemes based on implicit-explicit time integration," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    3. Busto, S. & Río-Martín, L. & Vázquez-Cendón, M.E. & Dumbser, M., 2021. "A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes," Applied Mathematics and Computation, Elsevier, vol. 402(C).
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    Cited by:

    1. Frolkovič, Peter & Gajdošová, Nikola, 2024. "Unconditionally stable higher order semi-implicit level set method for advection equations," Applied Mathematics and Computation, Elsevier, vol. 466(C).

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