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TVD-MOOD schemes based on implicit-explicit time integration

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  • Michel-Dansac, Victor
  • Thomann, Andrea

Abstract

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [1] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L∞ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

Suggested Citation

  • Michel-Dansac, Victor & Thomann, Andrea, 2022. "TVD-MOOD schemes based on implicit-explicit time integration," Applied Mathematics and Computation, Elsevier, vol. 433(C).
  • Handle: RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322004714
    DOI: 10.1016/j.amc.2022.127397
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    References listed on IDEAS

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    1. 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 & Žeravý, Michal, 2023. "High resolution compact implicit numerical scheme for conservation laws," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    2. Caballero-Cárdenas, C. & Castro, M.J. & Morales de Luna, T. & Muñoz-Ruiz, M.L., 2023. "Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    3. Gholamreza Farahmand & Taher Lotfi & Malik Zaka Ullah & Stanford Shateyi, 2023. "Finding an Efficient Computational Solution for the Bates Partial Integro-Differential Equation Utilizing the RBF-FD Scheme," Mathematics, MDPI, vol. 11(5), pages 1-13, February.

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