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Pré-Publication, Document De Travail Année : 2020

Large time step TVD IMEX Runge-Kutta schemes based on arbitrarily high order Butcher tableaux

Andrea Thomann

Résumé

The context of this work is the development of large time step TVD IMEX Runge-Kutta schemes 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 large-scale eigenvalues of the multi-scale PDE, as long as they are treated implicitly. However, a result from Gottlieb et al. [15] states that unconditionally stable implicit TVD or L∞ stable RK schemes can only be of first order. We show that this result is also valid for IMEX-RK schemes, which have a CFL restriction only depending on the explicitly treated scales. Therefore, the goal of this work is to improve the precision of a first-order IMEX-RK scheme, while retaining its L∞ stability and TVD properties. In this work we extend and generalize the approach introduced in [9] 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 method for a scalar multi-scale equation and we numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, we combine our TVD schemes with the MOOD framework to increase their applicability. For numerical validation, we apply the schemes to the isentropic Euler equations and compare the results with [9] where a second order scheme is used as a basis of the TVD scheme.
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Dates et versions

hal-02494767 , version 1 (29-02-2020)
hal-02494767 , version 2 (15-10-2020)
hal-02494767 , version 3 (25-03-2021)
hal-02494767 , version 4 (26-02-2022)
hal-02494767 , version 5 (04-04-2022)
hal-02494767 , version 6 (04-07-2022)

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  • HAL Id : hal-02494767 , version 3

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Victor Michel-Dansac, Andrea Thomann. Large time step TVD IMEX Runge-Kutta schemes based on arbitrarily high order Butcher tableaux. 2020. ⟨hal-02494767v3⟩
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