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Pré-Publication, Document De Travail Année : 2020

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

Résumé

The context of this work is the development of TVD Implicit-Explicit Runge-Kutta schemes to approximate the solution of hyperbolic multi-scale equations. A key feature of IMEX-RK schemes is that the resulting CFL condition does not depend on the stiff part of the considered equation, as long as this stiff part is treated implicitly. However, the negative result from Gottlieb et al. states that it is not possible to construct an implicit Runge-Kutta scheme of order higher than one that is either L ∞ stable or TVD. We show that this result is also valid for IMEX-RK schemes. Therefore, rather than building a high-order TVD scheme, the goal of this work is to provide a way of improving the precision of a first-order IMEX-RK scheme, while retaining its L ∞ stability and TVD properties. To that end, we introduce a convex combination between an oscillatory high-order IMEX-RK scheme and the stable but diffusive first-order IMEX-RK scheme. We derive generic conditions to ensure the TVD property for a convex combination based on an arbitrarily high order Butcher tableau. To increase the precision, we combine our new TVD schemes with an optimal order detection strategy inspired by the MOOD framework. We compare them to L-stable methods from the literature, applied on a two-scale hyperbolic equation where we test the performance on continuous and discontinuous solutions.
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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 2

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Victor Michel-Dansac, Andrea Thomann. TVD IMEX Runge-Kutta schemes based on arbitrarily high order Butcher tableaux. 2020. ⟨hal-02494767v2⟩
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