Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

Abstract

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme.We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff procedure. The former preserves the accuracy of the RK schemes and the latter guarantees the stability. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [12], which could not be derived for the IMEX schemes. In addition, the intermediate boundary conditions are only available for explicit RK schemes up to third order while our method applies to arbitrary order IMEX and explicit RK schemes. Moreover, the present boundary treatment method may be adapted to IMEX RK schemes solving many other partial differential equations. For a specific third-order IMEX scheme, we demonstrate the good stability and third-order accuracy of our boundary treatment through both 1D examples and 2D reactive Euler equations.

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

“…This is remedied for fourth-order schemes in [14,15], however, the methods therein only apply to one-dimensional (1D) scalar equations or systems with all characteristics flowing into the domain at the boundary. Different from this, we propose in our previous work [4] to use the RK schemes themselves at the boundary. This idea, combined with an inverse Lax-Wendroff (ILW) procedure in [5,6], preserves the accuracy and good stability of implicit-explicit (IMEX) RK schemes solving hyperbolic systems with source terms.…”

“…This idea, combined with an inverse Lax-Wendroff (ILW) procedure in [5,6], preserves the accuracy and good stability of implicit-explicit (IMEX) RK schemes solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, the method in [4] naturally applies to explicit methods as well.…”

“…In this paper, we examine the boundary treatment method in [4] for explicit high-order SSP RK schemes of hyperbolic conservation laws. Specifically, we use finite difference WENO schemes on a Cartesian mesh for the spatial discretization, where the corresponding downwind scheme is applied in the case of negative coefficients.…”

“…Additionally, when boundary conditions are present and the proposed boundary treatment is used, the SSP RK schemes with negative coefficients still allow larger time steps than those with all non-negative coefficients. In this regard, the boundary treatment method in [4] is an effective supplement to the SSP RK schemes with/without negative coefficients for initial-boundary value problems for conservation laws.…”

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In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well.

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