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

by   Weifeng Zhao, et al.

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, 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.



page 26

page 28

page 29

page 30


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

In <cit.>, we developed a boundary treatment method for implicit-explici...

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

In this article, we consider a class of finite rank perturbations of Toe...

A boundary-penalized isogeometric analysis for second-order hyperbolic equations

Explicit time-marching schemes are popular for solving time-dependent pa...

High-Order Multiderivative IMEX Schemes

Recently, a 4th-order asymptotic preserving multiderivative implicit-exp...

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the sum...

Very high-order Cartesian-grid finite difference method on arbitrary geometries

An arbitrary order finite difference method for curved boundary domains ...

A Dissipation Theory for Three-Dimensional FDTD with Application to Stability Analysis and Subgridding

The finite-difference time-domain (FDTD) algorithm is a popular numerica...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.