Runge-Kutta symmetric interior penalty discontinuous Galerkin methods for modified Buckley-Leverett equations

TL;DR

This paper develops a robust high-order symmetric interior penalty discontinuous Galerkin method combined with Runge-Kutta time integration to accurately solve the modified Buckley-Leverett equations in porous media flow, emphasizing shock capturing and stability.

Contribution

It introduces a novel combination of DG spatial discretization with SSP Runge-Kutta time stepping and evaluates shock-capturing limiters for improved accuracy in solving the modified Buckley-Leverett equations.

Findings

High-order schemes outperform lower order schemes.

Moment limiter by Moe et al. is most effective.

The proposed method achieves accurate and stable solutions.

Abstract

We present a robust and accurate numerical method to solve the modified Buckley-Leverett equation in two-phase porous media flow with dynamic capillary pressure effect. A symmetric interior penalty discontinuous Galerkin method is used to discretize the equation in the space direction. For accuracy and stability issues, the third-order strong stability preserving implicit-explicit Runge-Kutta method is adopted to solve the nonlinear semi-discrete system: the linear diffusion term is discretized implicitly while the nonlinear flux term is discretized explicitly. The spatial accuracy of the discontinuous Galerkin method depends on the limiters applied to the solution: we test a minmod-TVB limiter, a simple WENO limiter and a high-order shock-capturing moment limiter to demonstrate that a suitable shock capturing moment limiter leads to more accurate approximation of solution. A set of representative numerical experiments are presented to show the accuracy and efficiency of the proposed approach. The results indicate that the moment limiter proposed by Moe et al. [Arxiv:1507.03024, 2015] is the most suitable one to be used in solving the modified Buckley- Leverett equation, and high order schemes perform much better than lower order schemes. Our simulation results are consistent with the previous results in Kao et al.[J. Sci. Comput., 64(3) (2015), 837-857], Zhang and Zegeling [J. Comput. Phys., 345 (2017), 510-527 and Commun. Comput. Phys., 22(4) (2017), 935-964].