Computational multiscale methods for quasi-gas dynamic equations

TL;DR

This paper develops a multiscale numerical method combining CEM-GMsFEM and leapfrog time scheme to efficiently solve the quasi-gas-dynamic equations in heterogeneous media, with proven stability and convergence.

Contribution

It introduces a novel multiscale finite element approach for QGD models in multiscale media, addressing hyperbolic challenges and reducing computational costs.

Findings

The method is stable under a relaxed CFL condition.

Convergence of the scheme is theoretically established.

Numerical experiments confirm the effectiveness of the approach.

Abstract

In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications. This multiscale problem is interesting from a multiscale methodology point of view as the model problem has a hyperbolic multiscale term, and designing multiscale methods for hyperbolic equations is challenging. In the paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) combined with the leapfrog scheme in time to solve this problem. The CEM-GMsFEM provides a flexible and systematical framework to construct crucial multiscale basis functions for approximating the solution to the problem with reduced computational cost. With this approach of spatial discretization, we establish the stability of the fully discretized scheme under a relaxed version of the so-called CFL condition. Complete convergence analysis of the proposed method is presented. Numerical results are provided to illustrate and verify the theoretical findings.