Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

TL;DR

This paper introduces two implicit local time-stepping methods for advection-diffusion problems in heterogeneous media, enabling independent subdomain time steps and demonstrating convergence and effectiveness through numerical experiments.

Contribution

The paper develops and analyzes two novel implicit local time-stepping algorithms using domain decomposition for advection-diffusion equations, allowing flexible subdomain time discretizations.

Findings

Convergence of the OSWR algorithm with nonmatching time grids is proven.

Numerical results show the methods' effectiveness for high Peclét numbers and discontinuous coefficients.

The methods successfully simulate nuclear waste storage scenarios.

Abstract

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincar\'e type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Pecl\'et numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.