Operator Splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media (Communications in Computational Physics, 2023)
This paper is concerned with efficient numerical methods for the advection-diffusion equation in a heterogeneous porous medium containing fractures. A dimensionally reduced fracture model is considered, in which the fracture is represented asan interface between subdomains and is assumed to have larger permeability than thesurrounding area. We develop three global-in-time domain decomposition methodscoupled with operator splitting for the reduced fracture model, where the advectionand the diffusion are treated separately by different numerical schemes and with different time steps. Importantly, smaller time steps can be used in the fracture-interfacethan in the subdomains. The first two methods are based on the physical transmission conditions, while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions. A discrete space-timeinterface system is formulated for each method and is solved iteratively and globallyin time. Numerical results for two-dimensional problems with various Péclet numbers and different types of fracture are presented to illustrate and compare the convergenceand accuracy in time of the proposed methods with local time stepping.