Dual-grid mapping method for the advection-diffusion-reaction equation in a heterogeneous medium

Abstract

Processes described by the advection-diffusion-reaction (ADR) equation often arise in heterogeneous media. Of particular interest is the case of layered or otherwise composite media exhibiting many layers or other intricate geometric complexity. Numerical models for processes in these types of media are generally accompanied by a significant computational cost due to the large number of grid nodes required to capture the detail of the heterogeneous domain. An approach addressing this issue involves obtaining approximate solutions on a coarse grid comprised of a small number of nodes and reconstructing a solution on a fine grid comprising a large number of nodes. In this paper, we discuss a projection-based framework for coarse-to-fine-grid approximation methods, which is used to derive a dual-grid mapping method. We initially utilise an interpolation-based mapping, and then consider a new discretisation-based adaptive mapping defined through the solution of coupled problems on subsets of the fine grid. With our new adaptive dual-grid mapping method, we are able to accurately approximate complex solutions on very coarse grids and reconstruct much finer scale approximations on fine grids. Code implementing the developed methods is provided and also applied to several test cases, where accuracy and significantly reduced computational cost are demonstrated in comparison to a numerical solution applied directly on the fine grid.