Discontinuous in time Virtual Element method for Darcy equations coupled with Multi Species Transport with First Order Reaction Network
arXiv:2506.21326
The paper presents a numerical scheme that combines the Virtual Element Method for spatial discretization with a discontinuous Galerkin time integrator to solve Darcy flow coupled with multi‑species advection‑diffusion‑reaction equations featuring first‑order reactions.
Abstract
We study transport phenomena involving chemically reactive species, modeled by advection--diffusion--reaction systems coupled with flow fields governed by Darcy's law. Both the velocity field and the species concentrations are discretized using the Virtual Element Method, while time integration is performed through a discontinuous Galerkin scheme. This work represents a preliminary study, in which we introduce some simplifications of the full model. In particular, we assume a concentration-independent viscosity in the Darcy problem, constant diffusion tensors in the advection--diffusion--reaction systems, and first-order reaction networks with liquid-phase degradation. We derive an abstract error estimate by means of a technique that combines Gauss--Radau interpolation with numerical integration. The theoretical results are supported by numerical experiments that exhibit arbitrary-order accuracy in both space and time.