An exponential Integrator for finite volume discretization of nonlinear parabolic differential equation

TL;DR

This paper introduces an exponential integrator method combining finite volume discretization and exponential time differencing for solving nonlinear second-order semi-linear parabolic PDEs, with applications in geo-engineering.

Contribution

The paper develops a novel numerical scheme that integrates finite volume methods with exponential time differencing for improved accuracy in nonlinear parabolic PDEs.

Findings

Established $L^{2}$ stability estimates for the scheme

Validated theoretical results with numerical simulations

Applicable to transport problems in porous media

Abstract

We consider the numerical approximation of a general second order semi--linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media which is fundamental in many geo-engineering applications, including oil and gas recovery from subsurface. Using the finite volume with two-point flux approximation on regular mesh combined with exponential time differencing of order one (ETD1) for temporal discretization, we derive the $L^{2}$ estimate under the assumption that the non linear term is locally Lipschitz. Numerical simulations to sustain the theoretical results are provided.