A Memory-Efficient Finite Volume Method for Advection-Diffusion-Reaction Systems with Non-Smooth Sources

Author/Creator ORCID

Date

2014-06-20

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Citation of Original Publication

Jonas Schafer, Xuan Huang, Stefan Kopecz, Philipp Birken, Matthias K. Gobbert, Andreas Meister, A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources, Numerical Methods for Partial Differential Equations, Volume 31, Issue1, January 2015, Pages 143-167, https://doi.org/10.1002/num.21897

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This is the peer reviewed version of the following article: Jonas Schafer, Xuan Huang, Stefan Kopecz, Philipp Birken, Matthias K. Gobbert, Andreas Meister, A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources, Numerical Methods for Partial Differential Equations, Volume 31, Issue1, January 2015, Pages 143-167, https://doi.org/10.1002/num.21897, which has been published in final form at [Link to final article using the DOI]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.

Abstract

We present a parallel matrix-free implicit finite volume scheme for the solution of unsteady three-dimensional advection-diffusion-reaction equations with smooth and Dirac-Delta source terms. The scheme is formally second order in space and a Newton-Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix-vector product required is hardcoded without any approximations, obtaining a matrix-free method that needs little storage and is well suited for parallel implementation. We describe the matrix-free implementation of the method in detail and give numerical evidence of its second order convergence in the presence of smooth source terms. For non-smooth source terms the convergence order drops to one half. Furthermore, we demonstrate the method’s applicability for the long time simulation of calcium flow in heart cells and show its parallel scaling.