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A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources
Author(s) -
Schäfer Jonas,
Huang Xuan,
Kopecz Stefan,
Birken Philipp,
Gobbert Matthias K.,
Meister Andreas
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21897
Subject(s) - finite volume method , mathematics , advection , convergence (economics) , matrix (chemical analysis) , partial differential equation , nonlinear system , newton's method , mathematical analysis , mechanics , physics , quantum mechanics , economics , composite material , thermodynamics , economic growth , materials science
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 nonsmooth 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. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 143–167, 2015