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

作者：Schaefer Jonas; Huang Xuan; Kopecz Stefan^*; Birken Philipp; Gobbert Matthias K; Meister Andreas

来源：Numerical Methods for Partial Differential Equations, 2015, 31(1): 143-167.

DOI：10.1002/num.21897

• calcium waves
• Dirac delta distribution
• finite volume method
• matrix-free Newton-Krylov method
• parallel computing

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.

• 出版日期2015-1