Finite volume method, Dirac delta distribution, Matrix-free Newton-Krylov method, Calcium waves, Parallel computing, Naturvetenskap, Matematik, Natural Sciences, Mathematics

We present a parallel matrix-free implicit nite volume scheme for the solution of unsteady three-dimensional advection-diusion-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 ow in heart cells and show its parallel scaling.