A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

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     

Efficient computation of non-isothermal highly viscous incompressible flow

We consider the numerical solution of the non–isothermal incompressible Navier–Stokes equations using a discrete projection method. The computation of velocity and temperature subproblems is carried out on different meshes chosen with respect to the physical behavior of these quantities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled FEM/DG approach for the simulation of cavitating micro foams

We consider difficulties which arise during our simulations of cavitating micro foams. The compatibility of velocity and transport, satisfaction of a maximum principle and generalization of an incompressible Navier-Stokes solver. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast numerical approach for the simulation of highly viscous non-isothermal non-Newtonian fluids

This paper deals with the efficient simulation of polymer melts, as an example of highly viscous non-isothermal non-Newtonian fluids. In flow fields of our interest, which are characterized by small Reynolds numbers and large Prandtl numbers, steep gradients occur in thin boundary layers of the temperature distribution, whereas the boundary layers associated with the velocity field possess a considerable different length scale. In order to benefit from these properties, we introduce a physically motivated multigrid approach by computing velocity and temperature fields on different meshes. This new development is achieved by the modification of a discrete projection method. Numerical experiments are presented which confirm that the method decreases the computational effort considerably, while preserving the numerical accuracy.     

On the existence of three-stage third-order modified Patankar–Runge–Kutta schemes

Numerical Algorithms - Modified Patankar–Runge–Kutta (MPRK) schemes are modifications of Runge–Kutta schemes, which were developed to guarantee unconditional positivity and...