Numerical Integration of Damped Maxwell Equations

We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semi-discrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the curl terms. Central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two, in particular in the presence of an explicitly or implicitly treated damping term which models conduction

Krylov projection methods for linear Hamiltonian systems

We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.Comment: 16 pages, 17 figure

An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations

An implicit-explicit (IMEX) method is combined with some so-called Richardson extrapolation (RiEx) methods for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The results are applied to a model for determining the overpotential in a Proton Exchange Membrane (PEM) fuel cell

Efficient time-stepping-free time integration of the Maxwell equations

Solution of the time dependent Maxwell equations is an important problem arising in many applications ranging from nanophotonics to geoscience and astronomy. The problem is far from trivial, and solutions typically exhibit complicated wave properties as well as damping behavior. Usually, special staggered time stepping schemes are used [Botchev,Verwer,2009]. Although their time step may be severely restricted by the CFL condition, performance of these schemes is hard to beat by modern implicit or exponential time integration schemes [Verwer,Botchev,2009]. We show that in some cases so-called time-stepping-free schemes provide a very efficient alternative to the standard schemes. These schemes employ the matrix exponential function and can be implemented by special block Krylov subspace techniques [Botchev,Grimm,Hochbruck,2013],[Botchev,2013]. Numerical examples demonstrating the efficiency of the proposed approach are presented, coming from the fields of nanophotonics and geoscience

Time stepping free numerical solution of linear differential equations: Krylov subspace versus waveform relaxation

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation method based on block Krylov subspaces. Second, we compare this new implementation against Krylov subspace methods combined with the shift and invert technique

Component splitting for semi-discrete Maxwell equations

A time-integration scheme for semi-discrete linear Maxwell equations is proposed. Special for this scheme is that it employs component splitting. The idea of component splitting is to advance the greater part of the components of the semi-discrete system explicitly in time and the remaining part implicitly. The aim is to avoid severe step size restrictions caused by grid-induced stiffness emanating from locally refined space grids. The proposed scheme is a blend of an existing second-order composition scheme which treats wave terms explicitly and the second-order implicit trapezoidal rule. The new blended scheme retains the composition property enabling higher-order composition

Splitting and composition methods for explicit time dependence in separable dynamical systems

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Composition methods, Maxwell's equations and source terms

This paper is devoted to high-order numerical time integration of first-order wave equation systems originating from spatial discretization of Maxwell’s equations. The focus lies on the accuracy of high-order composition in the presence of source functions. Source functions are known to generate order reduction and this is most severe for high-order methods. For two methods based on two well-known fourth-order symmetric compositions, convergence results are given assuming simultaneous space-time grid refinement. Herewith physical sources and source functions emanating from Dirichlet boundary conditions are distinguished. Amongst others it is shown that the reduction can cost two orders. On the other hand, when a certain perturbation of a source function is used, the reduction is generally diminished by one order. In that case reduction is absent for physical sources and for Dirichlet sources the order is equal to at least three under stable simultaneous space-time grid refinement

Composition methods, Maxwell's equations, and source terms

This paper is devoted to high-order numerical time integration of first-order wave equation systems originating from spatial discretization of Maxwell's equations. The focus lies on the accuracy of high-order composition in the presence of source functions. Source functions are known to generate order reduction, and this is most severe for high-order methods. For two methods based on two well-known fourth-order symmetric compositions, convergence results are given assuming simultaneous space-time grid refinement. Herewith physical sources and source functions emanating from Dirichlet boundary conditions are distinguished. Among other things it is shown that the reduction can cost two orders. On the other hand, when a certain perturbation of a source function is used, the reduction is generally diminished by one order. In that case, reduction is absent for physical sources and for Dirichlet sources the order is equal to at least three under stable simultaneous space-time grid refinement