Runge-Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge-Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection-diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-stable Runge-Kutta methods are then very good candidates for time integration, in particular diagonally implicit one

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

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

Convergence and component splitting for the Crank-Nicolson--Leap-Frog integration method

A new convergence condition is derived for the Crank-Nicolson--Leap-Frog integration scheme. The convergence condition guarantees second-order temporal convergence uniformly in the spatial grid size for a wide class of implicit-explicit splittings. This is illustrated by successfully applying component splitting to first-order wave equations resulting in such second-order temporal convergence. Component splitting achieves that only on part of the space domain Crank-Nicolson needs to be used. This reduces implicit solution costs when for Leap-Frog the step size is severely limited by stability only on part of the space domain, for example due to spatial coefficients of a strongly varying magnitude or locally refined space grids

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

On the shift parameter in the backward beam method for parabolic problems for preceding times

AbstractWe consider the backward beam method of Buzbee & Carasso [Math. Comp. 27, 237–267. 1973] for the numerical computation of parabolic problems for preceding times. The performance of this method is strongly influenced by the choice of a spectral shift parameter. Using logarithmic convexity arguments Buzbee & Carasso derived an expression for the optimal value for linear problems. The main concern of this paper is to illustrate that this expression can also be found and explained via the numerical stability analysis of the forward and backward recurrence involved

On time staggering for wave equations

Grid staggering for wave equations is a validated approach for many applications, as it generally enhances stability and accuracy. This paper is about time staggering. Our aim is to assess a fourth-order, explicit, time-staggered integration method from the literature, through a comparison with two alternative fourth-order, explicit methods. These are the classical Runge-Kutta method and a symmetric-composition method derived from symplectic Euler