Approximate factorization for time-dependent partial differential equations

AbstractThe first application of approximate factorization in the numerical solution of time-dependent partial differential equations (PDEs) can be traced back to the celebrated papers of Peaceman and Rachford and of Douglas of 1955. For linear problems, the Peaceman–Rachford–Douglas method can be derived from the Crank–Nicolson method by the approximate factorization of the system matrix in the linear system to be solved. This factorization is based on a splitting of the system matrix. In the numerical solution of time-dependent PDEs we often encounter linear systems whose system matrix has a complicated structure, but can be split into a sum of matrices with a simple structure. In such cases, it is attractive to replace the system matrix by an approximate factorization based on this splitting. This contribution surveys various possibilities for applying approximate factorization to PDEs and presents a number of new stability results for the resulting integration methods

Implicit-explicit time stepping with spatial discontinuous finite elements

In this paper a combination of discontinuous, piecewise linear, finite elements with implicit-explicit time stepping is considered for convection-reaction equations. Combined with low order quadrature rules, this leads to convenient schemes. We shall consider the effect of such low order quadrature rules on accuracy and stability for one-dimensional problems

Accuracy and stability of splitting with stabilizing corrections

This paper contains a convergence analysis for the method of Stabilizing Corrections, which is an internally consistent splitting scheme for initial-boundary value problems. To obtain more accuracy and a better treatment of explicit terms several extensions are regarded and analyzed. The relevance of the theoretical results is tested for convection-diffusion-reaction equations

Application of the operator splitting to the Maxwell equations with the source term

Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations $w'(t)=Aw(t)+f(t),$ $A\in\mathbb{R}^{n\times n}$ split into two subproblems $w_1'(t)=A_1w_1(t)+f_1(t)$ and $w_2'(t)=A_2w_2(t)+f_2(t),$ $A=A_1+A_2,$ $f=f_1+f_2.$ First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems $w_1'=Aw_1$ and $w_2'=f$ (with the split-off source term $f$). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error

A Scale-selective Multilevel Method for Long-Wave Linear Acoustics

A new method for the numerical integration of the equations for one-dimensional linear acoustics with large time steps is presented. While it is capable of computing the "slaved" dynamics of short-wave solution components induced by slow forcing, it eliminates freely propagating compressible short-wave modes, which are under-resolved in time. Scale-wise decomposition of the data based on geometric multigrid ideas enables a scale-dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete-dispersion relations of some standard second-order schemes are analyzed, and their response to high wave number low frequency source terms are discussed. The performance of the new method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high wave number source term

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