On the convergence of Lawson methods for semilinear stiff problems

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schr\"odinger equation are included

A biconjugate gradient type algorithm on massively parallel architectures

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine

Error analysis of second-order locally implicit and local time-stepping methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong CFL condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time

Exponential integrators for quasilinear wave-type equations

In this paper we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. The analytical framework is an extension of the classical Kato framework and covers quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In contrast to earlier works, we do not assume regularity of the solution but only on the data. From this we deduce a well-posedness result upon which we base our error analysis. We include numerical examples to confirm our theoretical findings

On leap-frog-Chebyshev schemes

This paper is dedicated to the improvement of the efficiency of the leap-frog method for second order differential equations. In numerous situations the strict CFL condition of the leap-frog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed that exhibit a much weaker CFL condition than the leap-frog method. However, these methods do not even approximately conserve the energy of the exact solution which can result in a bad approximation quality. In this paper we propose two remedies to this drawback. For linear problems we show by using energy techniques that damping the Chebyshev polynomial leads to approximations which approximately preserve a discrete energy norm over arbitrary long times. Moreover, with a completely different approach based on generating functions, we propose to use special starting values that considerably improve the stability. We show that the new schemes arising from these modifications are of order two and can be modified to be of order four. These convergence results apply to semilinear problems. Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations

Finite element heterogeneous multiscale method for time-dependent Maxwell’s equation

We propose a Finite Element Heterogeneous Multiscale Method (FEHMM) for time dependent Maxwell’s equations in second-order formulation. This method can approximate the effective behavior of an electromagnetic wave traveling through a highly oscillatory material without the need to resolve the microscopic details of the material. To prove an a-priori error bound for the semi-discrete FE-HMM scheme, we need a new generalization of a Strang-type lemma for second-order hyperbolic equations. Finally, we present a numerical example that is in accordance with the theoretical results

Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems

This paper is concerned with the rigorous error analysis of a fully discrete scheme obtained by using a central fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford splitting scheme in time. We apply the scheme to a general class of wave-type problems and show that the resulting approximations as well as discrete derivatives thereof satisfy error bounds of the order of the polynomial degree used in the dG discretization and order two in time. In particular, the class of problems considered includes, e.g., the advection equation, the acoustic wave equation, and the Maxwell equations for which a very efficient implementation is possible via an alternating direction implicit (ADI) splitting