Marlis Hochbruck

Marlis Hochbruck, Alexander Ostermann

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 Schrödinger equation are included.

Tim Buchholz, Marlis Hochbruck

We propose and analyze a non-iterative domain decomposition integrator for the linear acoustic wave equation. The core idea is to combine an implicit Crank-Nicolson step on spatial subdomains with a local prediction step at the subdomain interfaces. This enables parallelization across space while advancing sequentially in time, without requiring iterations at each time step. The method is similar to the methods from Blum, Lisky and Rannacher (1992) or Dawson and Dupont (1992), which have been designed for parabolic problems. Our approach adapts them to the case of the wave equation in a fully discrete setting, using linear finite elements with mass lumping. Compared to explicit schemes, our method permits significantly larger time steps and retains high accuracy. We prove that the resulting method achieves second-order accuracy in time and global convergence of order $\mathcal{O}(h + τ^2)$ under a CFL-type condition, which depends on the overlap width between subdomains. We conclude with numerical experiments which confirm the theoretical results.

Marlis Hochbruck, Jonas Köhler, Malik Scheifinger

In this paper, we construct an efficient higher-order local time integration scheme for spatially discretized linear Friedrichs' systems. In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. The special combination of two methods like the leapfrog method on the coarse part of the mesh and the Crank-Nicolson method on the fine part as was done in Hochbruck, Sturm 2016 and Hochbruck, Köhler 2022 is not suitable for higher-order time integration. Therefore, we suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal 1991. The techniques developed here for linear problems also carry over to nonlinear problems, where linear systems of the same type arise within a Newton-type iteration. Motivated by the analysis of locally implicit methods by Hochbruck and Sturm 2016, we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory. The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient. We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauss-Legendre Runge--Kutta method.

Tim Buchholz, Marlis Hochbruck

We propose a novel non-iterative domain decomposition time integrator for acoustic wave equations using a discontinuous Galerkin discretization in space. It is based on a local Crank-Nicolson approximation combined with a suitable local prediction step in time. In contrast to earlier work using linear continuous finite elements with mass lumping, the proposed approach enables higher-order approximations and also heterogeneous material parameters in a natural way.