Alexander Ostermann

Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems

The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction.

RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE

We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the confolution kernal via a discrete operational calculus. 23 refs., 2 tabs.

Exponential Rosenbrock-Type Methods

We introduce a new class of exponential integrators for the numerical integration of large-scale systems of stiff differential equations. These so-called Rosenbrock-type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the Jacobian. In contrast to standard integrators, the methods are fully explicit and do not require the numerical solution of linear systems. We analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in Banach spaces. In particular, we derive an abstract stability and convergence result for variable step sizes. This analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods. We present a third-order method with two stages, and a fourth-order method with three stages, respectively. The application of the required matrix functions to vectors are computed by Krylov subspace approximations. We briefly discuss these implementation issues, and we give numerical examples that demonstrate the efficiency of the new integrators.

Multi-grid dynamic iteration for parabolic equations

We study the method which is obtained when a multi-grid method (in space) is first applied directly to a parabolic intitial-boundary value problem, and discretization in time is done only afterwards. This approach is expected to be well-suited to parallel computation. Further, time marching can be done using different time step-sizes in different parts of the spatial domain.

Runge-Kutta approximation of quasi-linear parabolic equations

We study the convergence properties of implicit Runge-Kutta meth- ods applied to time discretization of parabolic equations with time- or solution- dependent operator. Error bounds are derived in the energy norm. The con- vergence analysis uses two different approaches. The first, technically simpler approach relies on energy estimates and requires algebraic stability of the Runge- Kutta method. The second one is based on estimates for linear time-invariant equations and uses Fourier and perturbation techniques. It applies to A(9)- stable Runge-Kutta methods and yields the precise temporal order of conver- gence. This order is noninteger in general and depends on the type of boundary conditions.

Exponential splitting for unbounded operators

We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more then two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrodinger equation with an external magnetic field or with an unbounded potential.

Runge-Kutta methods for partial differential equations and fractional orders of convergence

We apply Runge-Kutta methods to linear partial differential equations of the form u¡(x, t) =5?(x, d)u(x, t)+f(x, t). Under appropriate assumptions on the eigenvalues of the operator 5C and the (generalized) Fourier coefficients of /, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the //-norm used to estimate the global error. The analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.

A minimisation approach for computing the ground state of Gross-Pitaevskii systems

In this paper, we present a minimisation method for computing the ground state of systems of coupled Gross-Pitaevskii equations. Our approach relies on a spectral decomposition of the solution into Hermite basis functions. Inserting the spectral representation into the energy functional yields a constrained nonlinear minimisation problem for the coefficients. For its numerical solution, we employ a Newton-like method with an approximate line-search strategy. We analyse this method and prove global convergence. Appropriate starting values for the minimisation process are determined by a standard continuation strategy. Numerical examples with two- and three-component two-dimensional condensates are included. These experiments demonstrate the reliability of our method and nicely illustrate the effect of phase segregation.

Explicit exponential Runge-Kutta methods of high order for parabolic problems

Exponential Runge-Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge-Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method. For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This allows us to obtain the conditions for a method of order 5 in an elegant way. After stating the conditions, we first show that there does not exist an explicit exponential Runge-Kutta method of order 5 with less than or equal to 6 stages. Then, we construct a fifth-order method with 8 stages and prove its convergence for semilinear parabolic problems. Finally, a numerical example is given that illustrates our convergence bound.

Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour

Abstract We derive error bounds for Runge-Kutta time discretizations of semilinear parabolic equations with nonsmooth initial data. The framework includes reaction-diffusion equations and the incompressible Navier-Stokes equations. Nonsmooth-data error bounds of the type given here are needed in the study of the long-time behaviour of numerical discretizations. As an illustration, we use these low-order error bounds in proving high-order convergence of invariant closed curves of a Runge-Kutta method to periodic orbits of the parabolic problem.