Exponential Runge Kutta Methods Research Articles

A quadrature based method for evaluating exponential-type functions for exponential methods

We present a quadrature-based method to evaluate exponential-like operators required by different kinds of exponential integrators. The method approximates these operators by means of a quadrature formula that converges like O(e−cK), with K the number of quadrature nodes, and it is useful when solving parabolic equations. The approach allows also the evaluation of the associated scalar mappings. The method is based on numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm, including examples with a mass matrix and the application of the method inside the MATLAB package EXP4, an adaptive solver based on an exponential Runge–Kutta method.

Exponential Runge–Kutta methods for delay differential equations

This paper deals with convergence and stability of exponential Runge–Kutta methods of collocation type for delay differential equations. It is proved that these kinds of numerical methods converge at least with their stage order. Moreover, a sufficient condition of the numerical stability is provided. Finally, some numerical examples are presented to illustrate the main conclusions.

Exponential Runge–Kutta methods for the Schrödinger equation

We consider exponential Runge–Kutta methods of collocation type, and use them to solve linear and semi-linear Schrödinger Cauchy problems on the d-dimensional torus. We prove that in both cases (linear and non-linear) and with suitable assumptions, s-stage methods are of order s and we give sufficient conditions to achieve orders s + 1 and s + 2 . We show and explain the effects of resonant time steps that occur when solving linear Schrödinger problems on a finite time interval with such methods. This work is inspired by [M. Hochbruck, A. Ostermann, Exponential Runge–Kutta methods for parabolic problems, Appl. Numer. Math. 53 (2–4) (2005) 323-339], where exponential Runge–Kutta methods of collocation type are applied to parabolic Cauchy problems. We compare our results with those obtained for parabolic problems and provide numerical experiments for illustration.

Unconditional stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations

In this paper we define unconditional stability properties of exponential Kunge-Kutta metnoas wnen tney are apphed to semi-linear systems or ordinary differential equations characterized by a stiff linear part and a non-stiff non-linear part. These properties are related to a class of systems and to a specific norm. We give sufficient conditions in order that an explicit method satisfies such properties. On the basis of such conditions we analyze some of the popular methods.

Positivity of exponential Runge–Kutta methods

In this article, we study positivity properties of exponential Runge–Kutta methods for abstract evolution equations. Our problem class includes linear ordinary differential equations with a time-dependent inhomogeneity. We show that the order of a positive exponential Runge–Kutta method cannot exceed two. On the other hand there exist second-order methods that preserve positivity for linear problems. We give some examples for the latter.

Exponential Runge–Kutta methods for parabolic problems

The aim of this paper is to construct exponential Runge–Kutta methods of collocation type and to analyze their convergence properties for linear and semilinear parabolic problems. For the analysis, an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities is chosen. This framework includes interesting examples like reaction–diffusion equations. It is shown that the methods converge at least with their stage order, and that convergence of higher order (up to the classical order) occurs, if the problem has sufficient temporal and spatial smoothness. The latter, however, might require the source function to fulfil unnatural boundary conditions. Therefore, the classical order is not always obtained and an order reduction must be expected, in general.