Exponential Integrators
Exponential integrators constitute an efficient tool for the numerical solution of stiff and highly oscillatory problems. They rely on the variation-of-constants formula and require the evaluation of the action of certain matrix functions. Such computations can be carried out efficiently on massively parallel systems.
Introduction:
Many problems from science and engineering are modelled by partial differential equations. Their solutions describes the temporal evolution of the modelled processes. In most cases, however, the arising equations are too complex to be studied analytically. Consequently, their solutions have to be approximated by numerical methods. In this regard, topics like accuracy, numerical stability, efficiency and required computer memory are of paramount importance.
Exponential integrators constitute a very efficient class of numerical methods for the solution of high-dimensional stiff or highly oscillatory problems. Therefore, they are perfectly suited for carrying out complex simulations.
Exponential Integrators:
Consider the system of stiff or highly oscillatory differential equations
with a sufficiently smooth solution u . Such systems arise from spatial discretizations of parabolic or hyperbolic partial differential equations. For its numerical solution we consider discrete times
Here,