## 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 $\begin{array}{cc}\phantom{\rule{6.0em}{0ex}}& u\text{'}\left(t\right)=F\left(u\left(t\right)\right),\phantom{\rule{1.00em}{0ex}}u\left({t}_{0}\right)={u}_{0}\hfill \end{array}$ 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 ${t}_{1},\dots ,{t}_{N}$ and compute for each of these times ${t}_{n}$ an approximation ${u}_{n}$ to the exact solution $u\left({t}_{n}\right)$ . Exponential integrators carry out these computations in an explicit way. A simple example is given by the exponential Rosenbrock-Euler method, which is a method of order 2: $\begin{array}{cc}\phantom{\rule{6.0em}{0ex}}& {u}_{n+1}={u}_{n}+{h}_{n}+{\varphi }_{1}\left({h}_{n}{J}_{n}\right)F\left({u}_{n}\right)\phantom{\rule{1.00em}{0ex}}\text{with}\phantom{\rule{1.00em}{0ex}}{\varphi }_{1}=\genfrac{}{}{0.1ex}{}{{\mathrm{e}}^{z}-1}{z}.\hfill \end{array}$ Here, ${J}_{n}$ denotes an appropriate approximation to the Jacobian matrix $F\text{'}\left({u}_{n}\right)$ . The implementation of this method requires the evaluation of the action of the matrix function on a vector which we perform by interpolation based on Leja points. Note that exponential integrators of arbitrarily high order exist. Their properties are studied in this research project.

© Numerical Analysis - University of Innsbruck 2015