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Wednesday 13 May 2015Local Exponential Methods: a domain decomposition approach to exponential time integration of PDEs

Luca Bonaventura -

Harrison 170 11:00-12:00

The application of exponential time integration methods (EM) to the time discretization of partial differential equations has been the focus of increasing attention over the last two decades. EM allow to eliminate almost entirely the time discretization error in the linear case and to reduce it substantially in many nonlinear cases.
EM possess stability properties that make them competitive with standard stiff solvers. Furthermore, when wave propagation problems are considered, they also allow to represent faithfully even the fastest linear waves, which are usually damped and distorted by conventional implicit and semi-implicit techniques. In this talk, results of an extensive comparison between IMEX and exponential methods will be briefly summarized.
An overlapping domain decomposition technique will then be proposed, that allows to replace the computation of a global exponential matrix by a number of independent local matrices. The obvious advantage of such a Local Exponential Method (LEM) is that each local problem can be solved independently in parallel, thus increasing the scalability of the resulting time discretization technique. Furthermore, if the number of degrees of freedom associated to each local domain is small enough, the local exponential matrices can be computed by Pade' approximation combined with scaling and squaring and can be actually stored in memory, thus bypassing the problems that result from having to compute the action of the exponential matrix rather than the exponential matrix itself. Numerical results obtained with preliminary implementations are presented that demonstrate the effectiveness of the proposed approach at least on some simple model problems.

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