Keywords: temporal integration, distributed algorithms, parallel algorithms, transformation methods..

Many engineering and applied science problems require the solutions of time dependent equations with nonlinear features. Usually a time-marching scheme, such as Euler's method, Runge-Kutta methods, multi-step methods, etc., with time step length restrictions is employed in any temporal integration procedure. Parallelisation of the time stepping becomes difficult and it is almost impossible to achieve a distributed/parallel algorithm that is able to yield a de-coupling of the original problem. On the other hand there are also many problems which require solution details not at each time step of the time-marching scheme, but only at a few crucial steps and the steady state. Therefore effort in finding fine details of the solutions using a temporal integration procedure with many intermediate time steps is considered being waste. This chapter presents the idea of inducing parallel properties into an otherwise sequential transient problem. A number of transformation methods and their relations to the possibility of providing concurrency in the solutions of partial differential equations are examined. Several examples related to the present approach are discussed, including convection diffusion problems and option pricing problems. In some cases numerical experiments are also included to support the concept. In other cases implementation issues are included to support the concept. Finally a two-level time domain parallel algorithm is presented with numerical tests from a nonlinear parabolic problem to demonstrate the viability of the method.

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