A simple proposal for parallel computation over time of an evolutionary process with implicit time stepping

Abstract

Evolutionary processes arise in many areas of applied mathematics, however since the solution at any time depends on the solution at previous time steps, these types of problems are inherently difficult to parallelize. In this paper, we make a simple proposal of a parallel approach for the solution of evolutionary problems with implicit time step schemes. We derive and demonstrate our approach for both the linear diffusion equation and the convection-diffusion equation. Using an allat- once approach, we solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method.