Toward an efficient parallel in time method for partial differential equations. (English) Zbl 1248.65106

Emmett, Matthew; Minion, Michael L.

Commun. Appl. Math. Comput. Sci. 7, No. 1, 105-132 (2012).

Summary: A new method for the parallelization of numerical methods for partial differential equations (PDEs) in the temporal direction is presented. The method is iterative with each iteration consisting of deferred correction sweeps performed alternately on fine and coarse space-time discretizations. The coarse grid problems are formulated using a space-time analog of the full approximation scheme popular in multigrid methods for nonlinear equations. The current approach is intended to provide an additional avenue for parallelization for PDE simulations that are already saturated in the spatial dimensions. Numerical results and timings on PDEs in one, two, and three space dimensions demonstrate the potential for the approach to provide efficient parallelization in the temporal direction.

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53	KdV equations (Korteweg-de Vries equations)
35Q35	PDEs in connection with fluid mechanics
65M55	Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65Y05	Parallel numerical computation

Keywords:

parallel computing; time parallel; deferred corrections; parareal; Burgers equation; Kuramoto-Silvashinsky equation; spectral method; multigrid method; numerical results