Proceedings of the 2001 ACM/IEEE conference on Supercomputing
A Fractional Splitting Algorithm for Non-overlapping Domain Decomposition
ICCS '02 Proceedings of the International Conference on Computational Science-Part I
A Highly Parallel Algorithm for the Numerical Simulation of Unsteady Diffusion Processes
IPDPS '05 Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDPS'05) - Papers - Volume 01
Journal of Computational and Applied Mathematics
International Journal of Computer Mathematics
Parallel domain decomposition procedures of improved D-D type for parabolic problems
Journal of Computational and Applied Mathematics
A generalization of Peaceman-Rachford fractional step method
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
Contractivity of domain decomposition splitting methods for nonlinear parabolic problems
Journal of Computational and Applied Mathematics
Locally linearized fractional step methods for nonlinear parabolic problems
Journal of Computational and Applied Mathematics
Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
Error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems
Journal of Computational Physics
Hi-index | 0.01 |
We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank--Nicolson method with the proposed method.