Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models

• Published in: Advanced modeling and simulation in engineering sciences 2016-12, Vol.3 (1), Article 8
• Main Author: de Vuyst, Florian
• Format: Article
• Language: English
• Subjects: Classical and Continuum Physics; Computational Science and Engineering; Computer Science; Engineering; Model order reduction: POD; Modeling and Simulation; PGD and reduced bases; Research Article; Theoretical and Applied Mechanics
• ISSN: 2213-7467; 2213-7467
• DOI: 10.1186/s40323-016-0063-y

In this paper, we introduce and comment some recent efficient solvers for time dependent partial differential or ordinary differential problems, considering both linear and nonlinear cases. Here “efficient” may have different meanings, for instance a computational complexity which is better than standard time advance schemes as well as a strong parallel efficiency, especially parallel-in-time capability. More than a review, we will try to show the close links between the different approaches and set up a general framework that will allow us to combine the different approaches for more efficiency. As a complementary aspect, we will also discuss ways to include reduced-order models and fast approximate exponential integrators as fast global solvers. For developments and discussion, we will mainly focus on the heat equation, in both linear and nonlinear form.