IMEX Spectral Deferred Correction (SDC) solver for Dedalus

📜 This project focus on the implementation of an IMEX-SDC solver for the Dedalus framework that can be used to solve partial differential equations using spectral methods (Fourier based, not polynomial).

Spectral Deferred Correction were originally developed by Dutt and Greengard in order to provide generic time integration method with arbitrary high order of accuracy. They received a particular attention with the development of time-parallel methods, in particular with Parallel Deferred Correction or PFASST allowing time-parallelization across the time-steps, or Parallel Diagonal SDC allowing parallelization across the stages within a time-step. This code intend to investigate the accuracy and performance of those methods when used in combination with the spectral method of Dedalus.

The IMEX-SDC solver can be used to solve initial value problem of the form :

$$ \mathcal{M} \cdot \partial_t \mathcal{X} + \mathcal{L} \cdot \mathcal{X} = \mathcal{F}(\mathcal{X}, t) $$

where $\mathcal{L}$ is a linear term, solved implicitly with Dedalus internal solvers, and $\mathcal{F}(\mathcal{X}, t)$ a (non-)linear term solved explicitly. See Dedalus documentation for more details and how to set up all those terms.

Structure of the repository

• dedalus_sdc : main code folder
• scripts : example scripts using IMEX-SDC
• dedalus_install.md : instruction to install Dedalus

Acknowledgements

This project has received funding from the European High-Performance Computing Joint Undertaking (JU) under grant agreement No 955701 (Time-X). The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Belgium, France, Germany, and Switzerland. This project also received funding from the German Federal Ministry of Education and Research (BMBF) grant 16HPC048.

                 

Bibliography

[Dutt & Greengard, 2000] Dutt, A., Greengard, L., & Rokhlin, V. (2000). Spectral deferred correction methods for ordinary differential equations. BIT Numerical Mathematics, 40, 241-266.

[Guibert & Tromeur-Dervout, 2007]Guibert, D., & Tromeur-Dervout, D. (2007). Parallel deferred correction method for CFD problems. In Parallel Computational Fluid Dynamics 2006 (pp. 131-138). Elsevier Science BV.

[Emmett & Minion, 2012] Emmett, M., & Minion, M. (2012). Toward an efficient parallel in time method for partial differential equations. Communications in Applied Mathematics and Computational Science, 7(1), 105-132.

[Speck, 2018] Speck, R. (2018). Parallelizing spectral deferred corrections across the method. Computing and visualization in science, 19, 75-83.