SDC-type methods based on Discontinuous Galerkin Time stepping

[tlunet]

🏷️ Proposal for a master / bachelor thesis

Collocation $Q$-generators currently implemented in qmat allow to generate various type of collocation methods, with different node distribution and quadrature types. As it produces dense $Q$ matrix, this is used as base method for Spectral Deferred Correction based time-stepping scheme Dutt, Greengard & Rokhlin 2000, that use a lower triangular preconditioner to solve the all-at-once problem, reducing the computational complexity to get to collocation accuracy (see tutorial on SDC).

Currently, the $Q$ matrices for collocation are built using a simple straightforward approach : consider the given (distinct) nodes, one simply build the Lagrange polynomial basis on those nodes, and integrate it on the intervals defined by the nodes to retrieve the $Q$ matrix (similar idea for the weights). However, this may not be the unique way to build a $Q$ matrix using a given set of nodes ...

🔍 Discontinuous Galerkin in Time (DGT) is another time-stepping scheme, based on polynomial approximation of the solution in time. While those were originally inspired from space discretization methods, a (quite ancient) paper has show that DGT is actually equivalent to a fully implicit Runge-Kutta method Lemma 1, Lesaint & Raviart 1974. It means that one could simply write the Butcher table of a DGT scheme, i.e its $Q$-coefficients.

The idea of this project is then as follow : if there is a generic way to generate DGT Butcher table, then it should become straightforward to generate all DGT $Q$-coefficients from a given set of nodes. Eventually, this may have a link to discontinuous collocation methods, such that everything should be formalized in a very generic way. Once it's done, then it would be again straightforward to investigate DGT-based SDC, in particular investigate time-parallel forms of SDC that would use DGT as basis method.

Proposed roadmap for this project :

1. Analyze DGT and how to generate their Butcher table (i.e $Q$-coefficients)
2. Implement new $Q$-generators in qmat allowing to generate $Q$-coefficients for DGT methods.
3. Investigate SDC-type time-stepping based on DGT and their potential time-parallel SDC sweepers.
4. Apply those newly developed (time-parallel) schemes on given ODE/PDE test problems (to be determined ...)

This can be subject to modifications before or during the project, and additional steps can eventually be added depending on the pace of the project.

Required :

• good knowledge in numerical method for scientific computing and ordinary differential equations
• good competences in Python programming
• intellectual curiosity and ability to independently search for information in the scientific literature

In case you are interested, please send a resume and brief motivation email to :

• [email protected] (@tlunet, TUHH, Hamburg, Germany)
• [email protected] (Cerfacs, Toulouse, France)