An ExCALIBUR Multigrid Solver Toolbox for ExaHyPE

Multigrid (MG) algorithms are among the fastest linear algebra solvers available and are used within simulators for time-dependent partial differential equations to invert the elliptic operators arising from implicit time stepping or constraint equations. Multigrid solvers are available off-the-shelf as black box software components. However, it is not clear whether a mathematically optimal MG algorithm will be able to deliver optimal performance on exascale hardware, and some scientists might not be able to use such black-box software as they have to stick to an existing (MG) software landscape. Most importantly, the optimality of off-the-shelf MG for one elliptic problem does not imply that the same algorithm behaves excellently for time-dependent PDEs, i.e. if we simply cast a time-dependent problem into a series of elliptic snapshots.


In MGHyPE, we will develop novel multigrid (MG) ingredients and implementation techniques to integrate state-of-the-art algorithms for time-dependent partial differential equations (PDEs) with
state-of-the-art MG solvers. A bespoke MG, co-designed with hardware and the PDE solver workflow, can unleash the potential of exascale: Without outsourcing the linear algebra to black-box libraries and with the right ingredients, skills and techniques, we can optimise the whole simulation pipeline in a holistic way rather than tuning individual algorithmic phases in isolation. To demonstrate the potential of our ideas, we will extend ExaHyPE, a code for purely hyperbolic (wave) equations that was behind ExCALIBUR's DDWG ExaClaw, to support elliptic constraints and implicit time stepping. An outreach and knowledge transfer program ensures that innovations find their way into the RSE community, align with vendor roadmaps, and can be used by other ExCALIBUR projects.