Fast multilevel solvers for computational fluid dynamics
Motivation and Background
Computational Fluid Dynamics (CFD) is a method for solving science and engineering problems that feature liquid or gas flows. Weather and climate forecasting, aircraft design and nuclear power generation are prime examples. We could make great progress in these fields if we could develop more accurate, efficient and robust numerical methods for CFD. High-order accurate discretisation schemes including Discontinuous Galerkin (DG) and Summation-By-Parts (SBP) in particular have a lot of potential for improving the accuracy of CFD simulations.
This project focuses on the development of efficient high-order accurate solution algorithms. For high performance on modern supercomputers, solution algorithms must have a high computation-to-communication ratio and low memory storage requirements. Multigrid (MG) is highly suitable in this context. MG incorporates 2 parts: a coarse-grid representation and a `smoother' or iterative method that converges the error on a particular grid. It has been very successful as a fast solver for steady-state CFD problems discretised by second-order methods. Improving the performance of MG for time-dependent and high-order accurate problems is the goal of the collaboration with researchers at Lund and Stanford.
Hierarchical algorithms including the Fast Multipole Method (FMM) and H-matrices are a less-explored alternative to MG. Functionally, they are equivalent: the full-resolution problem is recursively broken down into lower-resolution problems. However, FMM is less demanding in terms of memory for very large-scale computations. The goal of this part of the project is to incorporate hierarchical methods into a computationally efficient, high-order accurate CFD solver.
July 2016: A fast direct solver for the 2D advection-diffusion equation has been developed (see ICOSAHOM proceedings article). It serves as a prototype for a more advanced solver based on hierarchical methods.
June 2016: Several iterative methods were compared as the MG smoother for time-dependent advection-diffusion. The best ones were k-step GMRES and Symmetric Gauss-Seidel.