High-Order Preconditioners for Modern Computing Architectures: Line-Based Discontinuous Galerkin Method on GPUs
The goals of our project are to be able to overcome issues that plague computing in modern architecture such as limited memory bandwidth, badly conditioned Jacobians, and restrictive time steps. We begin by describing the line-based discontinuous Galerkin (line DG) method for a scalar two-dimensional problem. Line DG has been shown to provide a significant increase in sparsity for Jacobian matrix, especially in the diagonal block elements. In addition to this, the implicit time integration scheme helps alleviate the time step restriction but requires solving a (potentially badly conditioned) linear system. We can precondition the linear system to facilitate the solution. Since we want to solve the system on the GPU, the increase in sparsity from the line DG method is helpful to alleviate the memory issue. We successfully implemented a computation of the Jacobian on the CPU and are currently working on the GPU implementation. Finally, I discuss how this opportunity to intern at Lawrence Livermore National Laboratory through the National Science Foundation Mathematical Sciences Graduate Internship has affected my personal and professional growth.