Scalable Nonlinear Solvers

We are conducting research in the development of scalable parallel solution strategies for nonlinear differential equations discretized on three-dimensional meshes having upwards of one billion spatial zones. With the advent of more powerful supercomputers and highly effective solution methods, solving fully implicit formulations of many application models has become possible, whereas in the past, solutions to these formulations were thought to be impossible to attain. As one part of this project, we are looking at fully implicit formulations and solution methods for a diffusion approximation to radiation transport. In time dependent simulations, these implicit formulations allow for larger time steps than what current methods allow. Furthermore, these solution strategies respect the nonlinear couplings between problem parameters and iterate until these nonlinearities are converged. By focusing on the solution to the fully implicit formulation, we hope to improve over current methods in both allowing for larger time steps and providing more accuracy in the solution. Methods being investigated include Newton-Krylov algorithms preconditioned with multi-grid, nonlinear multi-grid methods, and nonlinear Schwarz methods.