Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

Implicit methods for finite-volume schemes on unstructured grids typically rely on a matrix-free implementation of GMRES and an explicit first-order accurate Jacobian for preconditioning. Globalization is typically achieved using a global timestep or a CFL based local timestep. We show that robustness of the globalization can be improved by supplementing the pseudo-timestepping method commonly used with a line search method. The number of timesteps required for convergence can be reduced by using a timestep that scales with the local residual. We also show that it is possible to form the high-order Jacobian explicitly at a reasonable computational cost. This is demonstrated for cases using both limited and unlimited reconstruction. This exact Jacobian can be used for preconditioning and directly in the GMRES method. The benefits of improvements in preconditioning and the elimination of residual evaluations in the inner iterations of the matrix-free GMRES method are substantial. Computational results focus on second- and fourth-order accurate schemes with some results for the third-order scheme. Overall computational cost for the matrix-explicit method is lower than the matrix-free method for all cases. The fourth-order matrix-explicit scheme is a factor of 1.6–3 faster than the matrix-free scheme while requiring about 50–100% more memory.