A Newton-Krylov solution to the porous medium equations in the agree code

In order to improve the convergence of the AGREE code for porous medium, a Newton-Krylov solver was developed for steady state problems. The current three-equation system was expanded and then coupled using Newton's Method. Theoretical behavior predicts second order convergence, while actual behavior was highly nonlinear. The discontinuous derivatives found in both closure and empirical relationships prevented true second order convergence. Agreement between the current solution and new Exact Newton solution was well below the convergence criteria. While convergence time did not dramatically decrease, the required number of outer iterations was reduced by approximately an order of magnitude. GMRES was also used to solve problem, where ILU without fill-in was used to precondition the iterative solver, and the performance was slightly slower than the direct solution. (authors)