Accuracy and Performance Testing of Three-Dimensional Unsaturated Flow Finite Element Groundwater Programs on the Cray XT3 Using Analytical Solutions

Because Richards´ equation for describing unsaturated groundwater flow is highly nonlinear, it is challenging to test the accuracy and efficiency of parallel three-dimensional finite element groundwater programs used to model such activities as remediation of military sites and computing flows and pressures in levees broken by Hurricane Katrina. To aid in this testing, the author has derived steady-state and transient analytical solutions for unsaturated flow. With these solutions, both accuracy of the numerical solutions and the performance of the parallel versions of the computer programs may be easily tested. The other significant feature of these analytical solutions is that the amount of nonlinearity of the equation may be easily adjusted by a parameter a. This paper first presents the equations for the analytical solutions of steady-state and transient flow into a rectangular block of initially dry soil. It then uses the program FEMWATER as a benchmark to compute the numerical solutions corresponding to these analytical solutions. Accuracy results for various sizes of the mesh and numbers of processing elements (PEs) on the Cray XT3 are computed for different values of a. The number of nonlinear iterations for the steady-state problem is also reported. The result of the accuracy assessment is that as a is increased, the errors also increase significantly with even 512 PEs not being enough compute power to adequately solve the problem. Overall parallel performance and performance of the message passing interface calls for updating the ghost nodes and are also tested with results given. The result of this study is that scaling of a 1,000,000-element problem was good up to 256 PEs (64 PEs is the typical sweet spot for FEMWATER), and the performance of the ghost-node updating routine met or exceeded expectations