FEFLOW solvers. Part 2 – Non-symmetrical solvers: ORTHOMIN, GMRES
|January 6, 2012||Posted by karmadsen under blog, Groundwater elevation, Groundwater Modeling, Groundwater Modeling Software, Modeling Software|
The FEFLOW Groundwater Modeling program contains several numerical solvers, these are described in more detail below.1 The system constructed for solving for the distribution of groundwater elevations is a symmetric system, while the system constructed for solving transport equations is non-symmetrical.
Part 2 – Non-symmetrical solvers: ORTHOMIN, GMRES
The mathematics of the non-symmetrical solvers are similar to the conjugate-gradient method. The ORTHOMIN solver is so-named because it requires that search directions be defined by orthogonal vectors. This guarantees a convergences, but a large number of search directions and results will need to be stored during the run of the algorithm.2
ORTHOMIN starts with an initial guess and searches the problem space for the best solution in different search directions, q. If x0 is the initially guess, each subsequent x will be defined as follows:
xk+1 = xk + wkqk
w is a relaxation parameter. Each new q value must be orthogonal to all previous q, and it should also be the best search direction available at the present moment. Because our types of linear systems have a large number of dimensions, there can be a large number of orthogonal search directions.
Thus, the restarted ORTHOMIN solver was developed, which runs for a shorter time period, achieving an approximation of the solution, and then restarts itself from scratch, using the approximated value of the previous run as the starting value.2
If the part of the preconditioned matrix that is symmetric is also positive definite, the method will definitely converge. The matrix must approximate a symmetric, positive definite matrix, and if it is too far away from this idea, the method may fail.3
The restarted GMRES produces the same iterates as the restarted ORTHOMIN, but it is more robust solver that is less likely to breakdown.3
GMRES is similar to ORTHOMIN , except that at each step the algorithm locates the optimal search direction within a subspace of the matrix. This subspace, called the Krylov subspace, is based on the previous iterations explored by the algorithm.
The restarting process is the same as described in the ORTHOMIN section.
1. DHI-WASY Software. 2010. FEFLOW Classic: Finite Element Subsurface Flow & Transport Simulation System. User Manual. Berlin, Germany: DHI-WASY GmbH. Available at: http://www.feflow.info/uploads/media/feflow_user_manual_classic.pdf
2. Letniowski, F.W. 1989. An Overview of Preconditioned Iterative Methods for Sparse Matrix Equations. Research Report CS-89-26. Department of Applied Mathematics, University of Waterloo.http://www.cs.uwaterloo.ca/research/tr/1989/CS-89-26.pdf
3. Carey, G.F., Wang, K.C. & Joubert, W.D. 1989. Performance of Iterative Methods for Newtonian and Generalized Newtonian Flows. International Journal for Numerical Methods in Fluids. 9: 127-150. Available: http://www.cfdlab.ae.utexas.edu/labstaff/carey/GFC_Papers/Carey075.pdf
4. Pettersen, O. 2006. Basics of Reservoir Simulation with the Eclipse Reservoir Simulator. Lecture Notes. Department of Mathematics, University of Bergen. Available: http://www.uib.no/People/fciop/Downloads/MAT354/Notes/ResSimNotes.pdf Available at: http://www.feflow.info/uploads/media/feflow_user_manual_classic.pdf