Methods for overcoming breakdown problems in the Unsymmetric Lanczos Reduction method

The Unsymmetric Lanczos Reduction (ULR) method is developed to solve the nite-element-based solution to the contaminant transport problem. The method sometimes su ers from breakdown when at some step division by a pivot which is zero or near zero, causes numerical instability. In this paper, the Maximum-Pivot New- Start Vector method is developed to overcome such breakdowns by constructing a new starting vector with the possible maximum pivot. Some cases of instability cannot be remedied by this approach (pathological breakdowns) and the Switch method is developed to complete the solution by changing the algorithm to an Arnoldi reduction approach. Investigation of some two-dimensional examples and eld problems illustrates the e ciency of the methods and substantial time savings over other existing solution methods.