The Unsymmetric Lanczos Reduction method has been recently developed to reduce the size of a large-scale linear system which is the discretized form of a time-dependent partial di erential equation problem with a large physical domain. This has been applied to solve the time-dependent advection-dispersion equation discretized by ÿnite element or ÿnite di erence methods. However, the reduced system sometimes su ers time instability because of relocation of the approximate eigenvalues into the left half plane. This paper develops a method for stabilizing the reduced system while preserving the accuracy of the solution. The unstable eigenvalues are translated from the left half complex plane to the right half, leaving eigenvalues in right half plane unchanged. The results of numerical simulations of the synthetic and practical ÿeld contaminant transport problems show the e ciency and accuracy of this method. ? 1998 John Wiley & Sons, Ltd.
The Unsymmetric Lanczos Reduction (ULR) method is developed to solve the ÿnite-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 ÿeld problems illustrates the e ciency of the methods and substantial time savings over other existing solution methods. ? 1998 John Wiley & Sons, Ltd.
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