Eigenvalue translation method for stabilizing an unsymmetric Lanczos reduction process

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 nite element or nite 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 eld contaminant transport problems show the e ciency and accuracy of this method