Date of Award
Doctor of Philosophy (PhD)
Temporal discretization methods for evolutionary differential equations that factorize the resolvent into a product of easily computable operators have great numerical appeal. For instance, the alternating direction implicit (ADI) method of Peaceman-Rachford for 2-D parabolic problems greatly reduces the simulation time when compared with the Crank-Nicolson scheme. However, just like many other factorized approximation methods that exhibit numerical stability, the ADI method is known to satisfy only the Von Neumann stability condition, a necessary condition that is usually surmised as sufficient in practical cases as pointed out by Lax and Richtmyer. Intensive efforts have been directed to understand the Von Neumann condition, e.g. by John, Lax and Richtmyer, Lax, Lax and Wendroof, and Strang. Their way of investigation is to find conditions under which the Von Neumann condition becomes sufficient for stability. Recently, we found a factorized (FAC) temporal approximation method and a well-posed problem for which the FAC method is unstable but satisfies the Von Neumann stability condition. However, the method still exhibits excellent numerical stability even for large time step sizes. Thus, to better understand the Von Neumann condition, we investigate the relation between stability and convergence in directions not covered by the Lax equivalence theorem which equates the stability with convergence for all initial values under some uniform consistency condition. To do that, we extend the Trotter-Kato theorem and the Chernoff product formula to possibly unstable "spatial" and "temporal" approximations and indicate how our results can be used for some unstable factorized approximation methods.
Zhuang, Yu, "Classically Unstable Approximations for Linear Evolution Equations and Applications." (2000). LSU Historical Dissertations and Theses. 7311.