Convergence Analysis of Pseudo-Transient Continuation

Driven cavity ows by eecient numerical techniques, J. An error estimate for the modiied newton method with applications to the solution of nonlinear two-point boundary value problems, Accurate and economical solution of the pressure head form of Richards' equation by the method of lines, Tech. Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code. and the conceptual framework of the theory, which was originally motivated by similar observations in the literature. There is a fairly long induction phase, in which the initial iterate is guided towards the Newton convergence domain by remaining close to the physical transient, with relatively small timesteps. There is a terminal phase which can be made as rapid as the capability of the linear solver permits (which varies from application to application), in which an iterate in the Newton convergence domain is polished. Connecting the two is a phase of moderate length during which the time step is built up towards the Newton limit of max , starting from a reasonably accurate iterate. The division between these phases is not always clear cut, though exogenous experience suggests that it becomes more so when the corrector of x 1 is iterated towards convergence on each time step. We plan to examine this region of parameter space in conjunction with an extension of the theory to mixed steady//tc systems (analogs of diierential-algebraic systems in the ODE context) in the future. update. Asymptotic convergence cannot be expected to be quadratic or superlinear, since we do not enforce n ! 0 in (3.5). However, linear convergence is steep, and our experience shows that overall execution time is increased if too many linear iterations are employed in order to enforce n ! 0 asymptotically. In the results shown in this section, the inner linear convergence tolerance was set at 10 ?2 for the defect correction part of the trajectory, and at 10 ?3 for the Newton part. The work was also limited to a maximum of 12 restart cycles of 20 Krylov vectors each. Examination of the pseudo-timestep history shows monotonic growth that is gradual through the defect correction phase (ending at n = 14), then more rapidly growing , and asymptotically at max (beginning at n = 20). Steps 21, 24, and 32 show momentary retreats from max in response to a reenement on the tc strategy that automatically cuts back the pseudo-timestep by a xed factor if a …