Arnoldi and Crank-Nicolson methods for integration in time of the transport equation
A comparison is made between the Arnoldi reduction method and the Crank-Nicolson method for the integration in time of the advection-diffusion equation. This equation is first discretized in space by the classic finite element (FE) approach, leading to an unsymmetric first-order differential system, which is then solved by the aforementioned methods. Arnoldi reduces the native FE equations to a much smaller set to be efficiently integrated in the Arnoldi vector space by the Crank-Nicolson scheme, with the solution recovered back by a standard Rayleigh-Ritz procedure. Crank-Nicolson implements a time marching scheme directly on the original first-order differential system. The computational performance of both methods is investigated in two- and three-dimensional sample problems with a size up 30000. The results show that in advection-dominated problems less then 100 Arnoldi vectors generally suffice to give results with a 10(-3)-10(-4) difference relative to the direct Crank-Nicolson solution. However, while the CPU time with the Crank-Nicolson starts from zero and increases linearly with the number of time steps used in the simulation, the Arnoldi requires a large initial cost to generate the Arnoldi vectors with subsequently much less expensive dynamics for the time integration. The break-even point is problem-dependent at a number of time steps which may be for some problems up to one order of magnitude larger than the number of Arnoldi vectors. A serious limitation of Arnoldi is the requirement of linearity and time independence of the flow field. It is concluded that Arnoldi can be cheaper than Crank-Nicolson in very few instances, i.e. when the solution is needed for a large number of time values, say several hundreds or even 1000, depending on the problem. Copyright (C) 2001 John Wiley & Sons, Ltd.
transport equation finite elements arnoldi method iterative and direct methods linear systems