.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 30 000. 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

Arnoldi and Crank-Nicolson methods for integration in time of the transport equation

GAMBOLATI, GIUSEPPE;PINI, GIORGIO
2001

Abstract

.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 30 000. 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
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/2459194
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