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Zbl 1012.65095
Douglas, Jim jun.; Kim, Seongjai
Improved accuracy for locally one-dimensional methods for parabolic equations. (English)
[J] Math. Models Methods Appl. Sci. 11, No.9, 1563-1579 (2001). ISSN 0218-2025

Summary: Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank-Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10\% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank-Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

MSC 2000:: *65M15 Error bounds (IVP of PDE)
80A20 Heat and mass transfer
65M06 Finite difference methods (IVP of PDE)
35K05 Heat equation
35L05 Wave equation

Keywords: alternating direction method; fractional step method; locally one-dimensional method; LOD; Crank-Nicolson procedure; splitting error; parabolic equations; backward Euler time-stepping scheme; second-order wave equations

Cited in: Zbl 1176.65090 Zbl 1135.65037 Zbl 1146.65068

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