Year: 2008
Numerical Mathematics: Theory, Methods and Applications, Vol. 1 (2008), Iss. 3 : pp. 321–339
Abstract
This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher-order time discretization and combine it with the ADI methods. We also discuss the discretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2008-NMTMA-6054
Numerical Mathematics: Theory, Methods and Applications, Vol. 1 (2008), Iss. 3 : pp. 321–339
Published online: 2008-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Partial differential equations operator-splitting methods evolution equations ADI methods LOD methods stability analysis higher-order methods.