Volume 1, Issue 3
Fourth-Order Splitting Methods for Time-Dependant Differential Equations

Jürgen Geiser

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 321-339.

Published online: 2008-01

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  • 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.

  • Keywords

Partial differential equations, operator-splitting methods, evolution equations, ADI methods, LOD methods, stability analysis, higher-order methods.

  • AMS Subject Headings

35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-321, author = {}, title = {Fourth-Order Splitting Methods for Time-Dependant Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {3}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/6054.html} }
TY - JOUR T1 - Fourth-Order Splitting Methods for Time-Dependant Differential Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 321 EP - 339 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/6054.html KW - Partial differential equations, operator-splitting methods, evolution equations, ADI methods, LOD methods, stability analysis, higher-order methods. AB -

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.

Jürgen Geiser. (2020). Fourth-Order Splitting Methods for Time-Dependant Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 1 (3). 321-339. doi:
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