Volume 26, Issue 6
Variable Step-Size Implicit-Explicit Linear Multistep Methods for Time-Dependent Partial Differential Equations

Dong Wang & Steven J. Ruuth

DOI:

J. Comp. Math., 26 (2008), pp. 838-855.

Published online: 2008-12

Preview Full PDF 110 814
Export citation
  • Abstract

Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-p, p-step VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers' equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.

  • Keywords

Implicit-explicit (IMEX) linear multistep methods Variable step-size Zero-stability Burgers' equation

  • AMS Subject Headings

65L06 65M06 65M20.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{JCM-26-838, author = {}, title = {Variable Step-Size Implicit-Explicit Linear Multistep Methods for Time-Dependent Partial Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {6}, pages = {838--855}, abstract = { Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-p, p-step VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers' equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8663.html} }
TY - JOUR T1 - Variable Step-Size Implicit-Explicit Linear Multistep Methods for Time-Dependent Partial Differential Equations JO - Journal of Computational Mathematics VL - 6 SP - 838 EP - 855 PY - 2008 DA - 2008/12 SN - 26 DO - http://dor.org/ UR - https://global-sci.org/intro/jcm/8663.html KW - Implicit-explicit (IMEX) linear multistep methods KW - Variable step-size KW - Zero-stability KW - Burgers' equation AB - Implicit-explicit (IMEX) linear multistep methods are popular techniques for solving partial differential equations (PDEs) with terms of different types. While fixed time-step versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit (VSIMEX) linear multistep methods for time-dependent PDEs. Families of order-p, p-step VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers' equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.
Dong Wang & Steven J. Ruuth. (2019). Variable Step-Size Implicit-Explicit Linear Multistep Methods for Time-Dependent Partial Differential Equations. Journal of Computational Mathematics. 26 (6). 838-855. doi:
Copy to clipboard
The citation has been copied to your clipboard