Method of lines

The method of lines (MOL, NMOL, NUMOL^[1]^[2]^[3]) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. By reducing a PDE to a single continuous dimension, the method of lines allows solutions to be computed via methods and software developed for the numerical integration of ordinary differential equations (ODEs) and differential-algebraic systems of equations (DAEs). Many integration routines have been developed over the years in many different programming languages, and some have been published as open source resources.^[4]

The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s.^[5] Many papers discussing the accuracy and stability of the method of lines for various types of partial differential equations have appeared since.^[6]^[7]

^ Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press. ISBN 0-12-624130-9.
^ Hamdi, S.; W. E. Schiesser; G. W. Griffiths (2007), "Method of lines", Scholarpedia, 2 (7): 2859, Bibcode:2007SchpJ...2.2859H, doi:10.4249/scholarpedia.2859
^ Schiesser, W. E.; G. W. Griffiths (2009). A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press. ISBN 978-0-521-51986-1.
^ Lee, H. J.; W. E. Schiesser (2004). Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java, Maple and Matlab. CRC Press. ISBN 1-58488-423-1.
^ E. N. Sarmin; L. A. Chudov (1963), "On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method", USSR Computational Mathematics and Mathematical Physics, 3 (6): 1537–1543, doi:10.1016/0041-5553(63)90256-8
^ A. Zafarullah (1970), "Application of the Method of Lines to Parabolic Partial Differential Equations With Error Estimates", Journal of the Association for Computing Machinery, vol. 17, no. 2, pp. 294–302, doi:10.1145/321574.321583, S2CID 15114435
^ J. G. Verwer; J. M. Sanz-Serna (1984), "Convergence of method of lines approximations to partial differential equations", Computing, 33 (3–4): 297–313, doi:10.1007/bf02242274, S2CID 30171258

[Schiesser1991-1] Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press. ISBN 0-12-624130-9.

[Hamdi2007-2] Hamdi, S.; W. E. Schiesser; G. W. Griffiths (2007), "Method of lines", Scholarpedia, 2 (7): 2859, Bibcode:2007SchpJ...2.2859H, doi:10.4249/scholarpedia.2859

[Schiesser2009-3] Schiesser, W. E.; G. W. Griffiths (2009). A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press. ISBN 978-0-521-51986-1.

[Lee2004-4] Lee, H. J.; W. E. Schiesser (2004). Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java, Maple and Matlab. CRC Press. ISBN 1-58488-423-1.

[Sarmin1962-5] E. N. Sarmin; L. A. Chudov (1963), "On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method", USSR Computational Mathematics and Mathematical Physics, 3 (6): 1537–1543, doi:10.1016/0041-5553(63)90256-8

[Zafarullah1970-6] A. Zafarullah (1970), "Application of the Method of Lines to Parabolic Partial Differential Equations With Error Estimates", Journal of the Association for Computing Machinery, vol. 17, no. 2, pp. 294–302, doi:10.1145/321574.321583, S2CID 15114435

[Verwer1984-7] J. G. Verwer; J. M. Sanz-Serna (1984), "Convergence of method of lines approximations to partial differential equations", Computing, 33 (3–4): 297–313, doi:10.1007/bf02242274, S2CID 30171258

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