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Method of lines transpose: An implicit solution to the wave equation


Authors: Matthew Causley, Andrew Christlieb, Benjamin Ong and Lee Van Groningen
Journal: Math. Comp. 83 (2014), 2763-2786
MSC (2010): Primary 65N12, 65N40, 35L05
DOI: https://doi.org/10.1090/S0025-5718-2014-02834-2
Published electronically: April 23, 2014
MathSciNet review: 3246808
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Abstract: We present a new method for solving the wave equation implicitly in one spatial dimension. Our approach is to discretize the wave equation in time, following the method of lines transpose, sometimes referred to as the transverse method of lines, or Rothe's method. We then solve the resulting system of partial differential equations using boundary integral methods.

Our algorithm extends to higher spatial dimensions using an alternating direction implicit (ADI) framework. Thus we develop a boundary integral solution that is competitive with explicit finite difference methods, both in terms of accuracy and speed. However, it provides more flexibility in the treatment of source functions and complex boundaries.

We provide the analytical details of our one-dimensional method herein, along with a proof of the convergence of our schemes in free space and on a bounded domain. We find that the method is unconditionally stable and achieves second order accuracy. Upon examining the discretization error, we derive a novel optimal quadrature method, which can be viewed as a Lax-type correction.


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Additional Information

Matthew Causley
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: causleym@math.msu.edu

Andrew Christlieb
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: christlieb@math.msu.edu

Benjamin Ong
Affiliation: Institute for Cyber-Enabled Research, Michigan State University, East Lansing, Michigan 48824
Email: ongbw@msu.edu

Lee Van Groningen
Affiliation: Department of Mathematics, Anderson University, Anderson, Indiana 46012
Email: glvangroningen@anderson.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02834-2
Keywords: Method of lines transpose, tranverse method of lines, implicit methods, boundary integral methods, alternating direction implicit methods, ADI schemes
Received by editor(s): January 30, 2012
Received by editor(s) in revised form: November 6, 2012, and March 8, 2013
Published electronically: April 23, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society