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

Causley, Matthew; Christlieb, Andrew; Ong, Benjamin; Van Groningen, Lee

doi:10.1090/S0025-5718-2014-02834-2

Method of lines transpose: An implicit solution to the wave equation
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by Matthew Causley, Andrew Christlieb, Benjamin Ong and Lee Van Groningen PDF

Math. Comp. 83 (2014), 2763-2786 Request permission

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.

References

Uri M. Ascher, Robert M. M. Mattheij, and Robert D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics, vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Corrected reprint of the 1988 original. MR 1351005, DOI 10.1137/1.9781611971231
H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys. 155 (1999), no. 2, 468–498. MR 1723309, DOI 10.1006/jcph.1999.6355
R. Coifman, V. Rokhlin, and S. Wandzura, The fast multipole method for the wave equation: A pedestrian prescription, IEEE Trans. Antennas and Propagation 35 (1993), no. 3, 7–12.
Zydrunas Gimbutas and Vladimir Rokhlin, A generalized fast multipole method for nonoscillatory kernels, SIAM J. Sci. Comput. 24 (2002), no. 3, 796–817. MR 1950512, DOI 10.1137/S1064827500381148
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348. MR 918448, DOI 10.1016/0021-9991(87)90140-9
Ulrich Hornung, A parabolic elliptic variational inequality, Manuscripta Math. 39 (1982), no. 2-3, 155–172. MR 675536, DOI 10.1007/BF01165783
S.D. Jackson and P.H. Muir, Theory and numerical simulation of $n$th-order cascaded Raman fiber lasers, JOSA B 18 (2001), no. 9, 1297–1306.
Jun Jia and Jingfang Huang, Krylov deferred correction accelerated method of lines transpose for parabolic problems, J. Comput. Phys. 227 (2008), no. 3, 1739–1753. MR 2450970, DOI 10.1016/j.jcp.2007.09.018
Leon Lapidus and George F. Pinder, Numerical solution of partial differential equations in science and engineering, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999. MR 1718868, DOI 10.1002/9781118032961
Peijun Li, Hans Johnston, and Robert Krasny, A Cartesian treecode for screened Coulomb interactions, J. Comput. Phys. 228 (2009), no. 10, 3858–3868. MR 2511077, DOI 10.1016/j.jcp.2009.02.022
Annamaria Mazzia and Francesca Mazzia, High-order transverse schemes for the numerical solution of PDEs, J. Comput. Appl. Math. 82 (1997), no. 1-2, 299–311. 7th ICCAM 96 Congress (Leuven). MR 1473548, DOI 10.1016/S0377-0427(97)00090-3
N. Muthiyalu and S. Usha, Eigenvalues of centrosymmetric matrices, Computing 48 (1992), no. 2, 213–218 (English, with German summary). MR 1169141, DOI 10.1007/BF02310535
Xiaodi Sun and Michael J. Ward, Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension, Stud. Appl. Math. 105 (2000), no. 3, 203–234. MR 1783547, DOI 10.1111/1467-9590.00149
Wen-Chyuan Yueh and Sui Sun Cheng, Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners, ANZIAM J. 49 (2008), no. 3, 361–387. MR 2440234, DOI 10.1017/S1446181108000102
A. Zafarullah, Application of the method of lines to parabolic partial differential equations with error estimates, J. Assoc. Comput. Mach. 17 (1970), 294–302. MR 280025, DOI 10.1145/321574.321583