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The solution of the sine-Gordon equation using the method of lines. (English) Zbl 1001.65512
Summary: For the semi-linear eigenvalue problem of the form \(Ax=\lambda F(x)\), where \(F\) is a nonlinear mapping, we present some methods for numerical solution. For this problem, we first describe a practical SOR method: in this method, the overrelaxation parameter automatically estimated is used instead of the optimum value, since the eigenvalue is not known a priori. We discuss the convergence of this Newton-like method. We then present a conjugate gradient (CG) method for the eigenvalue problem. We also discuss some preconditioning techniques for the present CG method. Finally, a comparison of the convergence rates for these methods is made with a numerical example.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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[1] DOI: 10.1016/0045-7825(87)90117-4 · Zbl 0624.76020
[2] Bratsos A., Numerical Solutions of Nonlinear Partial Differential Equations, Ph.D. (1993)
[3] DOI: 10.1080/00207169008803936 · Zbl 0732.65106
[4] DOI: 10.1016/0096-3003(89)90115-X · Zbl 0678.65046
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