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On the numerical solution of the sine-Gordon equation. II: Performance of numerical schemes. (English) Zbl 0874.65076
In the first part [J. Comput. Phys. 126, No. 2, 299-314, Art. No. 0139 (1996; Zbl 0866.65064)] the authors investigated the numerical behavior of a doubly-discrete, completely integrable discretization of the sine-Gordon equation. The phase space of this equation possesses tori and homoclinic structures, and it is important to determine how these structures are preserved by the numerical integration schemes.
In this second paper the nonlinear spectrum is used to compare the effectiveness of asymplectic (area preserving) and nonasymplectic scheme based integrators. It is studied, in particular, how the preservation of the nonlinear spectrum depends on the order of accuracy and the symplectic property of the numerical scheme. Extensive numerical experiments illustrate the theory.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65Y20 Complexity and performance of numerical algorithms
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