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Multilevel augmentation methods for solving the sine-Gordon equation. (English) Zbl 1213.35035

Authors’ abstract: We develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.

MSC:

35A35 Theoretical approximation in context of PDEs
35L71 Second-order semilinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
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