A symplectic high-order accurate numerical method for the sixth order Boussinesq equation

In this paper we consider the one-dimensional Boussinesq equation with a sixth order space derivative. The numerical method for its solution is constructed after representation of the equation as a generalized Hamiltonian system. Fourth order of approximation finite differences are used to replace t...

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Bibliographic Details
Main Authors: Vucheva, V., Kolkovska, N.
Format: Conference Proceeding
Language:English
Subjects:
Boussinesq equations
Hamiltonian functions
Numerical methods
Runge-Kutta method
Solitary waves
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Summary:In this paper we consider the one-dimensional Boussinesq equation with a sixth order space derivative. The numerical method for its solution is constructed after representation of the equation as a generalized Hamiltonian system. Fourth order of approximation finite differences are used to replace the space derivatives in the Hamiltonian and to obtain a semi-discrete finite-dimensional Hamiltonian system. For the time discretization we apply the symplectic partitioned Runge-Kutta method with 3-stage Lobatto IIIA and IIIB coefficients. The numerical solution preserves the discrete symplectic structure on every time level. Numerical experiments are provided for two specific problems with quadratic and cubic nonlinearities: for the propagation of a single solitary wave and for the interaction of two waves traveling toward each other. The numerical results show O(h4+τ4) order of convergence of the discrete solution to the exact one.
ISSN:0094-243X
1551-7616
DOI:10.1063/5.0180099