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Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation. (English) Zbl 1457.65147
Summary: This paper proposes three fast and high accuracy numerical methods for solving a nonlinear partial differential equation (PDE) describing water waves and called the Boussinesq (Bq) equation. We numerically solve the Bq equation with fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original PDE with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) which will be solved with fourth-order time-stepping methods. After transforming the equation to a system of ODEs, the linear operator is not diagonal, but we can implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented. Also we investigate the conservation of mass for Bq equation.

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
65L05 Numerical methods for initial value problems
65T50 Numerical methods for discrete and fast Fourier transforms
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Software:
Matlab
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References:
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