• 대한수학회논문집
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• 제13권4호
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• pp.889-901
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• 1998

Numerical approximations by pseudospectral method are obtained for the damped Boussinesq equation which is a modification of the good Boussinesq equation. The consistency and stability of the method are obtained using the extended Lax-Richtmyer equivalence theorem, which imply the convergence of the method. We obtain error estimates of O(h$^{s}$ ＋ k$^2$) for a fully discrete pseudospectral method.

• 대한조선학회논문집
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• 제59권2호
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• pp.64-71
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• 2022

In this paper, we simulate the collision of a solitary wave on a vertical wall in a uniform water channel and investigate the effect of damping on the amplitude attenuation. In order to take into account the damping effect, we introduce the Stokes damping whose dissipation is dependent on the velocity of wave motion on the surface of a thin layer of oil. That is, we use the improved Boussinesq equation with Stokes damping to describe the damped wave motion. Our work mainly focuses on the amplitude attenuation of a propagating solitary wave, which may depend on the Stokes damping together with the initial position and initial amplitude of the wave. We utilize the method of images and a powerful numerical tool (functional iteration method) for solving the improved Boussinesq equation, yielding an effective numerical simulation. This enables us to find the amplitudes of the incident wave and reflected one, whose ratio is a measure of the (wave) amplitude attenuation. Accordingly, we have shown that the reflection of a solitary wave by a vertical wall is dependent on not only the initial amplitude and position of a solitary but the Stokes damping.