Abstract
A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.
Highlights
Free surface flow is a fundamental problem in fluid dynamics
We develop a wavelet Galerkin method based on the anti-derivatives of Daubechies wavelets which can be applied to solve the forced Korteweg–de Vries (fKdV) model subjected to mixed boundary conditions
The wavelet Galerkin method based on the proposed basis for solving the fKdV model with mixed boundary conditions is developed in Sect
Summary
Free surface flow is a fundamental problem in fluid dynamics. It has been extensively studied from various points of view. Development is required of the application of the wavelet basis in the Galerkin method for solving a general type of boundary condition. We develop a wavelet Galerkin method based on the anti-derivatives of Daubechies wavelets (see [15, 19]) which can be applied to solve the fKdV model subjected to mixed boundary conditions. Gravity and surface tension forces are included This problem can be viewed as gravity-capillary wave flows due to an applied pressure distribution that is governed by the fKdV model as shown in equation (1); more details of derivations can be seen in [1]. 2. The wavelet Galerkin method based on the proposed basis for solving the fKdV model with mixed boundary conditions is developed in Sect.
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