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A finite element method for extended KdV equations

Karczewska A., Rozmej P., Szczeciński M., Boguniewicz B.

International Journal of Applied Mathematics and Computer Science

2016

Vol. 26, no. 3

555--567

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

Finite Element Method for Stochastic Extended KdV Equations

Karczewska A., Szczeciński M., Rozmej P., Boguniewicz B.

Computational Methods in Science and Technology

2016

Vol. 22, No. 1

19--29

The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.