Abstract
An efficient and accurate Legendre dual-Petrov-Galerkin spectral element method for solving the Kawahara-type equations is proposed. The Sobolev bi-orthogonal basis functions are constructed, which reduce the number of nonzero elements in the matrix and improve the computational efficiency. The generalized stability of the fully discrete spectral element scheme is analyzed. For applications, we consider the Kawahara, KdV-Kawahara and modified Kawahara equations, and discuss numerically the motion of single solitary wave, conservation laws, the phenomena of wave generation and interaction of two and three solitons. Numerical results illustrate the effectiveness of the suggested approach.
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