Dynamical Analysis and Numerical Computation of Shallow Water Wave Propagation

The b-family equation, which contains a general family of shallow water wave equations with the different values of b, has shown the so-called peaked wave solutions with the cases when b=2 (Camassa-Holm equation) and b=3 (Degasperis-Procesi equation). To explore whether a special case when b=0 exists the stable peaked solution, based on the multi-symplectic form, the multi-symplectic Box scheme to construct a new implicit scheme is applied focusing on this case. The numerical experiments show that the constructed scheme has well structure-preserving property and good long time numerical stability. Furthermore, we can also find that there do not exist the stable propagation of peaked solution from the numerical results in the special case when b=0.