Multi-symplectic Compact Difference Method For Two Kinds Of Shallow Water Equations With Singularity

Many physical processes can be represented by multi-symplectic Hamiltonian systems.Multi-symplectic methods can exactly preserve intrinsic multi-symplectic structures and some invariants of the original systems.On the basis of this scheme,combining compact finite difference method,we construct two kinds of multi-symplectic compact schemes to solve the classical shallow water equations with large singularity.A great many of numerical results show that this multi-symplectic compact difference method has distinct superiority and effective stability.The main works are as follows:1.The high-order multi-symplectic compact difference algorithm is proposed to solve the Camassa–Holm equation.The method combines the sixth-order or tenth-order compact difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization,which has corresponding discrete multi-symplectic conservation laws.2.Similarly,we construct the same multi-symplectic compact scheme for the twocomponent Camassa–Holm equation,which is extended from the Camassa–Holm equation by admitting a integrable multi-component generalization of density.The scheme is also precisely satisfy the discrete multi-symplectic conservation laws.3.Fourier analysis and numerical experiments reveal that the proposed algorithms can well simulate the singular problems in shallow water equations,and gathers various advantages,such as high-resolution,compact stencil and effectiveness in long-time numerical computation.