The Structure-preserving Methods For Degasperis-Procesi Equation

The symplecticity and energy conservation law are two crucial properties of con-servative mechanical systems. They play a significant role in the analysis of the sys-tems. We call the methods the structure-preserving methods which can preserve one of the two properties. With rapid development of science and technology, more and more attention has been paid to structure-preserving algorithms, which has developed very rapid in the last two decades. Now, the structure-preserving methods are used in lots of fields such as molecular simulation, celestial orbit computing and so on.Degasperis-Procesi (DP) equation is a partial differential equation to simulate nonlinear waves in dispersion mediums. It has attracted more and more attraction because of its peakon solution. However due to the complexity of its Hamiltonian structure, the symplectic scheme of the equation has not been constructed.In this thesis, we first investigate the bi-Hamiltonian structures and the structure-preserving properties of the DP equation. A symplectic scheme and two new energy-preserving schemes are constructed based on one of the bi-Hamiltonian structures. We prove that these three schemes obey the energy conservation law or the symplec-tic conservation law. Moreover we prove they can preserve another Hamiltonian of the system. Then, we do some numerical experiments on the three schemes with com-parison to the widely-used Crank-Nicolson method. Numerical results show that the three new schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.