Some New Energy-preserving Algorithms For The Camassa-Holm Equation And The Benjannn-type Equations

The energy conservation is a crucial property of mechanical systems and plays an significant role in the study of properties of solutions. In some examples, stability of a numerical method is proved by directly using energy conservative property. Since energy is the most important first integral of many evolutional equations, energy-preserving algorithms have naturally attracted researchers and therefore have very rapid development.In this thesis, we study the global energy-preserving property of two non-local partial differential equations. By applying Fourier pseudospectral method, finite element method and wavelet collocation method spatially and average vector field method and discrete partial derivation method temporally, we propose several new energy-preserving algorithms for the Camassa-Holm equation in the symplectic form (in Chapter 2) and the Benjamin-type equations in a modified multi-symplectic Hamiltonain system (in Chapter 3). We also discuss the relationship between AVF method and DPD method and in the Chapter 3. The numerical experiments in Chapter 4 are presented to support the theoretical analysis in the previous two chapters.