The Symplectic And Multi-symplectic Schemes For The Klein-Gordon Equation With Nonlinear Terms Of Any Degree

Since1984, Feng Kang, the founder of Chinese computational mathematics, putforward the symplectic geometric algorithm systemically which preserves symplecticstructure of the Hamiltonian system. In the1990s, Marsden, Bridges, Reichintroduced the concept of multi-symplectic algorithm. Rapid progress has been madein recent years. Many practical problems have been solved successfully. Besides, a lotof physical phenomena have been simulated.In the dissertation, we consider the algorithm of symplectic and multi-symplecticfor the nonlinear Klein-Gordon equation with nonlinear terms of any degree.According to the symplectic structure of the equation, we construct the symplecticschemes. Numerical experiments demonstrate the schemes can simulate the originalwaves in a long time. On the basis of multi-symplectic structure, we discuess themulti-symplectic conservation laws, construct the Preissmann schemes andmulti-symplectic Fourier pseudo-spectral schemes, and demonstrate the schemessatisfy the discrete multi-symplectic conservation law, the discrete multi-symplecticenergy conservation law and the discrete multi-symplectic momentum law. Thestability of the linear parts of Preissmann schemes is demonstrated. The correctnessof theoretical analysis is demonstrated by numerical experiment in the end.