The Multi-symplectic Schemes For The Generalized Modified Boussinesq Equation

This thesis illustrates the multi-symplectic algorithms for infinite dimensionalHamiltonian systems. The multi-symplectic algorithms maintain not only the sum ofsymplectic forms on discrete space under appropriate boundary conditions, but alsosymplectic forms in local representations. Therefore, multi-symplectic algorithmshave remarkable partial properties, and describe the system more essentially.The paper focuses on the multi-symplectic schemes and Fourier pseudo-spectralmethod to generalized modified Boussinesq equation. Multi-symplectic equations forgeneralized modified Boussinesq equation are presented by canonical transformations,thus concluding some conservation laws. Using the Gauss-Legendre Runge-Kuttamethod to disperse multi-symplectic equation, we summarize and demonstrate themulti-symplectic schemes for the generalized modified Boussinesq equation. Theypreserve discretic multi-symplectic conservation law exactly. We eliminate middlevariables and amount to a single variable multi-symplectic schemes for mid-pointschemes, and call them Preissmann schemes. We perform a lot of experiments toprove that the schemes preserve the multi-symplectic geometry structure precisely bysatisfying the discrete multi-syplectic conservation law and can simulate the originalwaves in a very long period of time. Numerical experiments demonstrate theconsistency between the theoretical analysis and the numerical results.In the dissertation we present multi-symplectic Fourier pseudo-spectral schemesfor the generalized modified Boussinesq equation, using the Fourier pseudo-spectralmethod and the mid-point method to disperse multi-symplectic equations in space andtime directions respectively. We also carry out numerous experiments to illustratetheir validity. Numerical results indicate that multi-symplectic Fourierpseudo-spectral schemes are correct.