Discrete Variational Methods For The Sine-Gordon Equation And Their Implementation

The symplectic algorithms for Hamiltonian systems have many advantages, such as, good stability, accuracy in long-time computation and so on. They have been applied to lots of fields in scientific computing. In recent years, the symplectic al-gorithms have been extended to the multi-symplectic algorithms for partial differ-ential equations. How to systematically construct multi-symplectic algorithms and smoothly apply them to some concrete differential equations have been hot topics in scientific computing.In this thesis, based on the discrete variational principle, we derive several new multi-symplectic algorithms for the sine-Gordon equation by discreting the corre-sponding Lagrangian functional. All the new algorithms can be proved to preserve the discrete multi-symplectic conservation law of the original equation. We use sev-eral different finite difference schemes to discrete the Lagrangian functional of the sine-Gordon equation and obtain a series of multi-symplectic algorithms and the cor-responding multi-symplectic conservation laws for the equation.Finally, we present some numerical results to illustrate the efficiency of the new algorithms and their advantages on controlling global errors of the numerical solutions and errors of the energy first integral.