ADI Methods For The Generalized Damped Sine-gordon Equation With Two-dimensional

In the modern research of qualitative theories of ordinary equations, dynamical systemhas been concerned by people. the limited dimensional dynamical systems has been studiedmore than thirty years, and also get more important achievements; compared with lim-ited dimensional dynamical system, the infinite dimensional dynamical system theory hassome new features, such as the soliton and spatial chaos phenomenon. Two-dimensionalgeneralized with damping term Sine-Gordon equation discussed in this article is a veryimportant model in the infinite dimensional dynamical system, which is a commonly usedmodel for the study of nonlinear wave equations in physics. For its vital significance inquantum physics, people have paid more and more attention to it with the gradual in-depthunderstanding.The first chapter briefly describes the relevant background knowledge and researchbackground of Sine-Gordon equation and alternating diference method.The second chapter introduces a number of related marks, concepts and lemma.The third chapter describes methods of ADI by separating operator. It will resolvethe nonlinear diferential operator into linearand nonlinear part. Using implicit scheme toapproach the linear part; display format to approach the nonlinear part. This method willtransform the multi-dimensional problem into one-dimensional problem, and have manyadvantages such as identical calculation structure in X and Y directions, three-diagonalmatrix form, easy to program and calculated. At last verify the theoretical analysis by numerical examples.The fourth chapter describes the method of constructing alternating direction finitediferences schemes by variable substitution. The main idea is reduction, first using thevariable v instead of the first derivative u, then discrete first derivative of v in the timelayer, at last construct alternating direction diference format. The theoretical analysiswill prove that this format is convergent and stable; the final numerical calculation showsthe calculation results are consistent with the theory.