Structure-Preserving Method For Maxwell's Equations

Since Maxwell established electromagnetism wave equations which described the evolutions of electromagnetic fields in space and time in 1837, Maxwell's equations have been developed for more than 100 years. They have been applied to a wide range of different physical situations. Due to the actual environment is complex, only some classical problems have exact solutions, we often describe characteristic of electromagnetic wave by numerical solutions.In 1980s, FengKang proposed symplectic algorithms, which were stable and effective in long time computation. Recently, Bridges and Marsden have extended the symplectic algorithms, and then proposed the concept of the multi-symplectic Hamilton system and the corresponding multisymplectic schemes. As the development of the theories, many researchers have computed famous equations with multisymplectic algorithms, such as nonlinear wave equation, nonlinear schrodinger equation, Korteweg-de Vries equation, Zakharov-Kuznetsov equation, Kadomtsev-Petviashvili equation. Numerical results indicate multisymplectic algorithm is useful.At present, there is no attempt of applying multisymplectic algorithm to Maxwell's equations. Motivation of the thesis is developing the multisymplectic theory and the multisymplectic algorithm of Maxwell's equations. We test whether the multisymplectic algorithm is effective. Based on the Bridges' multisymplectic form , we derive three multisymplectic schemes for Maxwell's equations.We derive a new multisymplectic scheme which is proved to be self-adjoint in time direction for Maxwell's equations. The multi-symplecticity of composition schemes based on the new self-adjoint scheme is also discussed. By the investigation of the multisymplectic Preissman scheme, we derive a new 102-point scheme which couples two time levels by eliminating the auxiliary variables for Maxwell's equations. Furthermore, the new derived 102-point scheme is proved to preserve the discrete local energy of the Maxwell's equations exactly. We also apply BEA for the multisymplectic Preissman scheme and get a modified system of equations which also have multisymplectic structure. According to the modified system of equations, we get the order of the scheme.For the self-adjoint scheme and the composition scheme respectively, two numerical examples are proposed to indicate that the derived multisymplectic schemes are effective when used to integrate the 2+1 dimensional Maxwell's equations. Numerical results can also show that the idea of composition is feasible. By numerical experiments for the multisymplectic Preissman scheme, we find that the Preissman scheme has higher order accuracy than the Yee's scheme in long time computation. Our numerical results can also indicate that the Preissman scheme keeps the discrete local energy and the wave form very well. Therefore, we may draw the conclusion that the multisymplectic schemes are powerful tools to integrate Maxwell's equations.