Multisymplectic Schemes And Error Analysis For The Conservation Laws

For conservative partial differential equations, such as various wave equations, Dirac equations, Schrodinger equations, coupled Schrodinger-Klein-Gordon system, generalized Zakharov system, it is often of interest to perform simulations over long times. It is generally believed that integration schemes should in some sense inherit the conservative properties of the continuous system in order to reproduce the correct numerical behavior of the exact solution. For finite dimensional Hamiltonian system, it is well known that symplectic integrators which conserve the symplectic structure of the Hamiltonian system have many favorable properties. The perhaps most obvious approach for generalizing these ideas to Hamiltonian PDEs is to discretize the Hamiltonian in space and then apply a symplectic integrator to the resulting finite dimensional Hamiltonain system.The limitations of such an approach is that the symplectic structure for the partial differential equation is global in space, such that the preservation of symplecticity is in some sense averaged in space and therefore it does not represent a sufficiently strong criterion for structure preservation. Bridges and Reich introduce the concept of a multi-symplectic formulation of a Hamiltonian partial differential equation. Along with such a formulation comes a local multisymplectic conservation law from which local conservation laws of energy and momentum can be derived.A multisymplectic integrator is a numerical scheme which satisfies a discrete version of the multisymplectic conservation law. Several multisymplectic schemes are known, in particular among the subclasses of finite difference schemes, finite volume schemes, finite element methods, and spectral discretization.It is showed that multisymplectic conservation does not imply preservation of other dynamical invariants of the original equation such as the local conservation laws of energy and momentum. However, extensive numerical tests have been performed with such schemes, showing that local conservation is well preserved over long times. Attempts have been made to explain this benign behavior through the use of backward error analysis and other technique, but as of now, only partial results are known for some selected Hamiltonian equations.The project of this thesis consists of three main components. The first component is to study the errors of discrete local conservation laws, when the multisymplectic Preiss-man scheme and the multisymplectic Fourier pseudospectral methods are used, and to present a straightforward method for obtaining the error formula of the discrete conser-vation laws of energy and momentum, which is applicable to many Hamiltonian PDEs. The second component is focused on novel multisymplectic method for the treatment of nonlinear solitary wave equations, which based on Sine collocation discretizations in space and Gauss-Legendre collocation discretizations in time. Thirdly, our obvious task is to conduct numerical tests with multisymplectic discretizations of some selected Hamil-tonian PDEs. This includes the numerical verification of some theoretical predications.