Study Of Theory And Application For High-order Preserving Energy Dissipation Method

There is a large of class for partial differential equations (PDEs) in mathematics and physics, such as Allen-Cahn equations, diffusion equations, Cahn-Hilliard equations, Ginzburg-Landau equations and so on. The system which described by these PDEs has dissipation property, that is to say the energy of the system which defined by these PDEs will decrease as the time go on. In numerical simulation, the numer-ical scheme which designed to conserve the energy dissipation in the differential system, has significant influence to accurately simulate the behavior of the system described by differential equations. In 1984, Feng Kang academician, and his research team proposed symplectic algorithm for Hamilton system. Further in 1997, based on the symplectic algorithm, Bridges and Reich et al, provided multisymplec-tic algorithm for partial differential equations. symplectic and multisymplectic algorithm have widely applied to non-linear optics, quantum physics, and calculation of plasma physics and electromagnetic field equations, since its accurate long-time calculate ability and approximately energy conservation. Lately numerical methods which can conserve energy in differential functions raise attention in research of structure preserving algorithms, while few domestic research focuses on this field. In 1999 Quispel and McLachlan et al, proposed energy-diminishing second-order average vector field method. Furihata and Matsuo provided energy-diminishing discrete variational derivative methods for differential equa-tions. Recently, Quispel et al constructed energy-diminishing high-order AVF method for differential equations based on modified AVF method.For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases mono-tonically along solutions. Algebraically stable Runge-Kutta methods are shown to also reduce the energy in each step under a mild step size restriction. In particular, Radau IIA methods can combine energy monotonicity and damping in stiff gradient systems. Discrete-gradient methods and averaged vector field collocation methods are unconditionally energy-diminishing, but can't achieve damping for very stiff gradient systems.In the first Chaper, we discuss the enengy diminishing methods of the gradient system for uncon-strained and constrained condition. These methods include the implicit middle Euler scheme and the average vector field method, etc.In the second chapter, we propose high-order energy-diminishing method for Cahn-Hilliard equa-tions. Firstly we show that the energy-diminishing of Cahn-Hilliard equations theoretically, then con-struct high-order energy-diminishing scheme by applying Fourier pseudo-spectral method in space, and applying high-order average vector field method in time, finally use this scheme to make numerical sim-ulation under different initial conditions. The numerical results show that this scheme can simulate the solution behavior of Cahn-Hilliard equations well and keep energy-diminishing.In the third chapter, we construct high-order energy-diminishing scheme for 2-dimensional Allen-Cahn equations, and use this scheme to numerical simulate the Allen-Cahn equations accurately, and the result finally shows that 2-dimensional Allen-Cahn equations is energy-diminishing.