Research On Numerical Methods For Several Classes Of Ordinary And Parabolic Differential Equations

Partial differential equations with delay are widely proposed as models for real-world problems, such as population ecology, cell biology, control theory etc. Numerical analysis and simulation for the equations play an important role in actual application. This is true not only because the progress has been made in the mathematical understanding and theory of partial differential equations with delay, but also because they may provide a powerful tool to control some dynamical systems. In the present thesis, we will introduce or develop some numerical methods to effectively solve these equations.This thesis consists of five chapters. In chapter 1, the background and source of the problems are firstly presented. Next, we summarize mainly about the development of the numerical methods for partial differential equations. Based on these research, we focus on some improvement on some full discrete numerical schemes for solving the equations.In chapter 2, a local discontinuous Galerkin method is introduced for solving nonlinear reaction-diffusion dynamical systems with time delay. Global stability of the local discon-tinuous Galerkin method is derived with the help of Halanay's inequality. The stability result implies that the perturbations of the numerical solutions are controlled by the ini-tial perturbations from the system and the method. Moreover, we show that if polynomials of degree k are used, the methods are (k+1)th order accurate in space. Meanwhile, we prove that the local discontinuous Galerkin method has the ability to preserve dissipativity of the underlying systems. Some confirmations of these are illustrated using the schemes on several biologic models. These results indicate that the algorithm is a good candidate to efficiently solve such problems.In chapter 3, some discontinuous Galerkin methods are applied to solve delay differ-ential equations. We derive that discontinuous Galerkin methods lead to global and analo-gously asymptotical stability for delay differential equations. These are followed by some convergence analysis for a first-order linear delay differential equations. Numerical experi-ments confirm the methods'effectiveness and the theoretical results. In chapter 4, we propose a family of predictor-corrector schemes based on IMEX meth-ods. Iterative methods become dispensable when a transformation of the discrete approxi-mation is given. Comparing with the usual implicit formulas, such as implicit linear multi-step methods and implicit Runge-kutta methods, the derived schemes significantly reduce the computational cost. Moreover, the presented numerical schemes possess higher effi-ciency and larger stability regions than the previously mentioned IMEX algorithm. These excellent properties of the methods are confirmed by theoretical results and the numerical examples.In chapter 5, inspired by some implicit-explicit linear multistep schemes and additive Runge-kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Global convergence and error estimates of the new method are also maintained. The proposed iterative method is shown to be a good candidate to solve such problems effectively. The reason we save a considerable computational CPU time is that, we don't have to update the Jacobian matrix all the time, even on the case that we solve the resulted nonlinear problems derived at different time level. In the future, we hope to exploit the spitting idea to solve some important real-world problems in science and technology.