Implicit-explicit Multistep Methods For Nonlinear Stiff Delay Differential Equations

Delay differential equations（DDEs） are a class of basic mathematical models, and they can also come from spatial discretization of delay partial differential equations. Their right hand functions often can be splitting into stiff and non-stiff terms. In order to improve the computation efficiency and to satisfy the stability requirement, implicit-explicit（IMEX） methods are often applied to these stiff problems, and the stiff and non-stiff terms are discretized by using implicit methods and explicit methods, respectively. This can obviously reduce the computation cost.In this thesis, we study four problems, i.e., error analysis of IMEX one-leg methods for the nonlinear stiff initial-value problems; stability and convergence analysis of IMEX one-leg methods for stiff DDEs; error analysis of IMEX time discretization coupled with finite element methods for delayed predator-prey reactiondiffusion systems; exact and numerical stability analysis of reaction-diffusion equations with distributed delays. The numerical results show that the IMEX one-leg methods are very effective, in particular, it avoid solving large-scale systems of nonlinear algebraic equations for the semi-linear stiff ordinary or delay differential equations.Firstly, IMEX one-leg methods combining implicit one-leg methods with explicit one-leg methods are applied to stiff ordinary differential equations（ODEs）.The order conditions and the efficient algorithms are obtained. And for the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, the corresponding convergence results of these methods are obtained.Moreover, the IMEX one-leg methods are applied to stiff DDEs which can be split into the stiff and nonstiff parts. For overcoming the difficulties caused by the discontinuous to the derivative of the solution（so-called breaking points）, we prove that the IMEX one-leg methods are stable and convergent with order 2.Next, we consider two classes of IMEX time discretization schemes coupled with the finite element methods for solving the delayed predator-prey reactiondiffusion systems with various functional responses and the Dirichlet boundary conditions. Both one-leg type and linear multistep type IMEX schemes are considered in time discretization, finite element methods are used to discretize the space variables. For overcoming the difficulties caused by the discontinuous to the partial time derivative of the solution, it shown that the error estimates are O(?t²+ h^r+1).In the last, we consider the stability analysis of the exact and numerical solutions of the reaction-diffusion equations with distributed delays. Asymptotic stability and dissipativity of the equations with respect to perturbations of the initial condition are obtained. Moreover, the fully discrete approximation of the equations is given, and we prove that the one-leg θ-method preserves stability and dissipativity of the underlying equations.