Piecewise-linearized And Linearized θ-Methods For Nonlinear Differential Equations

Fully and partial piecewise-linearized methods and linearized θ-methods are used to solve the regular and singular nonlinear equations, and the stability of two systems methods for the, Prothern and Robinson problem are researched. It is shown that piecewise-linearized methods for the Prothern and Robinson problem are A-stable, but not S-stable;while the linearized θ-methodsare S-stable so far as 1/2≤θ≤1.And all the methods described areapplied to the numerical examples .linearized θ- methods and Rosenbrock methods are compared.Centered difference θ- methods, compact difference of four-order accuracy θ-methods, and a new three-level θ-methods for one dimensional reaction-diffusion equations are presented;and all kinds of partial linearized θ-methods, which deduced by different ways of approximating the Jacobian matrix are presented. The methods above are tested and compared by a system of reaction-diffusion equations.To the end, a kind of non-standard linearized θ-methods , from which we can get piecewise continuous and piecewise differentiable solutions for reaction-diffusion equations are presented.