Abstract
The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.
Highlights
IntroductionBecause the semidiscrete ODE system obtained from spatial discretization, such as method of lines, of the nonlinear parabolic partial differential equation is highly stiff, the choices of time integration methods are limited to implicit methods only
Let us consider the following parabolic partial differential equation: ut = Duxx + f (u, x, t), (x, t) ∈ (a, b) × (0, T], (1)with the initial condition:u (x, 0) = u0 (x), x ∈ [a, b], (2)where D is a positive constant describing the diffusion property and f(u, x, t) is a function representing the reaction term, which is nonlinear on u
An efficient fourth-order numerical algorithm that combines the Padeapproximation in space and fourth-order accurate Rosenbrock method in time is proposed in this paper
Summary
Because the semidiscrete ODE system obtained from spatial discretization, such as method of lines, of the nonlinear parabolic partial differential equation is highly stiff, the choices of time integration methods are limited to implicit methods only. A-stability of the time stepping method is not sufficient for highly stiff problem To overcome these difficulties, it is desirable to construct new algorithms with strong A-stability or L-stability that are free of solving nonlinear equations. The objective is to develop a strongly A-stable Rosenbrock method to solve the semidiscrete stiff ODEs resulting from compact high-order finite difference approximation of a semilinear parabolic partial differential equation. Several numerical examples are used to demonstrate the accuracy and efficiency of the new algorithm in Section 4, which is followed by conclusions and possible future work
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