Partially implicit schemes for the numerical solutions of some non-linear differential equations

摘要

As numerical methods are one of the main tools used to discover the non-linear behaviour characteristics of continuous dynamical systems, a study on the use of numerical schemes which delineate the true physics from numerical artefact is desirable. As a step in that direction, partially implicit rational schemes based on first-order methods and a second-order method are derived and analysed (from the point of view of dynamical systems), for a scalar, autonomous ordinary differential equation. The second-order method is developed by taking a linear combination of first-order methods. Though implicit in nature, the methods are applied explicitly. The approach adopted is extended to obtain numerical schemes for the solution of a class of reaction-diffusion partial differential equations. The methods are tested on differential equations with cubic and quadratic reaction terms. It is found that for certain sets of initial conditions, the first-order partially implicit schemes converge monotonically to the correct fixed points for all time steps, unlike the Euler method and the second-order method. In other words, for the same steady-state solution the associated domain of attraction of the discrete problem and its continuous counterpart are exactly the same. However for other initial conditions, it is found that the domains of attraction of the second-order method mimic more closely the domains of attraction of the continuous differential equations. The comparative study of the various explicit Runge-Kutta methods of orders 2–6 shows that it is the discretized form with its specific algebraic form recursive relation which are the source of spurious steady-state solutions and numerical chaos. From these methods, it is found however that the use in general of fifth-order Runge-Kutta methods is seen to be preferred to other Runge-Kutta methods, as they have an extended stability range. In particular, the fifth-order Lawson method which has a far superior range of stability than the other Runge-Kutta methods.