ABSTRACT
In this paper we present a refinement to the single-step numerical methods of solving ordinary differential equations (ODEs). We propose an adaptive algorithm for generating the step length between successive numerical approximations to the solution of an ODE. The algorithm is designed such that more accurate approximations to the solution are made when the solution gradient is high, whilst accuracy is dropped in favour of economising computation effort when the gradient is low. It is demonstrated how this adaptive algorithm can be incorporated into the forward Euler method as well as all other single-step methods. An example is studied and simulations carried out comparing the single-step Euler, Heun and Runge-Kutta methods with their adaptive method counterparts. Analysis of the simulations is carried for each case. Through the simulations and analysis it is demonstrated that the proposed algorithm does indeed refine the existing single-step methods.
