Exponentially fitted one-step methods for the numerical solution of the scalar Riccati equation

Abstract

Using stability analysis and information from the constant coefficient problem, we motivate an explicit exponentially fitted one-step method to approximate the solution of a scalar Riccati equation ϵ y′ = c( x) y 2 + d( x) y + e( x), 0 < x ⩽ x, y(0) = y 0, where ϵ > 0 is a small parameter and the coefficients c, d and e are assumed to be real valued and continuous. An explicit Euler-type scheme is presented which, when applied to the numerical integration of the continuous problem, give solutions satisfying a uniform (in ϵ) error estimate with order one (where suitable restrictions are imposed on the coefficients c, d and e together with the choice of y(0)). Using a counterexample, we show that, for a particular class of problems, the solutions of the fitted scheme do not converge uniformly (in ϵ) to the corresponding solutions of the continuous problems. Numerical results are presented which compare the fitted scheme with a number of implicit schemes when applied to the numerical integration of some sample problems.