Using unity approximations to construct nonstandard finite difference schemes for Bernoulli differential equations

Abstract

We explore the use of nonstandard finite difference (NSFD) methods in numerically approximating solutions to Bernoulli ordinary differential equations (BDEs) with constant coefficients. Specifically, we investigate the numerical performance of NSFD schemes that deploy unity approximations. A unity approximation can be created by introducing and replacing the number 1 1 with an approximation using non-local terms of the form 2 x k + 1 x k + 1 + x k \displaystyle \frac {2x_{k+1}}{x_{k+1}+x_k} where x k ≈ x k + 1 x_k \approx x_{k+1} on a discrete grid. Our NSFD schemes derived from unity approximations produce solutions that compare favorably with numerical solutions obtained using standard finite difference (SFD) schemes. The NSFD schemes presented here demonstrate their utility by possessing one or more of the following properties: enhanced accuracy, preserving positivity, and maintaining dynamic consistency (i.e., other essential qualities of the ODE pertaining to its domain and the asymptotic properties of the equilibria such as stability).