Semi-analytic time differencing methods for singularly perturbed initial value problems

Abstract

We implement our new semi-analytic time differencing methods, applied to singularly perturbed non-linear initial value problems. It is well-known that, concerning the singularly perturbed initial problem, a very stiff layer, called initial layer, appears when the perturbation parameter is small, and this stiff initial layer causes significant difficulties to approximate the solution. To improve numerical quality of the classical integrating factor (IF) methods and exponential time differencing (ETD) methods for stiff problems, we first derived the so-called correctors, which are analytic approximations of the stiff part of the solution. Then, by embedding these correctors into the IF and ETD methods, we build our new enriched schemes to improve the IF Runge-Kutta and ETD Runge-Kutta schemes. By performing numerical simulations, we verify that our new enriched schemes give much better approximations of solutions to the stiff problems, compared with the classical schemes without using the correctors.