IMEX variable step-size Runge-Kutta methods for parabolic integro-differential equations with nonsmooth initial data

Abstract

We develop a class of implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving parabolic integro-differential equations (PIDEs) with nonsmooth initial data, which describe several option pricing models in mathematical finance. Different from the usual IMEX RK methods, the proposed methods approximate the integral term explicitly by using an extrapolation operator based on the stage-values of RK methods, and we call them as IMEX stage-based interpolation RK (SBIRK) methods. It is shown that there exist arbitrarily high order IMEX SBIRK methods which are stable for abstract PIDEs under suitable time step restrictions. The consistency error and the global error bounds for this class of IMEX Runge-Kutta methods are derived for abstract PIDEs with nonsmooth initial data. The related higher time regularity analysis of the exact solution and stability estimates for IMEX SBIRK methods play key roles in deriving these error bounds. Numerical experiments for European options under jump-diffusion models and stochastic volatility model with jump verify and complement our theoretical results.