Abstract: An efficient and accurate scalar auxiliary variable (SAV) scheme for numerically solving nonlinear parabolic integro-differential equation (PIDE) is developed in this paper. The original equation is first transformed into an equivalent system, and the k-order backward differentiation formula (BDFk) and central difference formula are used to discretize the temporal and spatial derivatives, respectively. Different from the traditional discrete method that adopts full implicit or full explicit for the nonlinear integral terms, the proposed scheme is based on the SAV idea and can be treated semi-implicitly, taking into account both accuracy and effectiveness. Numerical results are presented to
demonstrate the high-order convergence (up to fourth-order) of the developed schemes and it is computationally efficient in long-time computations.

Key words: Parabolic integro-differential equation, Scalar auxiliary variable, Fredholm equation, High-order BDF scheme