Compact Implicit-Explicit BDF Method For Parabolic Partial Integro-Differential Equations

Partial integro-differential equations(PIDEs)are widely used in physics,biology,mate-rials science and engineering,which can effectively simulate the memory effect in the above dynamic system.However,it is difficult to find the analytical solution for these equations,so we consider constructing an efficient and stable numerical scheme to solve themIn this paper,we mainly studies the numerical method and error analysis of a class of nonlinear parabolic PIDEs,that is,the compact difference method is used in the spatial direction,the implicit-explicit backward differentiation formula(IMEX BDF)method is used in the time direction,and the integral term is solved by trapezoidal quadrature formula.The above numerical method is called compact IMEX BDF method.At the same time,we make a theoretical analysis of the numerical scheme,and some numerical examples are given to verify the accuracy and effectiveness of the numerical schemeFirstly,this paper introduces the research background and development history of parabolic PIDEs and their numerical methodsSecondly,a compact IMEX BDF difference scheme is constructed for one-dimensional parabolic PIDEs,and the convergence order of the algorithm is o(?2+h4),where ? is the time step and h is the space step.Then,the numerical scheme is analyzed theoretically,and the effectiveness of the scheme is verified by some examplesFurthermore,by means of alternating direction implicit(ADI)method,the correspond-ing difference scheme in two-dimensional case is given,and the convergence order of the algorithm is O(?2+hx4+hy4),where ? is the time step,hx and hy are the space steps in the X and Y directions respectively.By means of a prior estimation and energy method,the stabil-ity and convergence of the numerical scheme are proved.At the same time,some numerical examples are given.Compared with the classical second-order implicit BDF method,the accuracy and effectiveness of the difference scheme are verifiedFinally,the numerical methods and theoretical analysis are summarized,and some prospects are proposed.