A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation

摘要

ABSTRACT Alternating direction implicit (ADI) method is widely utilized and computationally efficient for numerical approximation of the multidimensional evolution equations. This paper mainly formulates and analyzes a formally second order backward differentiation formula (BDF) ADI difference scheme for the three-dimensional time-fractional heat equation. The time derivative is approximated by second order BDF scheme and the integral term is treated by means of second order convolution quadrature. A fully discrete scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The stability and convergence are derived by the energy method. Numerical experiment in total agreement with our analysis is reported. Comparisons between the present schemes and the existing ones are included, showing that the present algorithms exhibit better performances than the existing ones.