The formally second-order BDF ADI difference/compact difference scheme for the nonlocal evolution problem in three-dimensional space

Abstract

This work formulates two kinds of alternating direction implicit (ADI) schemes for the parabolic-type three-dimensional evolution equation with a weakly singular kernel. The second-order backward differentiation formula (BDF2) and the second-order convolution quadrature (CQ) technique are applied to the discretization of the time derivative and the Riemann-Liouville (R-L) integral, respectively. Then, the fully-discrete BDF2 difference scheme and BDF2 compact difference scheme are constructed via the general centered difference and compact difference method, respectively. Meanwhile, the ADI algorithms are designed reasonably for two schemes to reduce the computational cost. The stability and convergence of two ADI schemes are derived via the energy method. Finally, several numerical examples are provided and tested to validate the theoretical analysis.