A BDF2 ADI difference scheme for a three-dimensional nonlocal evolution equation with multi-memory kernels

A second order backward differentiation formula (BDF2) alternating direction implicit (ADI) finite difference method (FDM) is proposed for the three-dimensional (3D) nonlocal evolution equation with multi-memory kernels in this paper. The second order convolution quadrature is applied to approximate the Riemann-Liouville (R-L) fractional integral term. The fully discrete scheme is obtained by using the standard central FDM in space. We strictly prove the stability and convergence of the BDF2 ADI difference scheme, whose convergence order can be \(\mathcal {O}(k^{1+\min \{\alpha ,\beta ,\gamma \}}+h^{2})\) $\mathcal{O}(k^{1+min\{\alpha ,\beta ,\gamma \}}+h^2)$ . At last, two numerical experiments are given. The ADI algorithms reduce the computational cost of 3D problems. We also compare our scheme with the scheme in Zhou et al. (Int J Comput Math 100(8):1719-36, 2023).