CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

Abstract

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.