CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics. (English) Zbl 07894914
Summary: 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 \(L^2\)-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached \(\mathcal{O}( \tau^2 + h_x^2 + h_y^2 + h_z^2)\). 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65R20 | Numerical methods for integral equations |
| 45K05 | Integro-partial differential equations |
Keywords:
evolution equation; Crank-Nicolson; alternating direction implicit; finite difference method; stability and convergenceReferences:
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