An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space

This paper addresses the numerical solution of the three-dimensional nonlocal evolution equation with a weakly singular kernel. The first order fractional convolution quadrature scheme and backward Euler (BE) alternating direction implicit (ADI) method, are proposed to approximate and discretize the Riemann-Liouville (R-L) fractional integral term and temporal derivative, respectively. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative. By using ADI scheme for the three-dimensional problem, the overall computational cost is reduced significantly. Two new approaches are adopted for theoretical stability analysis. The convergence behaviour of the proposed method is provided and the error bounds are proved. In addition, two test problems illustrate the validity and effectiveness of the methods. The CPU time of our scheme is extremely little.

Automated summary

This paper proposes a first-order fractional convolution quadrature scheme and backward Euler method for the numerical solution of the three-dimensional nonlocal evolution equation. By discretizing the second-order spatial derivative using the standard central finite difference approximation and using the alternating direction implicit method for three-dimensional problems, the computational cost is significantly reduced. The stability analysis and error bounds of the proposed method are provided. The effectiveness of the method is demonstrated through two test problems.