Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

• Authors: Ya-Nan Zhang;Zhi-Zhong Sun
• Venue: Journal of Scientific Computing
• Year: 2014

In this paper, a Crank---Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $$\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$$ O ( 驴 min { 2 - 驴 2 , 2 驴 } + h 1 4 + h 2 4 ) in standard $$H^1$$ H 1 norm, where $$\tau $$ 驴 is the temporal grid size and $$h_1,h_2$$ h 1 , h 2 are spatial grid sizes; the other is $$\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$$ O ( 驴 2 驴 + h 1 4 + h 2 4 ) in $$H^1_{\gamma }$$ H 驴 1 norm, a generalized norm which is associated with the Riemann---Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.