A higher-order unconditionally stable scheme for the solution of fractional diffusion equation

In this paper, a higher-order compact finite difference scheme with multigrid algorithm is applied for solving one-dimensional time fractional diffusion equation. The second-order derivative with respect to space is approximated by higher-order compact difference scheme. Then, Grunwald-Letnikov approximation is used for the Riemann-Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth-order accurate local truncation error. Fourier analysis is used to analyze the stability of higher-order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth-order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.

Automated summary

In this paper, a higher-order compact finite difference scheme with multigrid algorithm is used to solve the one-dimensional time fractional diffusion equation. The scheme achieves eighth-order accuracy in space, and its convergence is proven through Fourier analysis and matrix analysis. Numerical experiments confirm the performance and accuracy of the proposed scheme.