A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem
Provides a rigorous theoretical foundation for high-order numerical methods for fractional diffusion problems, addressing the initial singularity challenge.
The paper establishes stability and convergence of a second-order time-weighted scheme for linear reaction-subdiffusion equations using nonuniform time steps, achieving second-order accuracy on graded meshes with proper grading.
Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Grönwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.