Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations
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A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost \(O(MN^2)\) and storage O(MN) for the standard L1 formula to \(O(MN\log N)\) and \(O(M\log N)\), respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time \(t=0\), and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete \(H^2\) energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.
KeywordsSemilinear subdiffusion equation Two-level L1 formula Discrete fractional Grönwall inequality Discrete \(H^2\) energy method Unconditional convergence
The authors gratefully thank Professor Martin Stynes for his valuable discussions and fruitful suggestions during the preparation of this paper. Hong-lin Liao would also thanks for the hospitality of Beijing CSRC during the period of his visit.
- 12.Liao, H.-L., Sun, Z.Z., Shi, H.S.: Maximum norm error analysis of explicit schemes for two-dimensional nonlinear Schrödinger equations. Sci. China Math. 40(9), 827–842 (2010). (in Chinese) Google Scholar