High-order time stepping schemes for semilinear subdiffusion equations. (English) Zbl 1452.65252
High-order \(k\)-step backward differentiation formulae (BDFk) are developed for solving the initial-boundary value problem for the semilinear subdiffusion equation. The convolution quadrature is applied generated by BDFk to discretize the time-fractional derivative of order \(\alpha\in (0, 1)\) and the starting steps are modified in order to achieve an optimal convergence rate. The main result of the paper is to derive a convergence order \(\mathcal{O}(\Delta t^{\min (1,1 + 2\alpha -\epsilon)})\) for the corrected BDFk scheme without imposing further assumptions on the regularity of the solution. The optimal order convergence is achieved by splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part and using the generating function technique. Numerical examples are provided for the time-fractional Allen-Cahn equation to support the theoretical results.
Reviewer: Bülent Karasözen (Ankara)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
semilinear subdiffusion; convolution quadrature; \(k\)-step BDF; initial correction; error estimateReferences:
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