Some numerical extrapolation methods for the fractional sub-diffusion equation and fractional wave equation based on the \(L1\) formula. (English) Zbl 1513.65304
Summary: With the help of the asymptotic expansion for the classic \(L1\) formula and based on the \(L1\)-type compact difference scheme, we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation. Three extrapolation formulas are presented, whose temporal convergence orders in \(L_\infty \)-norm are proved to be 2, \(3-\alpha\), and \(4-2\alpha\), respectively, where \(0<\alpha <1\). Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal \(4-\gamma\) and \(6-2\gamma\) order in \(L_\infty\)-norm, respectively, where \(1<\gamma <2\). Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation, the fast extrapolation methods are obtained which reduce the computational complexity significantly while keeping the accuracy. Several numerical experiments confirm the theoretical results.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65B05 | Extrapolation to the limit, deferred corrections |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 60K50 | Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) |
Keywords:
\(L1\) formula; asymptotic expansion; fractional sub-diffusion equation; fractional wave equation; Richardson extrapolation; fast algorithmReferences:
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