An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data
By Bangti Jin, Raytcho D. Lazarov, Zhi Zhou. 2015
372 citations recorded by the source corpus.
The subdiffusion equation with a Caputo fractional derivative of order $\alpha\in(0,1)$ in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an $O(\tau^{2-\alpha})$ convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an $O(\tau)$ convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more general sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order $\beta\in(3/2,2)$ in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.
Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis
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Citations in the corpus
- fPINNs: Fractional Physics-Informed Neural Networks (2018, positive): “To approximate the time-fractional derivative in the equation, we adopt the commonly used finite difference L1 scheme [53, 54], as follows ∂ ũ(x, t) ∂tγ ≈ Lγ∆tũ(x, t) := 1 Γ(2− γ)(∆t)γ { −cdλte−1ũ(x, 0) + c0ũ(x, t)”
- High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II) (2015, neutral): “The analysis of L1 scheme for subdiffusion equation with nonsmooth data was recently studied in [6].”
- A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation (2019, background): “It is worth to mention that the solution of many time-fractional differential equations typically displays a weak singularity near the initial time [8, 9, 10, 19, 20, 21, 26, 29, 30], which leads to the loss of time accuracy for many related high-order numerical methods.”
- A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems (2018, background): “[11] B. Jin, B. Li, and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer.”
- An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017, background): “We choose the smooth and nonsmooth initial data (a) u0 = x(1− x) and the nonsmooth data (b) u0 = χ(0,1/2) we obtain the following results which are consistent with the Table 1 in [21].”
- Recovering an unknown source in a fractional diffusion problem (2017, positive): “As an alternative to the spectral representation we use a finite difference representation in space and the L 1 time-stepping method [7] to discretize the fractional derivative where τ is the step size of the uniform partition on t and b j = (( j + 1) 1 α − j 1 α ) / Γ(2 − α ) , j = 0 , 1 , . . . , N…”