Abstract
A two-dimensional time-fractional mobile-immobile diffusion problem with the Caputo time-fractional derivative of order \(\alpha \in (0,1)\) is considered. We show that the solution of the problem has a weak singularity at the initial time. Using the averaged L1 formula to approximate the Caputo time-fractional derivative and using a compact finite difference approximation to discretize the space derivatives, we propose a high-order averaged L1-type compact finite difference method on the uniform space-time mesh for the problem. We then base on this method to develop an averaged L1-type compact alternating direction implicit (ADI) finite difference method and a fast sum-of-exponentials compact ADI finite difference method, both of which significantly reduce the storage requirements and the computational costs while maintaining the same global convergence rate. By using the discrete energy analysis technique, we rigorously prove that all methods are unconditionally stable and convergent, and they have the spatial global fourth-order convergence rate and the temporal global convergence rate of order \(\min \{2, 3-2\alpha \}\). For the case of \(\alpha >1/2\), we use the discrete minimum-maximum principle to prove that the temporal second-order convergence rate can also be achieved in positive time. Numerical results confirm the theoretical analysis results and demonstrate the computational efficiency of the methods.
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The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of the paper.
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This work was supported in part by National Key R &D Program of China (No. 2022YFA1004402), and Science and Technology Commission of Shanghai Municipality (STCSM) (Nos. 22DZ2229014 and 21JC1402500).
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A Appendix: Proof of Theorem 11
A Appendix: Proof of Theorem 11
Proof
When the spatial mesh is isotropic, that is, \(h_{x}=h_{y}=h\), the operators \({{\mathcal {H}}}\) and \({{\mathcal {Q}}}\) can be written as
where
Then we can write (58a) in the form:
and for \(2\le n\le N\),
where
When the mesh conditions (98) and (99) are satisfied, \(\omega _{n,k+1}^{*}> \omega _{n,k}^{*}\) for \(1\le k\le n-1\) (see (40)) and all coefficients of the terms on the right-hand side of (119) and (120) are nonnegative. We also observe that \(\sum _{k_{1}, k_{2}=-1}^{1}p_{i,j}^{(k_{1},k_{2})}=1\) and \(\sum _{k_{1}, k_{2}=-1}^{1}q_{i,j}^{(k_{1},k_{2})}=0\) for all \((i,j)\in \Omega _{h}\).
Now suppose that \(u_{i,j}^{1}\) attains its maximum in \(\overline{\Omega }_{h}\) at \((i_{0}, j_{0})\in \Omega _{h}\). Then we have from (119) that
This shows \(u_{i_{0},j_{0}}^{1}\le \Theta _{\max }^{1}\). Since \(u_{i,j}^{1}=0\) for all \((i,j)\in \partial \Omega _{h}\), we obtain \(u_{i,j}^{1}\le \Theta _{\max }^{1}\) for all \((i,j)\in \overline{\Omega }_{h}\). An identical argument shows \(u_{i,j}^{1}\ge \Theta _{\min }^{1}\) for all \((i,j)\in \overline{\Omega }_{h}\). Assume, by induction, that \(\Theta _{\min }^{n}\le u_{i,j}^{n}\le \Theta _{\max }^{n}\) for all \((i,j)\in \overline{\Omega }_{h}\) and \(1\le n\le n_{0}-1\) \((n_{0}\ge 2)\). Suppose that \(u_{i,j}^{n_{0}}\) attains its maximum in \(\overline{\Omega }_{h}\) at \((i_{0}, j_{0})\in \Omega _{h}\). Then we use (120) and the inductive hypothesis to get a similar inequality (122), where subscript or superscript 1 is replaced by \(n_{0}\). By this inequality, we deduce \(u_{i_{0},j_{0}}^{n_{0}}\le \Theta _{\max }^{n_{0}}\) which implies \(u_{i,j}^{n_{0}}\le \Theta _{\max }^{n_{0}}\) for all \((i,j)\in \overline{\Omega }_{h}\) because \(u_{i,j}^{n_{0}}=0\) for all \((i,j)\in \partial \Omega _{h}\). The same argument can be applied to prove \(u_{i,j}^{n_{0}}\ge \Theta _{\min }^{n_{0}}\) for all \((i,j)\in \overline{\Omega }_{h}\). The principle of mathematical induction completes the proof. \(\square \)
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Zheng, ZY., Wang, YM. Fast High-Order Compact Finite Difference Methods Based on the Averaged L1 Formula for a Time-Fractional Mobile-Immobile Diffusion Problem. J Sci Comput 99, 43 (2024). https://doi.org/10.1007/s10915-024-02505-5
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DOI: https://doi.org/10.1007/s10915-024-02505-5
Keywords
- Time-fractional mobile-immobile diffusion problem
- Weak singularity
- Compact ADI finite difference method
- Fast algorithm
- Stability and convergence