Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations
Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations
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DOI：
10.48550/arxiv.2204.04673
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发表时间：
2022-04
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影响因子：
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通讯作者：
Dongling Wang;M. Stynes
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文献类型：
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作者：
Dongling Wang;M. Stynes
非线性初始值问题 $\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ 对于 $t>0$ 且 $y(0)>0$ 的解，其中 $\mathcal{D}_{t}^{\alpha}$ 是 $\alpha\in (0,1)$ 阶的 Caputo 导数，$\lambda, \gamma$ 是正参数，已知可表现出$O(t^{\alpha/\gamma})$ 衰减为 $t\to\infty$。先前尚未证明该问题的任何离散化的相应结果。本文表明，对于均匀网格 $\{t_n:=nh\}_{n=0}^\infty$ 上的完全单调性保持 ($\mathcal{CM}$-preserving) 方案（包括 L1 和 Gr\"unwald-Letnikov 方案）类，离散解也具有 $O(t_{n}^{-\alpha/\gamma})$ 衰减： $t_{n}\to\infty$。然后将该结果扩展到 $\mathcal{CM}$ - 保留某些时间分数非线性次扩散问题的离散化，例如时间分数多孔介质和 $p$-拉普拉斯方程。对于 L1 方案，$O(t_{n}^{-\alpha/\gamma})$ 衰减结果在非常一般的非均匀网格上仍然有效。使用离散比较原理与精心构造的离散子解和超解，以给出离散解的严格界限。提供数值实验来证实我们的理论分析。
The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^{\alpha}$ is a Caputo derivative of order $\alpha\in (0,1)$ and $\lambda, \gamma$ are positive parameters, is known to exhibit $O(t^{\alpha/\gamma})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Gr\"unwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-\alpha/\gamma})$ decay as $t_{n}\to\infty$. This result is then extended to $\mathcal{CM}$-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and $p$-Laplace equations. For the L1 scheme, the $O(t_{n}^{-\alpha/\gamma})$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.