In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion equation by incorporating a fractional diffusion term. We provide a rigorous analysis showing that the discretization preserves main properties of the continuous equations, including algebraic decay in the fractional porous medium case and the extinction phenomenon in the fractional fast diffusion case. The study is supported by extensive numerical simulations. In addition, we propose a novel method for accurately computing the extinction time for the fractional fast diffusion equation and illustrate numerically the convergence of rescaled solutions towards asymptotic profiles near the extinction time.

翻译：本文提出并分析了一种用于求解分数阶非线性扩散方程柯西-狄利克雷问题的数值格式。该方程通过引入分数阶扩散项，推广了多孔介质方程和快速扩散方程。我们通过严格分析证明，该离散格式保持了连续方程的主要性质，包括分数阶多孔介质情况下的代数衰减特性以及分数阶快速扩散情况下的熄灭现象。研究结果得到了大量数值模拟的支持。此外，我们提出了一种精确计算分数阶快速扩散方程熄灭时间的新方法，并通过数值模拟展示了在熄灭时间附近重标度解向渐近剖面的收敛过程。