We propose a new finite difference scheme for the degenerate parabolic equation \[ \partial_t u - \mbox{div}(|\nabla u|^{p-2}\nabla u) =f, \quad p\geq 2. \] Under the assumption that the data is H\"older continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable stability type CFL-condition. An important advantage of our approach, is that the CFL-condition makes use of the regularity provided by the scheme to reduce the computational cost. In particular, for Lipschitz data, the CFL-condition is of the same order as for the heat equation and independent of $p$.
翻译:我们为堕落的抛物线方程提出一个新的有限差别方案[\\part_t u -\mbox{div}( ⁇ nabla u ⁇ p-2 ⁇ nabla u) = f,\quad p\geq 2. = f,\ quad p\geq 2. 假设数据是 H\\\\"老的连续数据,我们为Cauchy问题确定明确的时间办法的趋同提供了适当的稳定型CFL条件。我们方法的一个重要优点是CFL条件利用了计划提供的规律性来降低计算成本。特别是对于Lipschitz数据来说,CFL条件与热方程的顺序相同,独立于$p美元。