论文标题
火星：一种自适应去除PDE的刚度的方法
MARS : a Method for the Adaptive Removal of Stiffness in PDEs
论文作者
论文摘要
最近开发了e（xplicit）i（隐式）n（null）方法是为了从pdes中删除数值不稳定性，添加和减去了任意结构的operator $ \ mathcal {d} $，在一种情况下隐含地处理了操作员，另一个情况下则明确处理了操作员。在这里，我们通过设计自适应过程来找到$ \ Mathcal {d} $的最佳近似来扩展这个想法。我们提出了一个数值误差的度量，该数值误差检测到所有波长的数值不稳定性，并将$ \ Mathcal {d} $的每个傅立叶组件调整为最小的值，从而抑制了数值不稳定。我们表明，对于许多非线性和非本地PDE，在一个和二维中，$ \ Mathcal {d} $的频谱自动而动态地适应了边际稳定性的理论结果。因此，我们的方法具有与完全隐式方法相同的稳定性属性，而仅需要与显式求解器相当的计算成本。自适应隐式部分在傅立叶空间中是对角线的，因此与显式方法相比，间接头顶最小。
The E(xplicit)I(implicit)N(null) method was developed recently to remove numerical instability from PDEs, adding and subtracting an operator $\mathcal{D}$ of arbitrary structure, treating the operator implicitly in one case, and explicitly in the other. Here we extend this idea by devising an adaptive procedure to find an optimal approximation for $\mathcal{D}$. We propose a measure of the numerical error which detects numerical instabilities across all wavelengths, and adjust each Fourier component of $\mathcal{D}$ to the smallest value such that numerical instability is suppressed. We show that for a number of nonlinear and non-local PDEs, in one and two dimensions, the spectrum of $\mathcal{D}$ adapts automatically and dynamically to the theoretical result for marginal stability. Our method thus has the same stability properties as a fully implicit method, while only requiring the computational cost comparable to an explicit solver. The adaptive implicit part is diagonal in Fourier space, and thus leads to minimal overhead compared to the explicit method.
