论文标题
一阶汉密尔顿 - 雅各比方程的多维平滑度指标
Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations
论文作者
论文摘要
缺乏平滑度是非线性双曲方程弱解的共同特征，并且在其近似中是至关重要的问题。这激发了基于近似解决方案的值定义适当指标的几项努力，以检测域中最麻烦的区域。此信息有助于调整近似方案，以避免使用高阶方案时伪造的振荡。在本文中，我们提出了WENO程序的真正多维扩展，以克服基于维截面的指标的局限性。我们的目的是获取2D中问题的新规则性指标，并将其应用于一类``自适应过滤''方案的一阶进化后的汉密尔顿 - 雅各比方程。根据通常的程序，通过简单的高阶方案和单调方案的简单耦合获得过滤的方案。混合物由过滤函数$ f $控制，并通过切换参数$ \ varepsilon^n = \ varepsilon^n（{{Δt，Δt，Δx}）> 0 $ 0 $，该> 0 $ as as as $（ΔT，ΔX）$均为0。 空间。介绍了有关1D和2D关键情况的几个数值测试，并确认提出的指标的有效性以及我们计划的效率。
The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of ``adaptive filtered'' schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function $F$ and by a switching parameter $\varepsilon^n=\varepsilon^n({Δt,Δx})>0$ which goes to 0 as $(Δt,Δx)$ is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter $\varepsilon^n_j$ in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme.
