We consider a class of “filtered” schemes for first order time dependent Hamilton–Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter $$\varepsilon =\varepsilon ({\Delta t,\Delta x})>0$$ ε = ε ( Δ t , Δ x ) > 0 which goes to 0 as the time and space steps $$(\Delta t,\Delta x)$$ ( Δ t , Δ x ) are going to 0 and does not depend on the time $$t_n$$ t n , for each iteration n . The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of $$\varepsilon =\varepsilon ^n (\Delta t, \Delta x)$$ ε = ε n ( Δ t , Δ x ) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold $$\varepsilon ^n$$ ε n . A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

中文翻译：

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一阶进化Hamilton-Jacobi方程自适应滤波方案的收敛性

我们考虑一阶时间相关的 Hamilton-Jacobi 方程的一类“过滤”方案，并证明此类方案的一般收敛结果。典型的滤波方案是根据滤波器函数 F 混合高阶方案和单调方案获得的，滤波器函数 F 决定方案必须从一种方案切换到另一种方案的位置。这种转换的一个关键作用是由参数 $$\varepsilon =\varepsilon ({\Delta t,\Delta x})>0$$ ε = ε ( Δ t , Δ x ) > 0 起到 0 的作用对于每次迭代 n ，时间和空间步长 $$(\Delta t,\Delta x)$$ ( Δ t , Δ x ) 将变为 0 并且不依赖于时间 $$t_n$$ tn 。代码中该参数的调整相当微妙，并且对滤波方案的全局精度有影响。在这里，我们在每次修改经典设置的迭代中引入了 $$\varepsilon =\varepsilon ^n (\Delta t, \Delta x)$$ ε = ε n ( Δ t , Δ x ) 的自适应和自动选择。适应性由平滑度指标控制，该指标选择我们修改规律性阈值 $$\varepsilon ^n$$ ε n 的区域。证明了新自适应滤波方案的收敛结果和一些误差估计，该分析依赖于方案和平滑度指标的特性。最后，我们提出了一些数值测试来比较自适应滤波方案与其他方法。适应性由平滑度指标控制，该指标选择我们修改规律性阈值 $$\varepsilon ^n$$ ε n 的区域。证明了新自适应滤波方案的收敛结果和一些误差估计，该分析依赖于方案和平滑度指标的特性。最后，我们提出了一些数值测试来比较自适应滤波方案与其他方法。适应性由平滑度指标控制，该指标选择我们修改规律性阈值 $$\varepsilon ^n$$ ε n 的区域。证明了新自适应滤波方案的收敛结果和一些误差估计，该分析依赖于方案和平滑度指标的特性。最后，我们提出了一些数值测试来比较自适应滤波方案与其他方法。