基于核的高阶"显式"无条件稳定方案用于时间依赖的哈密尔顿-雅可比方程
A Kernel Based High Order "Explicit" Unconditionally Stable Scheme for Time Dependent Hamilton-Jacobi Equations
AI总结：
本文提出了一种高阶数值方案用于求解哈密尔顿-雅可比方程，该方案基于特殊的核基 formulation 和连续卷积，具有高阶空间和时间精度，并且无条件稳定，能够使用更大的时间步长，从而节省计算成本。同时，采用高阶加权本质非振荡方法和新型非线性滤波器来捕捉正确的粘性解，并通过逆拉克斯-温德罗夫边界处理技术处理复杂几何和一般边界条件。
摘要：
在本文中，提出了一类用于求解哈密尔顿-雅可比（H-J）方程的高阶数值方案。这项工作被视为我们之前针对非线性退化抛物方程工作的延伸，见 Christlieb 等人 \emph{arXiv 预印本 arXiv:1707.09294}，该工作依赖于一种特殊的核基解的 formulation 和连续卷积。当应用于 H-J 方程时，新提出的方案在空间和时间上都实现了真正的高阶精度，并且更重要的是，它是无条件稳定的，因此允许比其他显式方案更大的时间步长演进，从而节省计算成本。进一步采用高阶加权本质非振荡方法和新型非线性滤波器来捕捉正确的粘性解。此外，通过耦合最近提出的逆拉克斯-温德罗夫边界处理技术，该方法在处理复杂几何和一般边界条件方面非常灵活。我们在一组数值示例上进行了数值实验，包括具有线性、非线性、凸或非凸哈密尔顿量的 H-J 方程。所提出方案在近似一般 H-J 方程的粘性解方面的有效性和效率得到了验证。
英文摘要：
In this paper, a class of high order numerical schemes is proposed for solving Hamilton-Jacobi (H-J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. \emph{arXiv preprint arXiv:1707.09294},, which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H-J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax-Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H-J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H-J equations is verified.
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