Second Order Numerical Methods for First Order Hamilton-Jacobi Equations
Second Order Numerical Methods for First Order Hamilton-Jacobi Equations
复制标题
一阶 Hamilton-Jacobi 方程的二阶数值方法
DOI：
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发表时间：
2002
影响因子：
2.9
通讯作者：
P. Dupuis
中科院分区：
文献类型：
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作者：
A. Szpiro;P. Dupuis
给出了一类定常一阶汉密尔顿-Jacobi偏微分方程的可证明二阶逼近的实用数值方法。利用概率方法，我们推导出一阶方法的高阶渐近展开式，然后利用这些结果来设计二阶方法。我们证明了二阶收敛的解决方案和其梯度的一个子集的解决方案是光滑的。虽然我们将注意力限制在二阶格式上，但原则上本文中的技术可以扩展到任意高阶方法。例子说明了收敛速度以及全球尖锐的分辨率的不连续性。我们考虑的汉密尔顿-雅可比方程对应于确定性最优控制问题，我们的收敛速度的结果是有效的值函数和最优反馈控制。
We present practical numerical methods which produce provably second order approximations for a class of stationary first order Hamilton--Jacobi partial differential equations. Using probabilistic methods, we derive high order asymptotic expansions for a first order method and then use those results to design second order methods. We prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth. Although we limit our attention to second order schemes, in principle the techniques in this paper can be extended to arbitrarily high order methods. Examples illustrate the rate of convergence as well as global sharp resolution of discontinuities. The Hamilton--Jacobi equations we consider correspond to deterministic optimal control problems, and our rate of convergence results are valid for the value functions and for the optimal feedback controls.