关于junction上Hamilton-Jacobi方程的收敛方案：应用于交通
A convergent scheme for Hamilton-Jacobi equations on a junction: application to traffic
AI总结：
本文提出一种有限差分方案用于junction上的Hamilton-Jacobi方程，证明了数值解的梯度和时间导数的界，并证明在网格尺寸趋于零时，数值解收敛于连续HJ问题的粘性解，应用于交通模型计算车辆密度。
摘要：
本文研究了定义在"junction"上的首次-order Hamilton-Jacobi (HJ) 方程，即有限多个半线的并集，具有唯一公共点。针对此连续HJ问题，我们提出一个有限差分方案，并证明两个主要结果。第一个结果是数值解的离散梯度和时间导数的界。第二个结果是当网格尺寸趋于零时，数值解（对于子序列）收敛于连续HJ问题的粘性解。当连续HJ问题的解唯一时，我们恢复了数值解的完整收敛性。我们将此方案应用于计算交通模型中的车辆密度。我们恢复了在junction点外的已知Godunov方案，并给出了数值示例。
英文摘要：
In this paper, we consider first order Hamilton-Jacobi (HJ) equations posed on a ``junction'', that is to say the union of a finite number of half-lines with a unique common point. For this continuous HJ problem, we propose a finite difference scheme and prove two main results. As a first result, we show bounds on the discrete gradient and time derivative of the numerical solution. Our second result is the convergence (for a subsequence) of the numerical solution towards a viscosity solution of the continuous HJ problem, as the mesh size goes to zero. When the solution of the continuous HJ problem is unique, we recover the full convergence of the numerical solution. We apply this scheme to compute the densities of cars for a traffic model. We recover the well-known Godunov scheme outside the junction point and we give a numerical illustration.
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