Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models
Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models
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DOI：
10.1137/130913961
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发表时间：
2014-01-01
影响因子：
2.1
通讯作者：
Starke, Jens
中科院分区：
文献类型：
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作者：
Marschler, Christian;Sieber, Jan;Starke, Jens
我们引入了慢-快系统设置中隐式无方程方法的一般公式。首先，我们给出了无方程分析的严格收敛结果，表明隐式定义的粗级时间步进器在慢流形上收敛到真实动力学，误差相对于测量时间尺度分离的小参数呈指数小。其次，我们将此结果应用于理想化的交通建模问题，即由在环形道路上行为一致的汽车产生的幻象拥堵问题。交通拥堵就像是逆着交通方向缓慢传播的波浪。无方程分析使我们能够在宏观层面上研究微观交通模型的行为。选择汽车车头时距的标准差作为基础动力学的宏观测量，使得行波解对应于无方程设置中宏观水平上的平衡。交通堵塞向自由流的崩溃对应于这种宏观平衡的鞍节点分叉。我们使用无方程分析在两个参数中继续这种分岔。
We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold within an error that is exponentially small with respect to the small parameter measuring time scale separation. Second, we apply this result to the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. The traffic jams are waves that travel slowly against the direction of traffic. Equation-free analysis enables us to investigate the behavior of the microscopic traffic model on a macroscopic level. The standard deviation of cars' headways is chosen as the macroscopic measure of the underlying dynamics such that traveling wave solutions correspond to equilibria on the macroscopic level in the equation-free setup. The collapse of the traffic jam to the free flow then corresponds to a saddle-node bifurcation of this macroscopic equilibrium. We continue this bifurcation in two parameters using equation-free analysis.