本篇論文透過模擬號誌化環狀道路模型，比較有限差分法與通量重建法的差異，環狀道路模型是屬於交通流模型的其中一種變形，而其中我們考慮密度與通量的關係為格林希爾茲模型。而數值方法的建構上，使用不連續有限元素法的原理對空間離散，按照相鄰網格上的通量決定在不連續介面上的數值通量，並使用通量重建法去重建介面上的數值通量，利用Runge-Kutta方法進行時間離散求得數值解，最後分別利用梯度限制器、WENO限制器修正其震盪現象。最初會先以連續與不連續的伯格斯方程、格林希爾茲模型當中的稀疏波及震波模型測試未加限制器的數值方法，再分別利用梯度限制器、WENO限制器消去震盪，並比較其精度、收斂階數
及計算時間，在數值實驗中，通量重建法可以利用較少的元素個數與計算時間，得到相對於有限差分法更高的精度，其中WENO限制器與梯度限制器在表現上大同小異，皆可以保持高階精度，但WENO限制器會花費較多的計算時間，而梯度限制器所需的時間大略介於無限制器及WENO限制器之間。最後運用在環狀道路模型上，比較有限差分法與兩種限制器在圖形上的差異，在最後的數值實驗中，從有限差分法觀察號誌化環狀道路模型的趨勢，使用通量重建法時，會在不連續的介面處產生震盪，WENO限制器可以消除大部分的震盪，但在不連續的介面會會留下一些輕微的震盪；梯度限制器可以消除所有的震盪，就圖形上來看，在這類出現許多震波的
數值模型中，梯度限制器的效果較WENO限制器的效果好。

This paper shows the difference between finite difference method and higher-order accuracy method for signalized ring road model. Signalized ring road is a special case of traffic flow model with put on a flux control function at the specific location. The Greenshield’s model is considered as an option to the relation between the density
and the flux. To construct the numerical method, we have spatial discretization based on discontinuous galerkin method and determine the numerical flux by near elements on the interface between two elements. And use the flux reconstruction method to
reconstruct the numerical flux on interface. Then compute the numerical solution by using Runge-Kutta method to discrete time. Finally, we use slope limiter and WENO limiter to fix the oscillation respectively. At the beginning, we will test the numerical method without limiter by linear burgers equation, non-linear burgers equation and shock wave, rarefraction wave in traffic flow model. Then use the slope limiter and WENO limiter to fix the oscillation respectively and show the error, order of accuracy and computational time. In the numerical experiment, flux reconstruction method can get high-order accuracy with less elements and computational time than finite difference method. Both WENO limiter and slope limiter keep the high-order accuracy but WENO takes the most time, then slope limiter. In the end, we apply on signalize ring road model and compare figures of finite difference method with figures of flux reconstruction method. We observe the trend of the signalized ring road by finite difference method. With flux reconstruction method, oscillations appear on the discontinuous interface. WENO limiter can deal with most oscillation but miss some slight ones; However, slope limiter can fix all the oscillation. On this numerical model which has many shock waves, the slope limiter works better than WENO limiter.

[1] Adimurthi, Jerome Jaffre, and G. D. Veerappa Gowda. Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM Journal on Numerical Analysis, 42(1):179-208, 2005.
[2] B. Cockburn, An introduction to the discontinuous galerkin method for convection-dominated problems. Springer, Berlin, Heidelberg, Page 177-180, 2008.
[3] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local rojection discontinuus Galerkin finite element method for conservation laws II: general framework. Mathematics of Computation, 50(1989), 411-435, 1989.
[4] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing. Journal of Computational Physic, 77(1998), 439-471, 1998.
[5] D.S. Balsara and C.-W. Shu, Monotonicity preserving weighted essentially nonoscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 160(2000), 405-452, 2000.
[6] Du, J. CPR method with limiters and study on traffic flow model. University of Science and Technology of China, China, Page 23-25, 2015.
[7] G. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. Journal of Computational Physics,126(1995),202-228, 1995.
[8] H. T. Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper, 2007-4079, 2007.
[9] H. T. Huynh, A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. AIAA Paper, 2009-403,2009.
[10] Hu, B. Computing effective flow rates for a signalized ring road. National Cheng Kung University, Page 1-32, 2020.
[11] Mark H.Holmes, Introduction to the Foundations of Applied Mathematics, Texts in Applied Mathematics, 56(2019), 255-260, 2019.
[12] Mauro Garavello, Benedetto Piccoli, Traffic flow model:Conservation laws model,AIMS 2006, Page 9-15, 2006.
[13] Randall J. LeVeque, Finite difference methods for ordinary and partial differential equations:steady-state and time-dependent problems. SIAM, Page 3-5, 2007.
[14] Randall J. LeVeque, Finite difference methods for ordinary and partial differential equations:steady-state and time-dependent problems. SIAM, Page 245-258, 2007.
[15] WenLong Jin and Yifeng Yu. Performance analysis and signal design for a stationary signalized ring road. https://arxiv.org/abs/1510.01216, 2015.
[16] X. Zhong and C.-W. Shu, A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods,Journal of Computational Physics,232(2013),397-415, 2013.