於此篇文章中我們主要探討並研究一個關於交通流的非線性雙曲型守恆定律。此交通流的模型包含著一個能夠表示車道變換之車輛的強度的參數。這模型可以重寫成一個含有源項的守恆定律的形式（亦即此守恆定律方程式之等號右側帶有非零項）。此文章亦會展示幾種不同案例的車道變換強度之數值計算結果。

In this thesis we study a nonlinear hyperbolic balance law arise from traffic flow. The model of traffic flow consists of a parameter representing the intensity of lane-changing of vehicles. The model is rewritten as a balance law with source terms. The numerical results are given for different cases of lane-changing intensities.

[1] G. Chen and J. Glimm, Global solution to the compressible Euler equations with geometrical structure, Comm. Math. Phys., 179(1996), page 153-193.
[2] R. Courant and K.O. Friedrichs, Supersonic flow and shock waves, John Wiley & Sons, New York, 1948.
[3] G. Dal Maso, P.G. LeFloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74(1995), pp. 483-548.
[4] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18(1965), pp. 697-715.
[5] S.K. Godunov, A difference method for numerical calculations of discontinu- ous solutions of the equations of hydrodynamics, Mat. Sb. 47(1959), in Rus- sion, pp. 271-306.
[6] J. Hong, The Glimm scheme extended to inhomogeneous systems, Doctoral Thesis, UC-Davis.
[7] E. Isaacson, Global solution of a Riemann problem for a non-strictly hyper- bolic system of conservation laws arising in enhanced oil recovery, Rockefeller University preprint.
[8] E. Isaacson, D. Marchesin, B. Plohr, and B. Temple The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Rie- mann problems I, SIAM J. Appl. Math.,48(1988), pp. 1009-1052.
[9] E. Isaacson, B. Temple, The structure of asymptotic states in a singular sys- tem of conservation laws, Adv. Appl. Math., 11(1990), pp. 205-219.
[10] E. Isaacson, B. Temple, Analysis of a singular hyperbolic system of conserva- tion laws, Jour. Diff. Equn., 65(1986), pp. 250-268.
[11] E. Isaacson, B. Temple, Examples and classification of non-strictly hyperbolic systems of conservation laws, Abstracts of AMS, January 1985. 
50
[12] E. Isaacson, B. Temple, Nonlinear resonance in systems of conservation laws, with E. Isaacson, SIAM Jour. Appl. Anal., 52, 1992, pp. 1260-1278.
[13] E. Isaacson, B. Temple, Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law, SIAM Jour. Appl. Math., 55, No. 3, pp. 625-640, June 1995.
[14] B. Keyfitz and H. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Rat. Mech. Anal., 72(1980), pp.219- 241.
[15] S.N. Kruzkov, First order quasilinear equations with several space variables, Mat. USSR Sb. 10 (1970), pp.217-243.
[16] P.D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10(1957), pp. 537-566.
[17] P.D. Lax and B. Wendroff, Systems of Conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217-237.
[18] L. Lin, J. Wang and B. Temple, A comparison of convergence rates for Go- dunov’s method and Glimm’s method in resonant nonlinear systems of con- servation laws, with L.Lin and J. Wang., SIAM J. Numer. Anal., 32, No. 3, pp. 824-840.
[19] L. Lin, J. Wang and B. Temple, Suppression of oscillations in Godunov’s method for a resonant non-strictly hyperbolic system, SIAM J. Numer. Anal., 32, No. 3, June 1995.
[20] T.P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys., 68(1979), pp. 141-172.
[21] T.P. Liu, Resonance for a quasilinear hyperbolic equation, J. Math. Phys. 28 (11), (1987), pp. 2593-2602.
[22] D. Marchesin and P.J. Paes-Leme, A Riemann problem in gas dynamics with bifurcation, PUC Report MAT 02/84, 1984.
[23] O.A. Oleinik, Discontinuous solutions of non-linear differential equations, Us- pekhi Mat. Nauk (N.S.), 12(1957),no.3(75), pp. 3-73 (Am. Math. Soc. Trans., Ser. 2, 26, pp. 195-172.)
[24] J. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, Berlin, New York, 1983. 
51
[25] B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 non- strictly hyperbolic conservation laws, Adv. in Appl. Math., 3(1982), pp. 335- 375.
[26] A. Tveito and R. Winther, Existence, uniqueness and continuous depen- dence for a system of hyperbolic conservation laws modelling polymer flooding, Preprint, Department of Informatics, University of Oslo, Norway, January, 1990.
[27] Wen-Long Jin *, A kinematic wave theory of lane-changing traffic flow, November 25, 2009.
[28] Wen-Long Jin *, A multi-commodity Lighthill-Whitham-Richards model of lane-changing traffic flow, Department of Civil and Environmental Engineering, University of California, 2012.