Numerical Methods For A Class Of Hyperbolic Conservation Law Equations

Systems engineering is a discipline that mainly studies systems in the fields of engineering,economy,environment and society,and uses mathematical methods and computer technology as tools to study the theories,techniques and methods of system modeling,analysis,design and implementation.In this paper,based on the mathematical theory of hyperbolic partial differential equations,a class of hyperbolic conservation laws in traffic flow is studied and the corresponding numerical simulations are presented to obtain the effective numerical method.In this paper,we derive the mass conservation law equation according to the physical background,then solve the Riemann solution of the conservation laws,and further study the numerical methods of the conservation laws with discontinuous flux.The interface flux plays an important role in designing numerical methods due to the discontinuity of the flux at x=0.By studying the properties of numerical flux and calculating some numerical examples,we obtain that results of Local Lax-Friedrichs interface flux are very smeared in regions near the discontinuities,the Modified Local Lax-Friedrichs interface flux is valid but not monotone.Therefore,based on the(A,B)-type entropy solutions introduced by Bürger,Karlsen,and Towers[SIAM J Numer Anal,2009,47:1684-1712],we propose a new modified Local Lax–Friedrichs interface flux to approximate this system and prove that the interface flux is Lipschitz continuous and monotone with respect to its two variables.Through numerical experiments and ~1L-error analysis,we demonstrate that our first-order accuracy scheme has good approximation effect.In order to improve the resolution of the numerical solution,we consider to design the second-order accuracy scheme,and the core of which is to construct the correction terms.In the numerical experiments,we found that the numerical solution calculated by the second-order Flux Total Variation Diminish(FTVD)scheme gives rise to the so-called dog-leg feature.The second-order Modified-FTVD(M-FTVD)scheme adds a new parameter Ch~?to the non-local limiter algorithm,and then by adjusting the value of this parameter,the dog-leg feature is improved or even vanished,but the correction terms of the second-order FTVD scheme and the second-order M-FTVD scheme are zero at the discontinuity x=0.Therefore,based on the definition of second order FTVD scheme introduced by Bürger,Karlsen,Torres and Towers[Numer Math,2010,116:579-617],we propose a new and efficient algorithm to construct the correction term at the discontinuity which can automatically choose whether it is zero or not,so that the numerical solution can achieve better resolution at the discontinuity x=0 in some special examples.In particular,in the numerical example,by piecewise value of the parameters in the new algorithm,not only can the dog-leg feature phenomenon disappear on the left side of the discontinuity x=0,but also the numerical solution of shock wave on the right side of the discontinuity x=0 can be better approximated to the exact solution.