Maximum-principle-satisfying And Positivity-preserving High Order Schemes For Traffic Flow Models

Hyperbolic conservation laws are a class of essential partial differential equations.They describe the conservation of nature and play significant roles in various area of applications such as computational fluid dynamics,computational astronomy and astrophysics,aerospace and shipbuilding,plasma simulation,population and traffic flow models.Because of the complexity of these equations,in general,it is not possible to derive their analytic solutions and leads us to use some appropriate numerical methods for their approximate solution in practical applications.In particular,when using classical high order numerical method to solve hyperbolic conservation laws,it is quite common to get numerical solutions violating some physical properties duo to numerical errors,for instance,the maximum principle,negative pressure and density and so on.In this thesis,we first introduce a maximum-principle-satisfying high order WENO method for the one-dimensional scalar conservation laws.We prove a sufficient condition for the cell averages of the numerical solutions in a WENO method with Euler forward time discretization with Gauss-Lobatto numerical integration formula to satisfy the maximum principle.Next,we use this finite volume framework to construct positivity-preserving WENO methods for compressible Euler equations in one dimensional space.Finally,we introduce the application of hyperbolic conservation laws,that is the traffic flow modes,and apply the maximum-principle-satisfying high order schemes to this model and obtain the desired results.