Extrapolation Algorithm Based On POD Method For Traffic Flow Model And Partial Differentio-Integral Equations

Proper Orthogonal Decomposition method (POD) is a way of simplifying the calculation of partial differential equations. The reduced-order extrapolation difference scheme based on POD method for solving partial differential equations can be used to reduce the degree of freedom, reduce the dimensions and reduce computing time while ensuring the accuracy of difference scheme. The advantages of POD method have been reflected in the solution of various forms of partial differential equations (such as two-dimensional parabolic equations and two-dimensional hyperbolic equations).As the applications of POD method, this paper mainly studies the reduced-order extrapolation difference scheme based on POD method for the anisotropic traffic flow model and the partial integro-differential equations. The second chapter mainly studies the POD method applied to solve the anisotropic traffic flow model. A extrapolation finite difference scheme with sufficiently high accuracy based on a few POD base is established, and the error estimates between the reduced and the classical finite difference solutions is provided. The conclusions obtained by numerical experiments which simulate actual traffic situation with two difference schemes are consistent with the theoretical results. Moreover, it shows the feasibility and effectiveness of applying POD method to the traffic flow model. The third chapter mainly studies the POD method applied to solve the partial integro-differential equations. A reduced-order extrapolation finite difference scheme with lower dimensions and sufficiently high accuracy based on POD is established, and the error estimates between the reduced and the backward Euler finite difference solutions is provided. Then the numerical experiments are used to illustrate the fact that the calculation results coincide with the theoretical results. Moreover, it shows the feasibility and effectiveness of applying POD method to the partial integro-differential equations.