Numerical Simulation Methods For Simplified Model Of Stochastic Conservation Laws With Discontinuous Flux

In recent years,statistical methods have penetrated into various fields,and the cross-application with computational mathematics and other disciplines has become more and more widespread.Influenced by uncertain variables,numerical simulations of stochastic conservation laws involving uncertainty,such as traffic flow and oil exploration,have received much attention.In order to design the numerical simulation method to solve the stochastic conservation law,this dissertation takes the simplified model of stochastic conservation law with intermittent circulation as the research object,constructs the numerical simulation method of the simplified model of stochastic conservation law,uses the least squares and nonlinear regression methods to regress the error data,and verifies the constructed numerical simulation method according to the regression equation and prediction graph.First,the intermittent flow volume will lead to jumps in the unknown function,thus creating difficulties in designing the conservation law numerical simulation method.To overcome this difficulty,a new and improved numerical method for splitting the flux function(denoted as MEO)is proposed in this dissertation,and the simulation results show that this method approximates the exact solution of the model better.Further,a more efficient numerical simulation method(denoted as MEO-WENO5)is obtained by inserting an improved numerical flux at the intermittent numerical flux in the MEO method with the help of a weighted essential oscillation-free reconstruction method combined with a third-order full-variance non-increasing Runge-Kutta time discretization.Second,under certain conditions,the MEO-WENO5 numerical simulation method has a pair of stable values at the discontinuity,so that two other numerical methods are proposed,which are denoted as DFLU-WENO5 and DFLU-WENO5 B numerical simulation methods.The simulation results of the three numerical methods are compared and analyzed by simulating relevant examples,and the results show that the MEO-WENO5 numerical method has good performance of stability,high accuracy and no oscillation.Again,since the numerical method constructed above requires too much information and thus generates more numerical dissipation.Thus,the MEO method is applied to the Hermite-type WENO reconstruction,and a new numerical simulation method(denoted as MEO-HWENO5)is proposed to approximate the model equations by using the evolution of the unknown function and its first-order derivatives.Through numerical experimental results,it is found that the MEO-HWENO5 numerical simulation method has better computational results.Finally,since the data simulated by several numerical methods mentioned above are limited,it is not yet possible to illustrate the results well.In this dissertation,the error data of different numerical simulation methods are regressed and analyzed using nonlinear regression and least squares,and the nonlinear models with smaller standard errors and better fitting effects are selected for comparison.The error data of MEO-1and DFLU-1 are fitted using the inverse model and power model,respectively;while the error data of MEO-WENO5,MEO-HWENO5 and DFLU methods are fitted using the logit model and cubic model,respectively.Comparing the regression equations and prediction results graphically,it can be seen that with grid encryption,the MEO-1 and MEO-WENO5 methods have smaller calculation errors,and the MEO-HWENO5 method with a grid density of 500 or less has more accurate calculation results.