Positivity-preserving Repair Techniques Of Finite Volume Element Solution For Nonlinear Degenerate Parabolic Equations

In mathematics,the nonlinear degenerate parabolic equation is often used to simulate the diffusion problem of porous media and the radiation heat conduction problem.Because the unknown quantity to be solved represents the concentration or temperature,its value should be positive,but when the finite volume element method is used to solve these problems,the numerical solution will be negative at some points,which is called non physical numerical oscillation,In some cases,the program will be terminated.Therefore,it is necessary to design some effective methods to keep the numerical solution positivity-preserve without changing a large amount of the existing program code.In this paper,three kinds of repair techniques are given for the corresponding finite volume element solution to keep positive which are simple repair,local repair and global repair.Among them,simple repair technique is to set the value of the negative numerical solution to zero before the next nonlinear iteration.This method is simple and easy,but the energy conservation is broken.Local repair technique distributes the value of the negative numerical solution to the adjacent positive numerical solution in a certain proportion.In this paper,a fast query method is designed to keep the energy conservation for local repair technique which called distance matrix.Global repair technique is to use the energy represented by the value of the positive numerical solution to absorb the negative energy represented by the value of all the negative numerical solutions.We present a large number of numerical experiments to verify the numerical performance of the three repair techniques and compare their advantages and disadvantages.