Research On Adaptive Adding Point Scheme For Singular Solutions Of Parabolic Equations

When solving the numerical solution of a partial differential equation,it is always required to discretize the partial differential equation into a difference equation and solve the algebraic equation on the discrete grid points.Traditional fixed isometric discrete meshes have many limitations,when the solution changes too fast in a small area,the computer will not be able to capture the behavior of the solution well.For example,when calculating the numerical approximation of the blowup problems,there is a significant difference between the continuous solution and the numerical solution.If the overall reduction of the grid is used to overcome this defect,the amount of calculation will be greatly increased.This paper introduces an adaptive adding point method,which automatically adjusts the density of the grid points,a large reduction in calculations and improved calculation accuracy.Firstly,for the semi-linear heat equation with Dirichlet boundary condition u_t=u_xx+u^p and the semi-linear parabolic equation with Neumann boundary condition u_t=u_xx-au^p,under the space semi-discrete condition,the integral formula satisfied by the equation is calculated from the mass lumping method and basis function property.According to the formula satisfied by the blowup node,the adding point condition is derived.The specific steps of the adaptive mesh adding method are introduced in detail,and the specific parameters needed to repair the gradual progress are given.Secondly,using scale transformation,two-grid,finite difference,Newton interpolation method,it is proved that the local convergence degree of the numerical solution can reach the 4/p order with assumption u?C^2,1.And it is also proved that the numerical blowup is at single point,that is,the numerical blowup set is B?u?=1?bounded before blowup?.The convergence of the numerical blowup time is proved by the comparison principle and the upper and lower bounds.Finally,the numerical comparison between the adaptive adding point method with the traditional fixed grid method and the error estimation of the adaptive grid addition method are given.Numerical examples show that the results obtained by adaptive adding point method are more accurate.The error estimate show that the adaptive adding point method can improve the local convergence order of the numerical solution.