Numerical Analysis Of Singular Solutions For A Class Of Degenerate Parabolic Equations

Because parabolic partial differential equations have a wide range of applications in physics,mechanics and engineering technology,the study of its theories and numerical algorithms have always been a hot topic for many mathematicians.For parabolic partial differential equations with Neumann boundary conditions,numerical algorithm research is also one of the research topics.Therefore,the research contents of this thesis are as follows:Firstly,the asymptotic behavior of numerical solutions of semilinear parabolic problems with singular Neumann boundary conditions and singular reaction terms is studied by using the finite-difference method with fully discrete scheme.Under the certain assumptions of the initial data,it is proved that the numerical quenching rate is consistent with the continuous quenching rate,and the numerical quenching time converges to the continuous quenching time.The correctness of the theoretical results is verified by some numerical experiments.Secondly,in this thesis,the asymptotic behavior of numerical solutions of quasilinear parabolic problems with singular Neumann boundary conditions and singular reaction terms is studied by using a semidiscrete finite element method.It is proved that the numerical quenching rate is consistent with the continuous quenching rate,and the numerical quenching time converges to the continuous quenching time.And we give numerical experiments to verify the correctness of the theoretical results.In short,the asymptotic behavior of the numerical solution is analyzed rigidly theoretically by two numerical methods to discretize the two problems,and the behavior of the understood solution is simulated perfectly numerically,which has a certain guiding significance for the application of these two problems in physics,mechanics and engineering technology.