Numerical Methods For Degenerate Nonlinear Parabolic Equations

When solving the degenerate nonlinear parabolic equation,some classical methods,such as finite element method,finite volume element method and discontinuous Glerkin method,will produce the so-called non-physical oscillation phenomenon in which the numerical solution exceeds the limit of physical quantity.Meanwhile,in order to solve the nonlinear problem effectively,nonlinear iteration has to be used,which leads to a very large amount of computation.In addition,due to coefficient degradation,some solutions will have the numerical thermal barrier phenomenon that the numerical solution cannot effectively propagate forward with time.For the non-physical oscillation phenomenon,this paper takes the finite element method as an example and proposes three repairing methods based on posterior correction without changing the existing program.The first method is the simple repair method that simply sets the value of all negative points to zero before each iteration step.The second method is the global repair method,which redistributes all the negative energy to all the nodes with positive energy at one time before each time step iteration.The third method is the local repair method,in which the energy of the node whose finite element solution is negative is distributed to the node whose function value is positive in a certain proportion before each iteration step.Numerical examples show that the finite element solution with these methods is positive.In this paper,a new method for nonlinear iterative problems is proposed,which transforms the original problem into a set of differential equations composed of two equations,one of which is a parabolic equation with respect to the vector-valued intermediate variable and the other is a hyperbolic equation with respect to the original variable,by introducing an auxiliary vector-valued intermediate variable.In this paper,an implicit difference method is designed to solve the first equation and the explicit equation is used to solve the second equation.In the actual calculation,the parabolic equation is implicitly solved only once at each time step,and the hyperbolic equation is explicitly solved once.This effectively avoids nonlinear iterations.It is worth noting that when solving the parabolic equations in the system of equations,when the approximation of the diffusion coefficient is taken as the harmonic mean of the unknown,the so-called“numerical thermal barrier”phenomenon will occur,so the arithmetic mean is used in the actual calculation to avoid “numerical thermal barrier”.Numerical experiments show that the new method can avoid both non-linear iteration and“numerical thermal barrier”,thus greatly improving the calculation efficiency.