Picard-Newton Iterative Method For Nonlinear Degenerate Parabolic Equations

Degenerate nonlinear parabolic equations are often used to describe many practical problems in science and engineering,such as porous media problems,seepage problems,phase transition problems,fluid diffusion problems,etc.Nonlinear degenerate parabolic equations have the characteristics of parabolic equations and hyperbolic equations.The true solution will change with the passage of time,the phenomenon of the mutation of the wave head interface,which brings great challenges to theoretical research.In terms of numerical calculations,the usual central methods,such as the standard finite volume method(FVM),the mixed finite element method,etc.Producing a so-called”numerical thermal barrier” phenomenon in which the wavefront of the numerical solution cannot propagate forward effectively.The main reason is that the inverse or harmonic mean approximation of the diffusion coefficient(or heat transfer coefficient)is used in these methods.The diffusion coefficient is equal to zero or tends to zero in some regions,resulting in a diffusion coefficient in the numerical format that is approximately zero,so non-diffusion.In addition,in order to ensure the convergence of the nonlinear iterative process,the Picard iteration method is often used,and this method generally converges slowly.In order to avoid the ”numerical thermal barrier” and low computational efficiency of the standard finite volume method in simulating nonlinear degenerate parabolic equations,this paper starts with the model equations and studies a variety of finite volume formats based on iterative methods,including:format ? : FV format based on the standard Picard iteration;format ? : FV format based on the modified Picard iteration;format ? : FV format based on the standard Picard-Newton iteration;format ? : FV format of Picard-Newton iteration based on diffusion term correction;format V : FV format of Picard-Newton iteration based on Newton term correction;format ? : FV format of Picard-Newton iteration based on diffusion term and Newton term;The advantages and disadvantages of these formats are compared through a large number of numerical experiments.The numerical results show that the modified diffusion term can avoid the ”numerical thermal barrier”.The introduction of Newton term can acceleratethe nonlinear iterative process.Therefore,from the perspective of calculation effect and iteration step,the format VI is the best.These studies are useful for efficient and stable numerical solution of nonlinear degenerate parabolic equations.