A Class Of Intrinsic Parallel Difference Methods For Solving Allen-Cahn Equations

Allen-Cahn equation was proposed by Allen and Cahn in 1979 to describe the motion of curved antiphase boundary in microscopic diffusion theory.It has been widely used in fluid dynamics and reaction-diffusion problems of materials.The research of Allen-Cahn equation has been focused by scholars since it was proposed.In recent years,two-dimensional Allen-Cahn equation has become a new research direction.In this thesis,two implicit parallel difference methods are used to solve Allen-Cahn equation.Respectively,one-dimensional Allen-Cahn equation was solved by Alternating segment Crank-Nicolson algorithm(ASC-N),and two-dimensional Allen-Cahn equation was solved by Alternating Band Crank-Nicolson algorithm(ABd C-N).It is well known that explicit difference method can be used for parallel computation,but the accuracy is poor and the condition is stable,while implicit difference method is not suitable for parallel computation.The two numerical methods mentioned in this thesis can solve the above limitations,with high accuracy,and the two numerical methods are parallel algorithms,which can accelerate the calculation speed of the equation.In 1992,Evance et al.proved that Allen-Cahn equation has two intrinsic properties,the maximum principle and energy stability.In this thesis,it is proved that the discrete numerical solution obtained by ASC-N scheme satisfies the maximum principle and energy stability in solving one-dimensional Allen-Cahn equation.The research is mainly divided into the following parts:1.The one-dimensional Allen-Cahn equation under Neumann boundary conditions was solved by Alternating segment Crank-Nicolson(ASC-N)algorithm.First,we calculated CrankNicolson format and four Saul’yev asymmetric formats,and then combined them skillfully to get the ASC-N format.As a kind of intrinsic parallel difference method,ASC-N algorithm can obtain the unique numerical solution.The numerical solutions obtained by ASC-N algorithm have second-order precision in space and second-order precision in time(approximate secondorder precision at the inner boundary points).At the same time,this difference method has obvious parallel property and is suitable for all kinds of parallel computing systems.It is proved that the solution of ASC-N scheme can satisfy two intrinsic properties of Allen-Cahn equation,the maximum principle and discrete energy stability,and the establishment condition of the maximum principle is not limited by the time-space step ratio.The numerical and analytical solutions are compared in numerical experiments,and it is verified that the scheme solutions have spatial and temporal second-order precision.Moreover,When the grid is finely divided,using ASC-N method to solve one-dimensional Allen-Cahn equation with Neumann boundary conditions can effectively improve the solving speed of the equation.It can be seen from the three-dimensional image of the numerical solution that this numerical solution obviously satisfies the maximum principle.2.The Allen-Cahn equation with Dirichlet boundary conditions was solved by the Alternate Band Crank-Nicolson(ABd C-N)algorithm,which matrix form was composed of eight Saul’yev schemes combined with a Crank-Nicolson scheme.In this chapter,eight Saul’yev schemes and a Crank-Nicolson scheme for the two-dimensional Allen-Cahn equation were obtained,and then ABd C-N schemes were obtained by proper combination.ABd C-N scheme is an implicit difference scheme with stability.And it’s also a parallel algorithm which can be used on parallel machines.Numerical experiments show that the numerical solution can be obtained by this method,and the maximum value of the numerical solution is less than 1.