1、Vo1.52,No.3DOI:10.3969/J.ISSN.1000-5137.2023.03.002Space-time spectral collocation method forJournal of Shanghai Normal University(Natural Sciences)Allen-Cahn equationJun.,2023JIA Mengshu,XIE Shanshan,JIAO Yujian*(Mathematics and Science College,Shanghai Normal University,Shanghai 200234,China)Abstr
2、act:Allen-Cahn equation is an important phase field model,which has been widely used to in-vestigate interfacial dynamic problems.In this paper,we propose a Legendre-Gauss spectral collocationscheme for the Allen-Cahn equation by using Legendre-Gauss-Lobatto nodes as the collocation pointsin both th
3、e time and the space directions.We use the fixed point iteration method to solve the resultednonlinear system.Ample numerical results demonstrate the efficiency of our new method.Key words:Allen-Cahn equation;nonlinear system;space-time spectral collocation methodCLC number:O 241.82Document code:AAr
4、ticle ID:1000-5137(2023)03-0285-10Allen-Cahn方程的时空谱配置法贾孟淑，谢姗姗，焦裕建(上海师范大学数理学院，上海2 0 0 2 34)摘要：Allen-Cahn方程是重要的相场模型，在界面动力学问题研究中得到广泛应用在时间和空间方向上使用Legendre-Gauss-Lobatto结点构造了Allen-Cahn方程的谱配置格式，并使用不动点选代法求解所得非线性系统。丰富的数值算例验证了新算法的有效性。关键词：Allen-Cahn方程；非线性问题；谱配置法Received date:2022-10-14Foundation item:The Natio
5、nal Natural Science Foundation of China(12271365,11771299,12171141);The Natural Sci-ence Foundation of Shanghai(22ZR1445400,20JC1413800)Biography:JIA Mengshu(1997-),female,graduate student,research area:numerical methods for partial differential equa-tions.E-mail:jia_*Corresponding author:JIAO Yujia
6、n(1968),male,professor,research area:numerical methods for partial differentialequations.E-mail:yj-jiao 引用格式：贾孟淑，谢姗姗，焦裕建.Allen-Cahn方程的时空谱配置法J.上海师范大学学报（自然科学版)，2 0 2 3,52(3)：285-294.Citation format:JIA M S,XIE S S,JIAO Y J.Space-time spectral collocation method for Allen-Cahn equation JJ.Journalof Sha
7、nghai Normal University(Natural Sciences),2023,52(3):285-294.2861IntroductionJ.Shanghai Normal Univ.(Nat.Sci.)Jun.2023Let A=a,b a,b.The perturbation factor e O is a parameter that represents the thickness of the interface,and F(u)=(ua2-1)is a double well potential.The two-dimensioal llen-Cahn equati
8、on s of the fllwingforml1:(u(s,)=2 u(s,)-(u(s,),s E(O,T),(c,y)E A,u(s,y)=0,(u(o,c,y)=uo(c,y),The Allen-Cahn equation was introduced by Allen and Cahn 2 for describing the inverted phase boundary mo-tion in crystals in 1979.Since then,it has been widely used to describe hydrodynamic problems and reac
9、tion diffu-sion problems,such as difusion phenomenon to describe the competition and exclusion of biological populations3,the migration process of riverbedsl4,and the interfacial dynamics in material science5.Therefore,the numericalmethods for solving the Allen-Cahn equation is of great significance
10、 in many areas of application.There are many scholars who have studied the two dimensional Allen-Cahn equation numerically.Some au-thors considered finite difference method in space for the two-dimensional Allen-Cahn equation6.A number ofresearchers proposed finite element methods for solving the tw
11、o-dimensional Allen-Cahn equation1,7.Many au-thors proposed spectral methods to solve the Allen-Cahn equation numerically8.The purpose of this paper is to propose a space-time collocation method on the Legendre-Guass type points forsolving the two dimensional Allen-Cahn equation.We use the Legendre-
12、Gauss-Lobatto points in directions of bothtime and space to construct the scheme.This brings great convenience to deal with the nonlinear term,and gives thespectral accurracy both in space and in time.s E(0,T),(,y)EO,(,y)E.(1)Nun(z)=Zijp;(2),j=02PreliminariesLet I=-1,1,Ln(z)be the Legendre polynomia
13、ls of degree n.For any non-negative integer N,the set(zi)-o are the Legendre-Gauss type points,and(0;)=o are the associated Lagrange basis polynomials.LetP(I)stand for the set of all algebraic polynomials of degree at most N.Obviously,for any un(z)e Pn(I),where a;=un(zj).For notational convenience,w
14、e denotedudzw(2)=(z z0)(z-z1).(z-z)=c(z2-1)0,L(z).dudz2LetVol.52,No.3According to the property of the Legendre polynomial9,we getNext,werecll someresults on pseudosectraldfferentalmatrix Let,=(d)i)1,(i(z)tbe the first-order dfferential matrix with entries104LN(2t)(2-zt)LN(z1)G1,z0,N(N+1)4Furthermore
15、,the mth-order diferential matrix Dkm):=(d,the first-order differential matrix,namely,JIA M S,XIE S S,JIAO Y J:Space-time spectral collocation method for Allen-Cahn287w(之)(z2-1)0,Ln(z)(z-2)0w(2)=N(N+1)(2-2i)L(2z)N(N+1)1I=lN-1,t=l=N.)oI.l/N=(om o(e)oI.ln is the m power oft=1=0,(2)N,zMoreover,(m）(2)=a
16、mun(z)=ijomd;(2).WNThe above equality can be expressed in the form of matrix-vector multiplication asm1.Nj=0U(m)=Dm)Un,m 1,WhereU(m)N,zuN),uN(21),um)(n),n=)(a,n).3 Space-time spectral collocation method for the Allen-Cahn equationLet a=-1,b=1 be in the definition of the domain A.With a slight abuse of notation,we also denoteA=-1,1 -1,1.Denoting f(u):=F(u)=u3-u as the nonlinear term in the Allen-Cahn equation(1),andtaking transformation t=s-1 for s e 0,T,then equations(1)is transformed to2u(t,c,y