空间分数阶偏微分方程非标准有限差分方法的稳定性和收敛性.pdf

应用数学MATHEMATICA APPLICATA2024,37(1):159-170Stability and Convergence ofNon-standard Finite DifferenceMethod for Space Fractional PartialDifferential EquationWANG Qi(王琦),LIU Ziting(刘子婷)(School of Mathematics and Statistics,Guangdong University of Technology,Guangzhou 510006,China)Abstract:In this paper the numerical solution of the space fractional partial differentialequation is derived by means of a non-standard finite difference method,and somecorresponding numerical studies are investigated.For the two space fractional derivatives,we adopt the Gr unwald-Letnikov formula and the shift Gr unwald-Letnikov formula todiscretize them,respectively.The non-standard finite difference scheme is constructedby denominator function with time and spacial steps.Furthermore,the stability and theconvergence of this scheme are studied by the method of von Neumann analysis.Some newresults are given.Numerical examples confirm that this scheme is effective for solving thespace fractional partial differential equation.Key words:Space fractional partial differential equation;Non-standard finite differencemethod;Stability;ConvergenceCLC Number:O242.21AMS(2010)Subject Classification:65M06;65M12Document code:AArticle ID:1001-9847(2024)01-0159-121.IntroductionFractional derivatives can effectively describe natural processes with memory and geneticcharacteristics because of their non-locality property.In recent decades,fractional partial dif-ferential equations have been widely used in many fields,such as control engineering,physicalmechanics,biology and computer science14.Space fractional partial differential equations(SFPDEs)are often used to model super-diffusion,where a particle plume spreads faster thanpredicted by the classical Brownian motion model.It is not easy to evaluate the fractionalderivative for most functions.Because of the analytic solutions of most of SFPDEs cannot beobtained,so it becomes important to develop numerical methods for this type of equations.CHEN et al.5considered a SFPDEs with fractional diffusion and integer advection term,afully discrete scheme was obtained by combining the Legendre spectral with Crank-Nicolsonmethods.They further proved that the scheme is unconditionally stable and convergent.Received date:2022-12-12Foundation item:Supported by the National Natural Science Foundation of China(11201084)Biography:WANG Qi,male,Han,Heilongjiang,professor,major in numerical analysis of differentialequation.160MATHEMATICA APPLICATA2024ZHAO and WANG6developed a finite difference method(FDM)for space-time fractionalpartial differential equations in three dimensions with a combination of Dirichlet and frac-tional Neumann boundary conditions,by analyzing the structure of the stiffness matrix innumerical discretization as well as the coupling in the time direction,a fast FDM was ob-tained.Saw and Kumar7combined the Chebyshev collocation method with FDM to studyone-dimensional SFPDEs,they attenuated the equations to a system of differential equationsand solved them by iterative method.Bansu and Kumar8established a meshless approach forsolving the space and time fractional telegraph equation,and the convergence of the schemewas discussed.Based on fourth-order matrix transfer technique for spatial discretization andfourth-order exponential time differencing Runge-Kutta for temporal discretization,Alzahraniet al.9developed two high-order methods for space-fractional reaction-diffusion equations.Owolabi10proposed an adaptable difference method for the approximation of the derivativesin fractional-order reaction-diffusion equations,the maximum error and relative error wereboth analyzed.Bolin et al.11studied a standard finite element method to discretize fractionalelliptic stochastic partial differential equations,the strong mean-square error was analyzedand an explicit rate of convergence was derived.Iyiola et al.12provided a novel exponentialtime differencing method for nonlinear Riesz space fractional reaction-diffusion equation withhomogeneous Dirichlet boundary condition,the numerical stability was examined.Takeuchiet al.13described a second-order accurate finite difference scheme for SFPDEs,it is provedthat the difference scheme is second-order convergent,and the conditions for the stabilityof the scheme were given.Recently,a spectral method based on the use of the temporaldiscretization by the Jacobi polynomials and the spatial discretization by the Legendre poly-nomials was proposed by ZHANG et al.14for the numerical solution of time-space-fractionalFokker-Planck initial-boundary value problem.Later,semi-implicit spectral approximationsfor nonlinear Caputo time-and Riesz space-fractional diffusion equations were investigatedin 15,the unconditional stability and the convergence of the fully discrete schemes wereproved.Recently,Zhang and Lv16introduced a kind of regularization method to get rid ofthe ill-posedness for a space-fractional diffusion problem,the convergence and stability of themethod were discussed.In the late 1980s,Mickens1718proposed a numerical method for solving differentialequations:the non-standard finite difference(NSFD)method.The NSFD method has thepotential to be dynamically consistent with the properties of the continuous model.Using thismethod,spurious solutions and numerical instabilities can be removed,and correct numericalsolutions are qualitatively achieved for every time-step size.The NSFD method has alreadybeen used in the numerical simulation of fractional differential equations1923.Differentfrom the above papers,we will study the stability and convergence of NSFD method for theSFPDEs with two fractional derivatives,the corresponding results are given.In this paper,we consider the following SFPDEsu(x,t)t=v(x)u(x,t)x+w(x)u(x,t)x+f(x,t),0 x L,0 t T,u(x,0)=s(x),0 x L,u(0,t)=u(L,t)=0,0 0,w(x)0,s(x)and f(x,t)are all given functions,u(x,t)xandu(x,t)xareRiemann-Liouville fractional derivatives24of order (0 1)and (1 j+1,jg()0,n=j+1,jg()0 jg()1,n=j,jg()jn jg()jn+1,n j 1.(2.6)3.Stability AnalysisIn this section,we will analyze the stability of the NSFD scheme by von Neumannmethod.Lemma 3.127For 0 1,g()ksatisfiesg()0=1,g()1=,g()2 g()3 0,m 1,for 1 g()3 0,andk=0g()k=0,mk=0g()k 0,m 1.Theorem 3.1The solution of(2.4)exists and is unique.No.1WANG Qi,et al.：Stability and Convergence of Non-standard Finite Difference Method163ProofAssuming that is the eigenvalue of matrix A and X=0 is the correspondingeigenvector,that is,AX=X.Define|xl|=max|xn|:n=1,2,m 1=0.(3.1)Sincem1n=1aj,nxn=xj,som1n=1al,nxn=xl,we have=al,l+m1n=1,n=lal,nxnxl,then,the use of(2.6)leads to=al,l+m1n=1,n=lal,nxnxl=lg()0 lg()1 lg()0 xl+1xl+(ll1n=1g()ln ll1n=1g()ln+1)xnxl=l(g()0+l1n=1g()lnxnxl)l(g()0 xl+1xl+g()1+l1n=1g()ln+1xnxl)=l(g()0+l1n=1g()lnxnxl)l(g()1+l+1n=1,n=lg()ln+1xnxl).From(3.1),we have|xn/xl|1,then by Lemma 3.1 we obtainl1n=1g()ln 0andg()0+l1n=1g()ln 0,g()1+l+1n=1,n=lg()ln+1 0,g()1+l+1n=1,n=lg()ln+1xnxl 0.Since is the eigenvalue of matrix A,so(1 )/(1+)is the eigenvalue of matrix(I+A)1(I A)and|(1 )/(1+)|1.Thus the spectral radius of(I+A)1(I A)isless than one,this completes the proof.Theorem 3.2The scheme(2.4)is unconditionally von Neumann stable for 0 1and 1 0,(3.6)according to Lemma 3.1 we have g()k 0(k=0,2,3,j+1),thenjk=0g()kcos(qkh)=g()0+jk=1g()kcos(qkh)g()0+jk=1g()k=jk=0g()k 0,andj+1k=0g()kcos(q(1 k)h)=g()1+j+1k=0,k=1g()kcos(q(1 k)h)g()1+j+1k=0,k=1g()k=j+1k=0g()k 0,No.1WANG Qi,et al.：Stability and Convergence of Non-standard Finite Difference Method165therefore,(3.6)is true.Next we prove that?1 bQ1+bQ?0,so 82 4(1+2+2)0,then?1 bQ1+bQ?2=(1 (2+2)2+42(1+)2+2)2 1,therefore(3.7)holds.The proof is finished.4.Convergence AnalysisIn this section,we will consider the convergence of the NSFD method.Theorem 4.1Assume that u(xj,tn)is the solution of(1.1)at(xj,tn)and unjis thesolution of(2.4)at(xj,tn),respectively.Then the difference scheme(2.4)is unconditionallyconvergent.ProofWe first compute the order of the local truncation error.In factR=u(xj,tn+12)t+v(xj)u(xj,tn+12)x w(xj)u(xj,tn+12)x f(xj,tn+12),andu(xj,tn+12)t=u(xj,tn+1)u(xj,tn)1()+O(2),(4.1)u(xj,tn+12)x=12(2(h)jk=0g()k(u(xj kh,tn+1)+u(xj kh,tn)+O(h),(4.2)u(xj,tn+12)x=12(2(h)j+1k=0g()k(u(xj(k 1)h,tn+1)+u(xj(k 1)h,tn)+O(h).(4.3)166MATHEMATICA APPLICATA2024Using(4.1)-(4.3),we obtainR=u(xj,tn+1)u(xj,tn)1()+O(2)+v(xj)2(2(h)jk=0g()k(u(xj kh,tn+1)+u(xj kh,tn)+O(h)w(xj)2(2(h)j+1k=0g()k(u(xj(k 1)h,tn+1)+u(xj(k 1)h,tn)+O(h)f(xj,tn+12),that isR=u(xj,tn+1)u(xj,tn)1()+v(xj)2(2(h)jk=0g()k(u(xj kh,tn+1)+u(xj kh,tn)w(xj)2(2(h)j+1k=0g()k(u(xj(k 1)h,tn+1)+u(xj(k 1)h,tn)f(xj,tn+12)+O(2+h),so the order of the local truncation error is O(2+h).Next,define enj=u(xj,tn)unj,j=1,2,m 1,n=1,2,.Substituting enjinto(2.5)leads toen+1j+jjk=0g()ken+1jk jj+1k=0g()ken+1jk+1=enj jjk=0g()kenjk+jj+1k=0g()kenjk+1+1()R,(4.4)in view ofe0j=0,1 j m 1,en0=enm=0,n 1,(4.4)can be rewritten as(I+A)En+1=(I A)En+1()R,(4.5)let K=(I+A)1(I A),P=(I+A)1,then(4.5)givesEn+1=KEn+PY,(4.6)whereEn=(en1,en2,enm1)T,andY=(1()O(2+h),1()O(2+h),1()O(2+h)T.Recursion of(4.6)yieldsEn+1=(Kn+Kn1+K+I)PY,by Theorem 3.2 we have(K)1 and(P)1,further(Kn)(K)n 1,(Pn)(P)n 1,for any given we obtainKn (Kn)+1+,Pn (Pn)+1+,then we haveEn+1 (Kn+Kn1+K+I)PY (n+1)(1+)21()O(2+h),No.1WANG Qi,et al.：Stability and Convergence of Non-standard Finite Difference Method167so En+1 0 when 0 and h 0,thus|enj|0.So the difference scheme(2.4)isunconditionally convergent with order O(2+h).5.Numerical ExperimentsIn this section,we give several numerical examples to illustrate the above theoreticalresults.Consider the following SFPDEsu(x,t)t=v(x)u(x,t)x+w(x)u(x,t)x+f(x,t),0 x 1,0 t T,u(x,0)=s(x),0 x 1,u(0,t)=u(1,t)=0,0 t T,(5.1)wherev(x)=(4 )x6,w(x)=(4 )x6,s(x)=x3(1 x),f(x,t)=4x4(4 )(5 )(4 )(5 )+4)et x3et.The analytic solution of(5.1)is given by u(x,t)=etx3(1 x).00.20.40.60.81024600.020.040.060.080.10.12xtu00.20.40.60.81024600.020.040.060.080.10.12xtunFig.5.1The analytic solution(left)and the numerical solution(right)of(5.1)with1()=,2(h)=h and T=6Tab.5.1The maximum error and error rate for(5.1)with 1()=,2(h)=h,T=1=h=0.2,=1.2=0.5,=1.5=0.8,=1.8Max ErrorError rateMax ErrorError rateMax ErrorError rate1/107.2899e-035.6731e-034.2503e-031/204.0691e-031.792.9616e-031.922.2225e-031.911/402.1501e-031.891.5545e-031.911.1449e-031.941/601.4501e-031.481.0534e-031.487.6952e-041.491/801.1066e-031.317.9596e-041.325.7923e-041.331/1009.1476e-041.216.3892e-041.254.6480e-041.25Firstly,let 1()=and 2(h)=h.Fig.5.1 shows the analytic solution and thenumerical solution of(5.1)for t 0,6.It is clear that,the numerical solution is in excellentagreement with the analytic solution.At the same time,both the analytic solution and thenumerical solution are asymptotically stable.In Tab.5.1 we list the maximum error and errorrate at T=1.From this table we can see that the NSFD method is convergent with orderO(2+h).Here we should note that the numerical method in the current situation is exactly168MATHEMATICA APPLICATA202400.20.40.60.81024600.020.040.060.080.10.12xtu00.20.40.60.81024600.020.040.060.080.10.12xtunFig.5.2The analytic solution(left)and the numerical solution(right)of(5.1)with1()=1 e,2(h)=h and T=6Tab.5.2The maximum error and error rate for(5.1)with 1()=1 e,2(h)=h,T=1=h=0.2,=1.2=0.5,=1.5=0.8,=1.8Max ErrorError rateMax ErrorError rateMax ErrorError rate1/105.2758e-033.8996e-032.6844e-031/203.1539e-031.672.1035e-031.851.4187e-031.891/401.7523e-031.801.1331e-031.867.4752e-041.901/601.2042e-031.467.7435e-041.465.0337e-041.491/809.2457e-041.305.8550e-041.323.8008e-041.321/1007.4866e-041.234.7137e-041.243.0521e-041.2500.20.40.60.81024600.020.040.060.080.10.12xtu00.20.40.60.81024600.020.040.060.080.10.12xtunFig.5.3The analytic solution(left)and the numerical solution(right)of(5.1)with1()=1 e,2(h)=eh 1 and T=6Tab.5.3The maximum error and error rate for(5.1)with 1()=1 e,2(h)=eh 1,T=1=h=0.2,=1.2=0.5,=1.5=0.8,=1.8Max ErrorError rateMax ErrorError rateMax ErrorError rate1/103.9635e-032.0903e-032.0482e-031/202.3690e-031.671.1450e-031.831.0114e-032.031/401.3185e-031.806.1120e-041.875.0289e-042.011/609.1246e-041.454.1662e-041.473.3474e-041.501/806.9762e-041.313.1509e-041.322.5086e-041.331/1005.6509e-041.232.5391e-041.242.0060e-041.25No.1WANG Qi,et al.：Stability and Convergence of Non-standard Finite Difference Method169the standard FDM,so the result is known to all.Furthermore,we find that the larger thevalues of and,the smaller the maximum error.Secondly,set 1()=1 eand 2(h)=h.The analytic solution and the numericalsolution of(5.1)for t 0,6 are shown in Fig.5.2.It can be seen from this figure that thenumerical solution is asymptotically stable.Then the maximum error and error rate are givenin Tab.5.2.From this table we know that the numerical method is also convergent with orderO(2+h).From comparing Tab.5.1 with Tab.5.2,it is easy to see that the maximum erroris decreasing.Thirdly,let the time denominator function remain unchanged and the spatial denom-inator function changes based on the second case.Specifically,let 1()=1 eand2(h)=eh 1.We plot the analytic solution and the numerical solution of(5.1)in Fig.5.3.Obviously,the numerical solution is asymptotically stable.Further,we list the maximumerror and error rate in Tab.5.3 at T=1.From this table we know that the NSFD methodis also convergent with order O(2+h).Comparing the three tables,we can see that themaximum error continue to decrease,which shows that the maximum error can be reducedby changing denominator function.In summary,the choice of the denominator function is not unique,and the selection ofappropriate denominator function can reduce the maximum error.From the above figuresand tables,it can be seen that the NSFD method is convergent and preserves the stability ofthe original equation.References:1 MACIASDIAZ J E,HENDY A S,DE STAELEN R H.A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave 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