# Linear Barycentric Rational Method for Solving Schrodinger Equation

Authors: Peichen Zhao; Yongling Cheng
Journal: Journal of Mathematics
 (2021)

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## Abstract

A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.


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## Body

## 1. Introduction

Schrodinger equation (SDE) is widely used in atomic physics, nuclear physics and solid physics, quantum mechanics, and so on. SDE is only applicable to nonrelativistic particles with low velocity, and there is no description of particle spin. In this paper, we are concerned with solving the numerical solution of the SDE:(1)ih∂φx,t∂t=−h22m∂2φx,t∂x2+Vx,tφx,t+fx,t,(2)φx,0=g1x,φx,t=g2x,x∈a,b,(3)φa,t=h1t,φb,t=h2t,0<t<T,where h is reduced Planck constant and m denotes quality. In [1], the fractional Schrodinger–Choquard equation with blow-up criteria and instability of normalized standing waves is studied. In [2], the finite-difference time-domain (FDTD) method is studied to solve SDE. In [3], nonlinear magnetic Schrodinger–Poisson type equation is studied. In [4], high-order multiscale discontinuous Galerkin method for one-dimensional stationary SDEs with oscillating solutions is presented. In [5], sixth-order nonlinear SDE is concerned by factorization formula and an analytical method. In [6], nonlinear SDEs are solved by the iterative method. In [7], the two-dimensional Klein–Gordon SDEs are solved by linear compact alternating direction implicit (CADI) scheme.For getting the equidistant node of the barycentric formula, Floater [8–10] has proposed a reasonable interpolation method; in particular, equidistant distribution nodes and the quasi-equidistant nodes have high numerical stability and accuracy of interpolation [11, 12]. In [13, 14], the linear barycentric rational collocation method (LBRCM) have been used to solve the integro-differential equation. Wang et al. [15–17] have expanded the application fields of the collocation method (CM), such as initial value problems, plane elasticity problems, and nonlinear problems. LBRCM for solving heat conduction equation and biharmonic equation are studied in [18, 19].In this paper, a LBRCM for solving SDE is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method that is easy to program is obtained. The convergence rate of the LBRC method for solving the telegraph equation is proved from the convergence rate of linear barycentric rational interpolation (LBRI). Finally, a numerical example verifies the correctness of the theoretical analysis.The remaining of this paper is planned as follows. Section2 presents the differentiation matrices, CM for SDE, and the matrix form of CM. In Section 3, the convergence rate is proved. Finally, a numerical example verifies the theoretical analysis.


## 2. Differentiation Matrices of SDE

We partition the intervala,b and 0,T into a=x0<x1<⋯<xm=b and 0=t0<t1<⋯<tn=T with hi=xi−xi−1,i=0,1,…,m, and τj=tj−tj−1,j=0,1,…,n, for the uniform partition with h=b−a/m and τ=T/n. For Ω=a,b×0,T with xi,tj,i=0,1,…,mandj=0,1,…,n will be the uniform partition.Consider the barycentric interpolation function (BIF) as(4)φxi,t=φxi,i=0,1,…,m,and its barycentric interpolation approximation is(5)φx,t=∑i=0m∑j=0nrixrjtφij,where(6)rix=wi/x−xi∑k=0nwk/x−xk,wi=∑k1∈Ji−1i−d1∏h1=k1,h1≠ik1+d11xi−xh1,where Ji=k1∈Im:i−d1≤k1≤i,Im=0.,…,m−d1, and(7)rjt=wj/t−tj∑k=0nwk/t−tk,wj=∑k2∈Jj−1j−d2∏h2=k2,h2≠jk2+d21tj−th2,where Jj=k2∈In:j−d2≤k2≤j,In=0.⋯,n−d2 is the basis function, and φij is the value at point φxi,tj. Combining equations (1) and (5), we obtain(8)iIm⊗D1φ0⋮φm+C2⊗Inφ0⋮φm−Inφ0⋮φm=f0⋮fm,and then, we have(9)iIm⊗D1+C2⊗In−Im⊗InΦ=F,and(10)LΦ=F,where(11)L=iIm⊗D1+C2⊗In−Im⊗In,and(12)Dij1=wj/witi−tj,i≠j,−∑k≠iDik1,i=j,and(13)Cij1=wj/wixi−xj,i≠j,−∑k≠iCik1,i=j,Cij2=2Cij1Dii1−1xi−xj,i≠j,−∑k≠iCik2,i=j,and ⊗ is Kronecher product of matrix. In the following, we define the Kronecher product of matrix A=aijm×n and B=bijk×l as(14)A⊗B=aijBm⋅k×n⋅l,where(15)aijB=aijb11aijb12⋯aijb1laijb21aijb22⋯aijb2l⋮⋮⋮⋮aijbk1aijbk2⋯aijbkl.


## 3. Convergence Rate and Error Analysis

The barycentric rational interpolants of function (BRIF)ϕx with rx and its error convergence rate is(16)ex≔φx−rnx=x−xi⋯x−xi+dφxi,xi+1,…xi+d,x,and(17)ex=∑i=0n−dλixφx−rnx∑i=0n−dλix=AxBx=Ohd+1,where(18)Ax≔∑i=0n−d−1iφxi,xi+1,…xi+d,x,and(19)Bx=∑i=0n−dλix,where(20)λix=−1ix−xi⋯x−xi+d.The following Lemma was proved by Jean-Pau Berrut in [11].Lemma 1.
(see [11]), For ex defined in (16), we have(21)ex≤Chd+1,φ∈Cd+2a,b,e′x≤Chd,φ∈Cd+3a,b,e″x≤Chd−1,φ∈Cd+4a,b,d≥1.
For the BRIFφx,t with rx,t, we can get the barycentric rational interpolation (BRI):(22)rnx,t=∑i=0m∑j=0nwij/x−xit−tjφij∑i=0m∑j=0nwij/x−xit−tj,where(23)wij=∑k1∈Ji∑k2∈Jj−1i−d1+j−d2∏h1=k1,h1≠ik1+d11xi−xh1∏h2=k2,h2≠jk2+d21tj−th2,and Ji=k1∈Im:i−d1≤k1≤i,Im=0.…,m−d1,(24)Jj=k2∈In:j−d2≤k2≤j,In=0.…,n−d2.
By the error term of Newton–Cotes rule for two-dimensional function, we have(25)ex,t≔φx,t−rx,t=φx,t−rnx,t=x−xi⋯x−xi+d1φxi,xi+1,⋯xi+d1,x+t−tj⋯t−tj+d2φtj,tj+1,⋯tj+d2,t.
The following theorem has been proved in reference by Li in [18].Theorem 1.
Forex,tdefined in (25) andφx,t∈Cd1+2a,b×Cd2+20,T, we have(26)ex,t≤Chd1+1+τd2+1.Corollary 1.
Forex,tdefined in (25),(27)exx,t≤Chd1+τd2+1,φx,t∈Cd1+3a,b×Cd2+20,T,etx,t≤Chd1+1+τd2,φx,t∈Cd1+2a,b×Cd2+30,T,exxx,t≤Chd1−1+τd2+1,φx,t∈Cd1+4a,b×Cd2+20,T,d1≥1.
This corollary can be obtained similarly as Theorem1, where we omit it.
Letφx,t be the solution of (1) and φxm,tn be the numerical solution; then, we have(28)Dφxm,tn=fx,t,and(29)limm,n⟶∞Dφxm,tn=fx,t.
According to the above lemma, the following theorem can be proved.Theorem 2.
Letφxm,tn:Dφxm,tn=fx,tandfx∈Ca,b; we have(30)φx,t−φxm,tn≤Chd1−1+τd2.Proof.
As(31)Dφx,t−Dφxm,tn=iφtx,t−φx,t+iφxxx,t−iφtxm,tn−φxm,tn+iφxxxm,tn=iφtx,t−φtxm,tn+−φx,t+φxm,tn+iφxxx,t−φxxxm,tn≔iR1x,t+R2x,t+iR3x,t,we have(32)R1x,t=φtx,t−φtxm,tn,R2x,t=φx,t−φxm,tn,R3x,t=φxxx,t−φxxxm,tn.
As, forR3x,t, we have(33)R3x,t=φxxx,t−φxxxm,tn=φxxx,t−φxxxm,t+φxxxm,t−φxxxm,tn=∑i=0m−d1−1iφxxxi,xi+1,…,xi+d1,x,t∑i=0m−d1λix+∑j=0n−d2−1jφxxtj,tj+1,…,tj+d2,xm,t∑j=0n−d2λjt=exxx,tn+exxxm,tn.
By the corollary, we obtain(34)R3x,t≤exxx,tn+exxxm,tn≤Chd1+1+τd2+1.
Similarly, forR2x,t and R1x,t, we have(35)R1x,t=φtx,t−φtxm,tn=etx,tn−etxm,tn,(36)R1x,t≤etx,tn+etxm,tn≤Chd1+1+τd2,and(37)R2x,t=βφx,t−φxm,tn≤Chd1+1+τd2+1.
Combining the identity equations (31), (34), (36), and (37), the conclusion of theorem is obtained.


## 4. Numerical Examples

Example 1.
The SDEa=0,b=1,t=2,and(38)fx,t=0,under condition g1x=2sinπx; the analysis solution is(39)φx,t=2e−iπ2t/2sinπx.
Tables1 and 2 show the errors of the LBRCM for equidistant nodes of space variables and time variables.
Tables3 and 4 show the errors of the LBRCM for quasi-equidistant nodes of space variables and time variables.Table 1
Convergence rate of equidistant nodes with differentd1=7 and t=2.
nd2=2d2=3d2=4d2=58×86.1237e− 018.7824e− 017.4433e− 011.1222e+ 0016×161.7534e− 011.80421.0471e− 013.06823.0341e− 024.61664.9307e− 024.508432×323.3624e− 022.38266.6239e− 033.98262.3415e− 033.69577.0714e− 046.123664×646.0006e− 032.48634.5724e− 043.85671.1369e− 044.36431.0298e− 056.1016Table 2
Convergence rate of equidistant nodes with differentd2=7 and t=2.
nd1=2d1=3d1=4d1=58×81.67711.67161.67951.678216×161.5313e− 026.77511.6007e− 026.70641.6217e− 026.69441.6091e− 026.704532×321.4665e− 033.38437.1109e− 057.81448.4703e− 057.58097.2416e− 057.795764×641.8040e− 043.02312.9783e− 064.57756.9138e− 076.93682.4334e− 078.2172Table 3
Convergence rate of quasi-equidistant nodes with differentd1=7 and t=2.
nd2=2d2=3d2=4d2=58×85.1334e− 013.7631e− 011.1156e− 011.2259e− 0116×162.8133e− 024.18961.1572e− 025.02324.8063e− 034.53682.5873e− 035.566332×322.8913e− 033.28255.4800e− 044.40031.4573e− 045.04352.1084e− 056.939264×643.2935e− 043.13403.0820e− 054.15234.1594e− 065.13082.5797e− 076.3528Table 4
Convergence rate of quasi-equidistant nodes with differentd2=7 and t=2.
nd1=2d1=3d1=4d1=58×82.8707e− 012.8566e− 012.8297e− 012.8323e− 0116×161.0263e− 024.80595.7980e− 048.94455.0888e− 049.11915.0644e− 049.127432×326.1791e− 044.05392.9283e− 067.62931.1254e− 068.82071.0428e− 068.923764×647.9511e− 052.95821.3835e− 061.08171.4398e− 06—4.3847e− 06—Example 2.
The SDEa=0,b=1,t=1,and(40)fx,t=x2−2eiπ2t/4cosπx2+2πsinπx2,under condition g1x=x2cosπx/2; the analysis solution is(41)φx,t=e−tsinx.
In Figures1 and 2, the error estimate of equidistant and quasi-equidistant nodes with t=2,m=n=19,andd1=d2=8 is presented. It can be seen from Figure 2 that the barycentric rational interpolation collocation method has higher accuracy in both quasi-equidistant and equidistant nodes conditions.
Tables5 and 6 show the errors of the LBRCM for equidistant nodes of space variables and time variables.
Tables7 and 8 show the errors of the LBRCM for quasi-equidistant nodes of space variables and time variables.Figure 1
Error estimate of equidistant nodes witht=2,m=n=19,andd1=d2=8.Figure 2
Error estimate of quasi-equidistant nodes witht=2,m=n=19,andd1=d2=8.Table 5
Convergence rate of equidistant nodes with differentd1=7 and t=1.
nd2=2d2=3d2=4d2=58×81.7995e− 034.6557e− 041.5050e− 041.4417e− 0416×165.5795e− 041.68946.5769e− 052.82351.1394e− 053.72351.5421e− 066.546732×321.1570e− 042.26985.2988e− 063.63374.9129e− 074.53552.8309e− 085.767564×642.1482e− 052.42914.7396e− 073.48282.3083e− 084.41175.7039e− 105.6332Table 6
Convergence rate of equidistant nodes with differentd2=7 and t=1.
nd1=2d1=3d1=4d1=58×81.6168e− 035.1505e− 041.7042e− 049.0483e− 0516×161.9952e− 043.01852.1403e− 054.58888.2906e− 064.36151.2726e− 066.151832×322.5369e− 052.97541.2042e− 064.15162.4768e− 075.06491.4595e− 086.446164×643.4030e− 062.89827.6096e− 083.98428.0774e− 094.93843.4093e− 105.4198Table 7
Convergence rate of quasi-equidistant nodes with differentd1=7 and t=1.
nd2=2d2=3d2=4d2=58×82.1572e− 032.1324e− 042.9588e− 051.1861e− 0516×161.7804e− 043.59896.5585e− 065.02309.5692e− 074.95057.8014e− 087.248232×321.2864e− 053.79074.1551e− 073.98043.2481e− 084.88071.0198e− 096.257464×641.3676e− 063.23362.2121e− 084.23148.0913e− 092.00512.4516e− 08-Table 8
Convergence rate of quasi-equidistant nodes with differentd2=7 and t=1.
nd1=2d1=3d1=4d1=58×82.1937e− 035.6367e− 047.8737e− 056.3534e− 0616×161.6769e− 043.70943.6712e− 067.26255.9101e− 077.05778.7629e− 086.180032×321.4478e− 053.53391.2449e− 074.88211.2882e− −085.51984.5441e− 107.591364×647.0525e− 061.03766.1937e− 07—2.2631e− 07—1.0710e− 06—


## 5. Conclusion

In this paper, the LBRCM have been constructed to solve SDE, while the time variable and space variable are obtained at the same time. Numerical solution confirms the theorem analysis.



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