# Barycentric Rational Collocation Method for Burgers’ Equation

Authors: Qingli Zhao; Ruiwen Wang; Zhaoqing Wang; Xiaoping Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, barycentric rational collocation method is introduced to solve Burgers’ equation. The algebraic equations of the barycentric rational collocation method are presented. Numerical analysis and error estimates are established. With the help of the barycentric rational interpolation theory, the convergence rates of the barycentric rational collocation method for Burgers’ equation are proved. Numerical experiments are carried out to validate the convergence rates and show the efficiency.


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## Body

## 1. Introduction

Burgers’ equation involves the convection term, diffusion term, and kinetic viscosity coefficient whose characteristic is same as the structure of the Navier–Stokes equation without the stress term. It describes the phenomena such as dispersion in porous media, weak shock propagation, heat conduction, acoustic attenuation in fog, compressible turbulence, gas-dynamics, continuous stochastic processes, and even continuum traffic simulation. Burgers’ equation is as follows [1–6]:(1)Dux,t≔∂ux,t∂t+ux,t∂ux,t∂x−ν∂2ux,t∂x2=fx,t,x∈α,β,t>0,with boundary conditions as(2)uα,t=ϕ1t,uβ,t=ϕ2t,t>0,and initial condition as(3)ux,0=ψx,x∈α,β,where ν>0 be the kinetic viscosity. Boundary conditions sometimes are presented as periodic boundary conditions uα,t=uβ,t=0.In view of the universality of Burgers’ equation in describing lots of important physical phenomena, many numerical methods were introduced to solve it such as the finite difference method, finite element method, mixed finite element method, characteristics mixed finite element, spectral method, and meshless method; see [1–6] and the references therein.With the help of Lagrange interpolation, the barycentric rational interpolation method is obtained [7–9]. A rational interpolation scheme with equidistant and special distributed nodes has been proposed by Floater and Hormann [10]. Compared with Lagrange interpolation, the barycentric rational interpolation has the advantages of stability. Abdi et al. [11, 12] have used the barycentric rational collocation method to solve Volterra and Volterra integro-differential equation. With the further expansion of the application fields, the barycentric rational collocation method has been successfully applied to solve some initial value problems and boundary value problems by Wang et al. [13–15]. The relevant calculation results show the stability advantages and high accuracy of the barycentric rational collocation method. The research of the barycentric rational collocation method for the heat-conduction equation, biharmonic problem, second-order Volterra integro-differential equation, third-order two-point boundary value problem, beam force vibration equation, telegraph equation, and incompressible Forchheimer flow in porous media has been presented in recent papers by Li et al. [16–22]. In these papers, error estimation and numerical simulation are given.The main goal of the present paper is to solve the nonlinear Burgers’ equation with the barycentric rational collocation method.Ohd1−1+τd2 error estimates are proved. Numerical experiments are carried out to show the convergence rates. Remaining part of the paper is structured as follows. In Section 2, the barycentric rational interpolation formula is given. In Section 3, convergence analysis of the barycentric rational collocation method for the nonlinear Burgers’ equation is presented. Section 4 reports some test examples to show the accuracy, effectiveness, and efficiency.


## 2. Notations and Barycentric Rational Interpolation

Define the partition of space intervalα,β as(4)α=x0<x1<⋯<xm−1<xm=β,h=max1≤i≤mxi−xi−1.Polynomial(5)Pix=∑k=ii+d∏j=i,j≠ki+dx−xjxk−xjyk,denotes the d-order Lagrange interpolation with yk=yxk(6)Pixk=yxk,k=i,i+1,…,i+d.The barycentric interpolation functionRxd=0,1,…,m is presented as(7)Rx=∑i=0m−dμixPix∑i=0m−dμix,where μix denotes the blending function as follows:(8)μix=−1ix−xi,…,x−xi+d.According to the definition ofμix, it can be deduced that(9)∑i=0m−dμixPix=∑i=0m−d−1i∑k=ii+d∏i,j≠ki+d1xk−xjyk=∑k=0mϖkx−xkyk,where ϖk denotes the interpolation weight function as follows:(10)ϖk=∑i∈Jk∏j=i,j≠ki+d1xk−xj,Jk=i∈I;k−d≤i≤k,I=0,1,…,m−d.Through simple derivation, we know(11)∑k=ii+d∏j=i,j≠ki+dx−xjxk−xj=1,∑i=0m−dμix=∑k=0mϖkx−xk.Combining (5)–(11), the barycentric rational interpolation function Rx is presented as(12)Rx=∑j=0mϖj/x−xjyj∑j=0mϖj/x−xj=∑j=0mRjxyj,where weight function ϖj is given in (10) and interpolation basis function Rjx is defined by(13)Rjx=ϖj/x−xj∑j=0mϖj/x−xj,j=0,1,…,m.Thes-order differential function at the mesh-point xi is obtained as(14)ysxi≔yis=dsyxidxs=∑j=0mRjsxiyj=∑j=0mDijsyj,s=1,2,….Itss-order differential matrices formulation can be written into(15)ys=Dsy,s=1,2,…,where(16)ys=y0s,y1s,…,yns,s=1,2,…,(17)Dijs=Rjsxi,s=1,2,….According to definition ofRjx in (13), we get the first-order derivative of interpolation basis function Rjx as(18)Rj′xi=ϖj/ϖixi−xj,j≠i,(19)Ri′xi=−∑j≠iPj′xi.Combining equations (17)–(19) together, the s-order differential recurrence formula of Dijss=1,2,… is(20)Dijs=mDiis−1Dij1−Dijs−1xi−xj,i≠j,Diis=∑j≠iDijs.For the nonlinear Burgers’ equation withΩ=α,β×0,T, we partition the space region α,β into(21)α=x0<x1<⋯<xi<⋯<xm=β,h=max1≤i≤mxi−xi−1,and time interval 0,T into(22)0=t0<t1<⋯<tj<⋯<tn=T,τ=max1≤j≤ntj−tj−1.Functionux,t is approximated by its barycentric rational interpolation as follows:(23)ux,t≈∑j=0mRjxujt,where(24)ujt=uxj,t,j=0,1,…,m.Taking (23) into equation (1), we see(25)∑j=0mRjxu˙jt+∑j=0mRjxujt⋅∑j=0mRj′xujt−ν∑j=0mRj′′xujt=fx,t,where u˙jt is the first-order derivative of the function ujt.Takingx=xi in equation (25), we get(26)∑j=0mRjxiu˙jt+∑j=0mRjxiujt⋅∑j=0mRj′xiujt−ν∑j=0mRj′′xiujt=fxi,t.Note thatRjxi=δij; after further simplification of equation (26), we know(27)∑j=0mδiju˙jt+∑j=0mδijujt⋅∑j=0mCij1ujt−ν∑j=0mCij2ujt=fxi,t,where(28)Cij1=Rj′xi,Cij2=Rj′′xi.Combining equations (26)–(28), the matrix form is presented as(29)u˙0t⋮u˙mt+u0t⋱umtC001⋯C0m1⋮⋯⋮Cm01⋯Cmm1u0t⋮umt−νC002⋯C0m2⋮⋯⋮Cm02⋯Cmm2u0t⋮umt=f0t⋮fmt.Further, matrix equation (29) can be rewritten into a simple vector form as follows:(30)u˙t+diagutC1ut−νC2ut=ft,where(31)ut=u0t,u1t,…,umtT,u˙t=u˙0t,u˙1t,…,u˙mtT,ft=f0t,f1t,…,fmtT.Through similarly derivation, the discrete scheme of time variablet is obtained as(32)uitj=uxi,tj=uij,i=0,1,…,m,j=0,1,…,n,(33)uit=∑k=0nRktuik,i=0,1,…,m.According to equations (27)–(33), we have j=0,1,…,n(34)∑k=0nR˙ktju0k⋮∑k=0nR˙ktjumk+∑k=0nRktju0k⋱∑k=0nRktjumkC001⋯C0n1⋮⋯⋮Cm01⋯Cmn1∑k=0nRktju0k⋮∑k=0nRktjunk−νC002⋯C0n2⋮⋯⋮Cm02⋯Cmn2∑k=0nRktju0k⋮∑k=0nRktjumk=f0tj⋮fmtj.Equation (34) can be written into vector form as follows:(35)Im⊗D1U+diagUC1⊗InU−νC2⊗InU=F,which can be restated as a simple form:(36)LU=F,with(37)L=Im⊗D1−diagUC1⊗In−νC2⊗In,U=u00,u01,…,u0n,u10,u11,…,u1n,…,um0,um1,…,umnT,F=f00,f01,…,f0n,f10,f11,…,f1n,…,fm0,fm1,…,fmnT.Here, operation symbol⊗ represents the Kronecker product.Then, we get thes-order differential at the mesh-point xi as(38)usxi:=uis=dsuxidxs=∑j=0nRjsxiuj=∑j=0nCijsuj,s=1,2,….Its matrices formulation is(39)us=Csu,where(40)us=u0s,u1s,…,uns,Cijs=Rjsti.The elementRj′′xi of the second-order differentiation matrices is defined by(41)Rj′′xi=−2ϖj/ϖiti−tj∑k≠iϖk/ϖixi−xk+1xi−xj,j≠i,Ri′′xi=−∑j≠iRi′′xi.Then, we get the 1-order time differentiation matrix as follows:(42)Dij1=Rj′ti.Similarly, the 1-order and 2-order space differentiation matrices are obtained:(43)Cij1=Rj′xi,Cij2=Rj′′xi.Thes-order differential matrix recurrence formula is presented as follows:(44)Cijs=sCiis−1Cij1−Cijs−1ti−tj,i≠j,Ciis=−∑j=0,j≠inCijs.


## 3. Convergence Analysis and Error Estimates

Define the error betweenux and Rx as follows:(45)Ex=ux−Rx.According to the error theory of interpolation, it is well known that(46)Ex=x−xi,…,x−xi+duxi,xi+1,…,xi+d,x.In the light of the definition of barycentric rational interpolation functionRx, combining (46) with (45), we have(47)Ex=∑i=0n−dμixux−Pix∑i=0n−dμix=ξxηx,where(48)ξx=∑i=0n−d−1iuxi,xi+1,…,xi+d,x,ηx=∑i=0n−dμix.Define(49)Ex=maxα≤x≤βEx.The following lemma has been proved by Berrut et al. in [7].Lemma 1.
For the errorEx defined in (45), if function ux satisfies certain smoothness conditions on interval α,β, we have(50)Ex≤Chd+1,ux∈Cd+2α,β,E′x≤Chd,ux∈Cd+3α,β,E″x≤Chd−1,ux∈Cd+4α,β.Now, we research the rational interpolationRm,nx,t to approximate the function ux,t as follows:(51)Rm,nx,t=∑i=0m∑j=0nωij/x−xjt−tjuij∑i=0m∑j=0nωij/x−xjt−tj.Note that the weight functionωij is defined by(52)ωij=−1i−d1+j−d2∑k1∈J1∑k2∈J2∏s1=k1,s1≠ik1+d11xi−xs1∏s2=k2,s2≠jk2+d21tj−ts2,J1=k1∈Im:i−d1≤k1≤i,Im=0,1,…,m−d1,J2=k2∈In:j−d2≤k2≤j,In=0,1,…,n−d2.Here, parametersd1andd2 represent the space interpolation parameter and time interpolation parameter, respectively.The error functionEx,t between ux,t and Rm,nx,t is defined by(53)Ex,t=ux,t−Rm,nx,t=x−xix−xi+1,…,x−xi+d1uxi,xi+1,…,xi+d1,x,t+t−tjt−tj+1,…,t−tj+d2utj,tj+1,…,tj+d2,x,t.Based on Lemma1, we get the following theorem.Theorem 1.
For the error functionalEx,t, if ux,t∈Cd1+2α,β×Cd2+20,T, we have(54)Ex,t≤Chd1+1+τd2+1.Proof.
By equation (53), we have(55)Ex,t=ux,t−Rm,nx,t=x−xix−xi+1,…,x−xi+d1uxi,xi+1,…,xi+d1,x,t+t−tjt−tj+1,…,t−tj+d2utj,tj+1,…,tj+d2,x,t=∑i=0m−d1−1iuxi,xi+1,…,xi+d1,x,t∑i=0m−d1μix+∑j=0n−d2−1jutj,tj+1,…,tj+d2,x,t∑j=0n−d2μjt.
Note that(56)∑i=0m−d1μix≥1d1!hd1+1,(57)∑j=0n−d2μjt≥1d2!τd2+1.
Combining equations (55)–(57) together, the proof of Theorem 1 is completed.Theorem 2.
For the error functionalEx,t defined as (53), if function ux,t satisfies certain smoothness conditions on Ω=α,β×0,T, we have(58)Exx,t≤Chd1+τd2+1,ux,t∈Cd1+3α,β×Cd2+20,T,(59)Etx,t≤Chd1+1+τd2,ux,t∈Cd1+2α,β×Cd2+30,T,(60)Exxx,t≤Chd1−1+τd2+1,ux,t∈Cd1+4α,β×Cd2+20,T.Proof.
By equation (53), we know(61)Exx,t=uxx,t−dRm,nx,tdx=x−xix−xi+1,…,x−xi+d1uxxi,xi+1,…,xi+d1,x,t+t−tjt−tj+1,…,t−tj+d2uxtj,tj+1,…,tj+d2,x,t=∑i=0m−d1−1iuxxi,xi+1,…,xi+d1,x,t∑i=0m−d1μix+∑j=0n−d2−1juxtj,tj+1,…,tj+d2,x,t∑j=0n−d2μjt.
Combining equations (56), (57), and (61) together, the error estimate (58) is obtained. The proof of (59) and (60) is similar.
Letuxm,tn be the numerical solution of function ux,t as follows:(62)Duxm,tn=fx,t,(63)limm,n⟶∞Duxm,tn=fx,t.Theorem 3.
Letuxm,tn:Duxm,tn=fx,t and fx,t∈Cα,β×C0,T, we have(64)ux,t−uxm,tn≤Chd1−1+τd2.Proof.
Following equations (1) and (62), we have(65)Dux,t−Duxm,tn=utx,t+ux,tuxx,t−νuxxx,t−utxm,tn+uxm,tnuxxm,tn−νuxxxm,tn=utx,t−utxm,tn+ux,tuxx,t−uxm,tnuxxm,tn−νuxxx,t−νuxxxm,tn=A1+A2+A3,where(66)A1=utx,t−utxm,tn,A2=ux,tuxx,t−uxm,tnuxxm,tn,A3=−νuxxx,t−νuxxxm,tn.
As for the first termA1 in equation (65), we know(67)A1=utx,t−utxm,tn=utx,t−utxm,t+utxm,t−utxm,tn=∑s1=0m−d1−1s1utxs1,xs1+1,…,xs1+d1,x,t∑s1=0m−d1μs1x+∑s2=0n−d2−1s2utts2,ts2+1,…,ts2+d2,xm,t∑s2=0n−d2μs2t=A11+A12.
Then, we get(68)A1=A11+A12≤Etx,t+Etxm,t≤Chd1+1+τd2.
Considering the second termA2 of equation (65), we have(69)A2=ux,tutx,t−uxm,tnutxm,tn=ux,tutx,t−uxm,tutx,t+uxm,tutx,t−uxm,tutxm,t+uxm,tutxm,t−uxm,tutxm,tn+uxm,tutxm,tn−uxm,tnutxm,tn=A21+A22+A23+A24=∑s1=0m−d1−1s1utx,tuxs1,xs1+1,…,xs1+d1,x,t∑s1=0m−d1μs1x+∑s1=0m−d1−1s1uxm,tutxs1,xs1+1,…,xs1+d1,x,t∑s1=0m−d1μs1x+∑s2=0n−d2−1s2uxm,tutts2,ts2+1,…,ts2+d2,xm,t∑s2=0n−d2μs2t+∑s2=0n−d2−1s2utxm,tnuxs2,xs2+1,…,xs2+d1,xm,t∑s2=0n−d2μs2x.
Then, we see(70)A2=A21+A22+A23+A24≤CEx,t+Etx,t+Etxm,t+Exm,t≤Chd1+1+τd2.
To the third termA3 of equation (65), we get(71)−A3=νuxxx,t−νuxxxm,tn=νuxxx,t−νuxxxm,t+νuxxxm,t−νuxxxm,tn=A31+A32=∑s1=0m−d1−1s1νuxxxs1,xs1+1,…,xs1+d1,x,t∑s1=0m−d1μs1x+∑s2=0n−d2−1s2νuxxts2,ts2+1,…,ts2+d2,xm,t∑s2=0n−d2μs2t.
Then, we have(72)A3=A31+A32≤A31+A32≤CExxx,t+Exxxm,t≤Chd1−1.
Combining results (68), (70), and (72), the proof is finished.Remark 1.
In the programming of numerical simulation, to deal with the nonlinear characteristic of Burgers’ equation, we adopt the following iteration algorithm:(73)Im⊗D1Uj+diagUj−1C1⊗InUj−νC2⊗InUj=Fj,j=1,2,3,…,or the Newton–Rapson iteration algorithm.


## 4. Numerical Experiments

In this section, some numerical experiments with the barycentric rational collocation method are carried out for Burgers’ equation.Example 1.
Consider the following Burgers’ equationν=1,α=−4,β=4,T=1:(74)∂ux,t∂t+ux,t∂ux,t∂x−ν∂2ux,t∂x2=0,−4<x<4,0<t<1,u−4,t=12−12tanh−1−18t,0<t<1,u4,t=12−12tanh1−18t,0<t<1,ux,0=12−12tanh14x,−4<x<4.
The analysis solution is chosen to be(75)ux,t=12−12tanh14x−18t.
In Table1, the errors of the barycentric rational collocation method with Chebyshev nodes with α=−4,β=4,m=20,T=1,andn=20 are presented. The absolute error(76)E1=ux,t−uxm,tn,E2=ux,t−uxm,tn2,and relative error(77)Er1=ux,t−uxm,tnux,t,Er2=ux,t−uxm,tn2ux,t2,are listed.
We can see from Table1 that the minimum absolute error E1andE2 can reach 3.3373∗10−13 and 2.4989∗10−12. The minimum relative error Er1andEr2 can reach 3.6891∗10−13 and 2.7623∗10−12, respectively. The calculation results show that the proposed method has high accuracy feature.
In Table2, in order to test the convergence rates of space variable in the case of equidistant subdivision, we take the time interpolation parameter with d2=9. In Table 3, adopting equidistant subdivision, we take the space interpolation parameter with d1=9 to test the convergence rates of time variable. In Table 4, in order to test the convergence rate of space variable in the case of Chebyshev nodes, we take the time interpolation parameter with d2=9. In Table 5, adopting Chebyshev nodes, we take the space interpolation parameter with d1=9 to test the convergence rates of time variable. Taking m=40,n=40,andd1=d2=4, Figure 1 shows the exact solution, numerical solution, and error with equidistant nodes for the barycentric rational collocation method.Table 1
Errors of the barycentric rational collocation methods in the case of Chebyshev nodes withm×n=20×20 for Example 1.
d1×d2ux,t−uxm,tnux,t−uxm,tn/ux,tux,t−uxm,tn2ux,t−uxm,tn2/ux,t21×13.2384e − 043.5797e − 042.7503e − 033.0402e − 032×21.4959e − 061.6536e − 061.0386e − 051.1481e − 053×31.4776e − 061.6334e − 061.2311e − 051.3609e − 054×43.0983e − 073.4249e − 072.5724e − 062.8436e − 065×58.4484e − 099.3388e − 093.9362e − 084.3510e − 086×62.4148e − 082.6693e − 081.8519e − 072.0471e − 077×76.3028e − 096.9671e − 094.3475e − 084.8057e − 088×82.3539e − 092.6020e − 091.5647e − 081.7296e − 089×91.5231e − 091.6837e − 099.7012e − 091.0724e − 0810×104.0990e − 104.5310e − 102.2026e − 092.4347e − 0911×113.8695e − 104.2774e − 102.0091e − 092.2208e − 0912×121.1561e − 101.2779e − 106.3747e − 107.0466e − 1013×136.1870e − 116.8391e − 112.9800e − 103.2941e − 1014×143.5908e − 113.9693e − 111.9690e − 102.1765e − 1015×153.2042e − 123.5419e − 121.2566e − 111.3890e − 1116×167.2335e − 127.9960e − 124.3264e − 114.7824e − 1117×173.1590e − 123.4919e − 122.1361e − 112.3613e − 1118×183.3373e − 133.6891e − 132.4989e − 122.7623e − 1219×191.3369e − 121.4778e − 121.1956e − 111.3216e − 11Table 2
Absolute errors and convergence rates in the case of equidistant nodes with time interpolation parameterd2=9 for Example 1.
m×nd1=1hαd1=2hαd1=3hαd1=4hα10×101.0477e − 03—4.0347e − 05−1.6121e − 04−3.7427e − 05—20×203.1401e − 041.743.2019e − 063.668.8537e − 064.192.0848e − 064.1740×401.0667e − 041.566.7471e − 072.255.2818e − 074.077.8919e − 084.72Table 3
Absolute errors and convergence rates in the case of equidistant nodes with space interpolation parameterd1=9 for Example 1.
m×nd2=1ταd2=2ταd2=3ταd2=4τα10×104.9734e − 05—4.8510e − 05—4.8182e − 05—4.8190e − 05—20×202.9590e − 064.072.9872e − 077.342.4811e − 0810.922.4818e − 0810.9240×401.0505e − 061.495.3110e − 082.494.7796e − 119.024.6032e − 119.07Table 4
Absolute errors and convergence rates in the case of Chebyshev nodes with time interpolation parameterd2=9 for Example 1.
m×nd1=1hαd1=2hαd1=3hαd1=4hα10×101.2075e − 03—1.5664e − 05—6.9893e − 05—2.6630e − 05—20×203.2381e − 041.901.4962e − 063.391.4776e − 065.563.0960e − 076.4340×409.5077e − 051.772.1888e − 072.775.4753e − 084.754.5608e − 096.08Table 5
Absolute errors and convergence rates in the case of Chebyshev nodes with space interpolation parameterd1=9 for Example 1.
m×nd2=1ταd2=2ταd2=3ταd2=4τα10×102.5041e − 06—8.4087e − 07—5.9037e − 07—5.9032e − 07—20×205.7915e − 072.113.4124e − 084.621.5666e − 098.561.5257e − 098.6040×401.3292e − 072.124.0109e − 093.093.5787e − 128.771.5124e − 116.66Figure 1
Exact solution, numerical solution, and error of Example1m=n=40,d1=4,d2=4.Example 2.
Consider the following Burgers’ equationν=0.01,α=0,β=1,T=1:(78)∂ux,t∂t+ux,t∂ux,t∂x−ν∂2ux,t∂x2=0,0<x<1,0<t<1,u0,t=0.1e−0.05/ν∗−0.5+4.95t+0.5e−0.25/ν−0.5+0.75t+e−0.5/ν−0.375e−0.05/ν∗−0.5+4.95t+e−0.25/ν−0.5+0.75t+e−0.5/ν−0.375,0<t<1,u1,t=0.1e−0.05/ν∗0.5+4.95t+0.5e−0.25/ν0.5+0.75t+e−0.5/ν1−0.375e−0.05/ν∗0.5+4.95t+e−0.25/ν0.5+0.75t+e−0.5/ν1−0.375,0<t<1,ux,0=0.1e−0.05/ν∗x−0.5+0.5e−0.25/νx−0.5+e−0.5/νx−0.375e−0.05/ν∗x−0.5+e−0.25/νx−0.5+e−0.5/νx−0.375,0<x<1.
The analysis solution is set to be(79)ux,t=0.1e−0.05/ν∗x−0.5+4.95t+0.5e−0.25/νx−0.5+0.75t+e−0.5/νx−0.375e−0.05/ν∗x−0.5+4.95t+e−0.25/νx−0.5+0.75t+e−0.5/νx−0.375.
In Table6, in order to test the convergence rates of space variable in the case of equidistant nodes, we take the time interpolation parameter d2=9. In Table 7, adopting Chebyshev nodes, we take the time interpolation parameter with d2=7 to test the convergence rates of space variable. In Table 8, in order to test the convergence rates of time variable in the case of equidistant nodes, we take the space interpolation parameter with d1=7. In Table 9, adopting Chebyshev nodes, we take the space interpolation parameter d1=7 to test the convergence rates of time variable. Figure 2 shows the exact solution, numerical solution, and error in the case of equidistant nodes for the barycentric rational collocation method with m=80,n=80,d1=3,andd2=3.Table 6
Errors and convergence rates in the case of equidistant nodes with time interpolation parameterd2=9 for Example 2.
m×nd1=1hαd1=2hαd2=3hαd2=4hα10×103.4018e + 00—8.5575e − 01—4.8427e − 01—9.6832e − 01—20×202.9437e − 013.533.1314e − 011.453.6812e − 010.402.9374e − 011.7240×401.2990e − 037.828.3680e − 048.552.0446e − 037.495.0362e − 035.8780×801.9663e − 042.727.4317e − 053.494.5646e − 055.491.5620e − 058.33Table 7
Errors and convergence rates in the case of Chebyshev nodes with time interpolation parameterd2=7 for Example 2.
m×nd1=1hαd1=2hαd1=3hαd1=4hα10×101.4110e − 01—2.4877e − 01—3.9195e − 01—3.8549e − 01—20×202.0857e − 022.762.1041e − 023.562.3790e − 024.043.3643e − 023.5240×401.3842e − 033.914.6871e − 045.493.9810e − 045.904.0548e − 046.3780×801.9691e − 042.814.0052e − 053.557.5968e − 052.392.2850e − 054.15Table 8
Errors and convergence rates in the case of equidistant nodes with space interpolation parameterd1=7 for Example 2.
m×nd2=1ταd2=2ταd2=3ταd2=4τα10×103.4168e + 01—2.4147e + 00—6.7459e − 01—4.9529e + 00—20×206.3355e − 015.756.4652e − 011.906.4123e − 010.071.5455e + 001.6840×401.6567e − 025.261.8626e − 025.121.9236e − 025.061.9257e − 026.3380×805.0144e − 031.728.0557e − 044.531.7440e − 046.795.6014e − 058.43Table 9
Errors and convergence rates in the case of Chebyshev nodes with space interpolation parameterd1=7 for Example 2.
m×nd2=1ταd2=2ταd2=3ταd2=4τα20×202.6326e − 01—6.1290e − 01—3.0737e − 01—2.5485e − 01—40×402.1381e − 036.948.1962e − 049.558.4712e − 048.509.0261e − 048.1480×805.8129e − 041.889.3689e − 053.131.8191e − 055.545.5216e − 067.35Figure 2
Exact solution, numerical solution, and error of Example2m×n=80×80,d1=4,d2=4.Example 3.
Consider the following Burgers’ equationν=0.1,α=−2,β=2,T=1:(80)∂ux,t∂t+ux,t∂ux,t∂x−ν∂2ux,t∂x2=xcosxt+t2sin2xt+t210sinxt,x∈−2,2,0<t<1,u−2,t=sin−2t,0<t<1,u2,t=sin2t,0<t<1,ux,0=0,x∈−2,2.
The analysis solution is chosen to be(81)ux,t=sinxt.
In Table10, in order to test the convergence rates of space variable in the case of equidistant subdivision, we take the time interpolation parameter with d2=9. In Table 11, adopting equidistant subdivision, we take the space interpolation parameter with d1=9 to test the convergence rates of time variable. In Table 12, in order to test the convergence rate of space variable in the case of Chebyshev nodes, we take the time interpolation parameter with d2=9. In Table 13, adopting Chebyshev nodes, we take the space interpolation parameter with d1=9 to test the convergence rates of time variable. Taking m=40,n=40,andd1=d2=4, Figure 3 shows the exact solution, numerical solution, and error with equidistant nodes for the barycentric rational collocation method.Table 10
Absolute errors and convergence rates in the case of equidistant nodes with time interpolation parameterd2=9 for Example 3.
m×nd1=1hαd1=2hαd1=3hαd1=4hα10×103.2588e − 04—4.8980e − 04—1.5235e − 04—1.3054e − 05—20×203.8769e − 053.072.6580e − 054.208.2946e − 064.202.1865e − 075.9040×404.7871e − 063.023.6399e − 062.871.7144e − 062.272.1392e − 083.35Table 11
Absolute errors and convergence rates in the case of equidistant nodes with space interpolation parameterd1=9 for Example 3.
m×nd2=1ταd2=2ταd2=3ταd2=4τα10×101.0508e − 02—1.9404e − 03—1.3672e − 04—3.2741e − 05—20×204.4923e − 031.233.8848e − 042.322.0154e − 052.761.9964e − 064.0440×401.7248e − 031.387.4068e − 052.392.0977e − 063.269.4904e − 084.39Table 12
Absolute errors and convergence rates in the case of Chebyshev nodes with time interpolation parameterd2=9 for Example 3.
m×nd1=1hαd1=2hαd1=3hαd1=4hα10×102.3715e − 02—1.5651e − 03—8.6548e − 04—6.6468e − 05—20×205.5071e − 032.111.3604e − 043.526.7864e − 053.671.0970e − 065.9240×406.8410e − 043.016.6176e − 064.362.2923e − 064.891.1683e − 086.55Table 13
Absolute errors and convergence rates in the case of Chebyshev nodes with space interpolation parameterd1=9 for Example 3.
m×nd2=1ταd2=2ταd2=3ταd2=4τα10×103.2411e − 03—3.2588e − 04—2.9469e − 05—4.1816e − 06—20×208.1489e − 041.993.8769e − 053.072.0032e − 063.881.2569e − 075.0640×401.9254e − 042.084.7871e − 063.021.2704e − 073.983.8042e − 095.05Figure 3
Exact solution, numerical solution, and error of Example3m=n=40,d1=4,d2=4.Remark 2.
Numerical experiments using the barycentric rational collocation method for Burgers’ equations show the consistency of the convergence rates with the theoretical analysis. Note that if we further increase the space interpolation parameterd1 and time interpolation parameter d2, we can get more accurate results. Some numerical results show that the convergence rate is higher than our theorem which is out of our goal of this paper and will be presented in other papers. In the future, we will research the (1 + 2) dimensional and (1 + 3) dimensional Burgers’ equations.



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