new finite element algorithm for the coupled · finite element method and the finite differences...

FINITE ELEMENT APPROXIMATION FOR THE COUPLED

TWO-DIMENSIONAL BURGERS’ EQUATION

ION AUREL CRISTESCU

Politehnica University of Bucharest, Applied Sciences Faculty,

Bucharest, Romania

E-mail: [email protected]

Received July 24, 2017

Abstract. In this work we analyze a technique to determine the approximate

solution of a system of nonlinear partial differential equations: the two-dimensional

coupled Burgers’ equations. The proposed method combines the finite elements and

homotopy analysis. It can be applied to many other nonlinear problems arising from

fluid mechanics, elasticity, electrostatics, geometry, etc. It provides the possibility to

consider complex geometric forms of the problem’s domain. The numerical solutions

are successfully compared with the exact ones. The computation reveals very good

accordance between the convergence regions for absolute and relative errors. This fact

is really important for the case when the exact solutions are unknown.

Key words: nonlinear partial differential equations, coupled two-dimensional

Burgers’ equation, finite element method, homotopy analysis

method, variational methods.

1. INTRODUCTION

In this paper, we investigate a method to solve the two-dimensional (2D)

coupled Burgers’ equation. The technique is useful for a lot of problems arising

from natural sciences and modeled by nonlinear partial differential equations. The

original nonlinear problem is replaced by linear sub-problems with the aid of

homotopy analysis method. The geometric complexity of the domain is overcome

by the finite element method. We proceed with a short description of our problem.

Burgers’ equation, both in steady and unsteady case, describes the turbulent

motion of a viscous fluid in a channel under the interaction of convection and

diffusion, the structure of shock waves, traffic flow problems, the acoustic

transmission [1, 2]. An analytic solution of the 2D coupled Burgers equation is

obtained in [3] using the Hopf-Cole transformation. A comparison between the

finite element method and the finite differences solutions of the 2D problem is

investigated in [4].

Romanian Journal of Physics 63, 105 (2018)

Article no. 105 I.A. Cristescu 2

In [5], algorithms based on cubic spline function technique are analyzed for

the one-dimensional (1D) Burgers’ equation. Fletcher [6] examined the finite

element method based on Newton’s algorithm to resolve the steady coupled 2D

equations. In [7], Boules accomplished a finite element approach based on the

least-squares conjugate gradient method to obtain the numerical solution of steady

2D Burgers’ equation. A meshless method using thin plate spline approximation

for solving the steady Burgers-type equations was developed in [8]. Kakuda and

Tosaka [9] studied the generalized boundary element method for the numerical

solution of Burgers’ equation. They used the fundamental solution for the

linearized differential operator of the equation. In [10], the differential quadrature

method is applied for 2D time-dependent Burgers’ equation. Solutions of 2D

Burgers’ equation by means of the modified extended tanh-function method are

studied in [11]. Abazari and Borhanifar [12] implemented the differential

transformation method to obtain the numerical/analytical solutions of coupled

Burgers’ equation. In [13] it is analyzed a meshfree technique for the 2D coupled

Burgers’ equation, which combines the collocation method using the radial basis

functions with first order accurate forward difference approximation. A

combination of 2D Hopf-Cole transformation and local discontinuous Galerkin

finite element method is investigated in [14]. The Ansatz method and Lie

symmetry analysis are involved in [15] to deduce soliton solutions and

conservation laws of the coupled Burgers’ equation. Our work is referring to

computational methods in physics. Some recent articles [16–18] contain very

interesting results in this research domain.

This paper is organized in the following way: Sec. 2 is referring to the

statement of the problem, a short description of the homotopy analysis method, and

the formulation of approximate variational problems. In Sec. 3 we accomplish the

applications that involve different types of finite elements. In Sec. 4 the

conclusions are formulated.

2. MAIN RESULTS

The governing coupled 2D Burgers’ equations in the stationary case are:

2122

2

21

2

2122

2

21

2

x

vv

x

vuRe

x

v

x

v

x

uv

x

uuRe

x

u

x

u

221 R),( Dxxx , (1)

21, gvgu . (2)

3 Finite element approximation for Burgers’ equation Article no. 105

Here ),(),,( 2121 xxvxxu are the velocity components to be calculated;

1Re is the Reynolds number, is the coefficient of viscosity;

2RD is an open bounded set with Lipschitz boundary ;

)(, 21

21 Hgg (we apply the usual notations for Sobolev spaces [19]).

The method we implement for the numerical resolution of (1), (2) combines

the homotopy analysis method (HAM) and the finite elements method. Liao created

the HAM, one of the most efficient techniques used in nonlinear sciences [20, 21].

The HAM provides the tools to solve a lot of nonlinear problems: viscous flow

[22], MHD-problems for non-newtonian fluids [23], to obtain approximate

solutions for the Zakharov-Kuznetsov equations [24], to solve integral equations

[25], to obtain the analytical solution of the nonlinear cubic-quintic Duffing

oscillators [26], nonlinear heat transfer [27], to determine the numerical solutions

of nonlinear Poisson equations [28, 29], to examine a fractional model of

convective radial fins [30], to determine the numerical solution of Burgers’

equations using finite differences [31], etc. An optimal variant of HAM was

developed in [32] to analyze the stagnation point flow. In [33] Liao developed the

general boundary element method based on the HAM.

We consider the nonlinear equation with boundary conditions:

0)]([ xuN (3)

0)]([1 xuN . (4)

The method constructs a homotopy series

ss

s

pxupx )(),(0

(5)

so that when p varies from 0 to 1, the series solution varies from the initial guess

)()0,( 0 xux to the exact solution )()1,( xux . The zero-order deformation

equation associated to (3) will be [20]:

)],([)()](),([)1( 0 pxNxHpxupxLp (6)

and similar for the boundary conditions (4). The method allows different selections

for the auxiliary linear operator L, the initial approximation 0u and the auxiliary

function 0)( xH , )(DCH . The convergence-control parameter 0 can

always be selected to provide convergent homotopy series for 1p . It represents a

solution of the problem:


)()1,()(0

xuxxu s

s

. (7)

The s-th order homotopy derivative )0,(!

1)( x

psD

s

s

s

in Eq. (6) provides

the s-th order deformation equation:

)]),([()()]()([ 11 pxNDxHxuxuL ssss (8)

where

2,1

1,0

s

ss .

We carry out the HAM for Burgers’ equations in a vectorial form:

)],(),,([),( 21 pxpxpx

where 0],,[,)(),(0

svuUpxUpx ssss

s

s

ss

s

pxupx )(),(0

1

(9)

ss

s

pxvpx )(),(0

2

.

Taking into consideration the Eqs. (1), we select in (6) the linear operators

2,1),,()],(),,([ 21 ipxpxpxL ii (10)

and the nonlinear operators:

2,1

,),(),(),(),(),()],(),,([2

21

121

i

pxx

pxpxx

pxRepxpxpxN iiii

(11)

Equation (8) yields

2,1)]),,([()()]()([ 11 ipxNDxHxUxUL issssi (12)

The 1-st order deformation equations arise from (10), (11), and (12):

2

00

1

0001 )(

x

uv

x

uuReuxHu (13)


2

00

1

0001 )(

x

vv

x

vuRevxHv . (14)

We compute

js

js

j

mjm

jm

jm

ss ux

upu

x

uDpx

xpxD

1

1

1

0100

1

1

111 )(),(),(

(15)

and similar. Equations (12), (15) give for 2s :

js

js

j

js

js

j

ss vx

uu

x

uRexHuxHu 1

2

1

0

1

1

1

0

1 )())(1( (16)

js

js

j

js

js

j

ss vx

vu

x

vRexHvxHv 1

2

1

0

1

1

1

0

1 )())(1( . (17)

We apply the homotopy technique for the boundary conditions (2) selecting

2,1 ,),()],([

),(),()],([

1

1

ipxpxL

xgpxpxN

ii

iii

and 00 ,vu so that

2010 , gvgu . (18)

The method provides

1 ,0 and 0 svu ss . (19)

Further, we introduce the Sobolev spaces:

}2,1),(/)({)( 221

iDL

x

uDLuDH

i

and }0/)({)( 110 uDHuDH .

The approximate variational problems formulated in the approximate

vectorial spaces hV will be established. The construction of hV is accomplished by

using appropriate finite elements, which depend on the geometry of the problem’s

domain D: triangles, rectangles, isoparametric elements, etc. Let hT be a grid of the

domain D:

hTK

KD

.

We state the function spaces:


})(,)(/)({ )()(0h

Ki

Kih

i

Khhh TKpauuDCuW

}0/{ hhhh uWuV (20)

– iKia )( )( are the nodes of interpolation in K;

– iKip )( )( is the local polynomial basis.

These yield [34] )(1 DHWh and )(10 DHVh .

Integration on D and Gauss formula provide the following approximate

variational problems associated to (13), (14), and (19) at the step 1s :

– find hhh Vvu 11, so that

hhhhhh VwwFwua )(),(),( 11 (21)

hhhhhh VwwGwva )(),(),( 11 (22)

where:

– RVVa hh : is the bilinear, continuous, coercive form

2

1 2

1

( , ) d dh hh h

i i iD

u wa u w x x

x x

,

– RVGF hhh :, 11 are the linear, continuous forms on hV :

The approximate variational equations at step 2s and related to Eqs. (16),

(17), and (19) mean to find hhshs Vvu , such that:

hhhhshhs VwwFwua )(),(),( (23)

hhhhshhs VwwGwva )(),(),( . (24)

We have RVGF hhshs :, the linear, continuous forms defined by


The variational solutions )1(, svu hshs of the problems (21), (23) and

respectively (22), (24) give the approximate homotopy series solutions:

shs

s

h pxupxpx );();,(),(0

11

shs

s

h pxvpxpx );();,(),(0

22

.

We deduce the following approximations of the exact solutions:

);();();1,();1,()1,()(

);();();1,();1,()1,()(

0

,222

0

,111

xvxvxxxxv

xuxuxxxxu

hN

def

hs

N

s

Nhh

hN

def

hs

N

s

Nhh

(25)

In fact, )1(, svu hshs also depend on the parameter , so we have to take it

into account.

The initial approximations hhh Wvu 00 , will be constructed by interpolation.

We prescribe their values in the nodes ia of the grid:

)( for),(~for),(

)(1

10

DIntaag

aagau

ii

iiih

)( for),(~for),(

)(2

20

DIntaag

aagav

ii

iiih (26)


The functions 21~,~ gg will be selected according to the rule of solution

expression [20], which takes into account the governing differential equations and

the boundary conditions. We proceed with applications of the proposed method.

3. APPLICATIONS

In the numerical tests we shall compare the approximate solutions performed

by finite elements-homotopy analysis technique and the exact solutions generated

by using Hopf-Cole transformation:

1 2 1 2

1 21 2 1 2

1 2 1 2

2 ( , ) 2 ( , )

( , ) , ( , )Re ( , ) Re ( , )

x x x xx x

u x x v x xx x x x

. (27)

It results 0 . The following exact solution of the above equation

provides considerable control over the corresponding velocity ],[ vu [6]:

)cos())(cosh(2),( 20152142312121 xxxaxxaxaxaaxx .

3.1. APPLICATION 1

We consider Burgers’ equations (1) in the domain )1.0,0()1,1( D . The

boundary conditions (2) are prescribed by the exact solutions (27) provided by

01 a , 100432 aaa5 0, 1, 10, 5,Re 500a x . In general, for the

rectangle ),(),( dcbaD we divide the interval ],[ ba in m subintervals and

],[ dc in n subintervals. We draw parallels to 1Ox and 2Ox through the division

points and obtain a grid hT of the domain D:

hTK

KD

.

The approximate function spaces in Eq. (20) are defined by

})(,)(/)({ )()(4

1

0h

Ki

Kih

i

Khhh TKpauuDCuW

}0/{ hhhh uWuV .

Considering K the rectangle with the nodes 4,1),,()(

2)(

1)(

iaaaKi

Ki

Ki and

)(KA = the area of K, we deduce:


))(()(

1),(),)((

)(

1),(

))(()(

1),(),)((

)(

1),(

)(1221

)(31214

)(122

)(11121

)(3

2)(

32)(

11121)(

2)(

322)(

31121)(

1

KKKKK

KKKKKK

axxaKA

xxpaxaxKA

xxp

xaaxKA

xxpaxaxKA

xxp

According to the rule of solution expression, the exact solution )(xu in the domain

D can be expressed as

jiij

ji

dcx

baxaxxu )

2()

2(),( 21

0,

21

and similar for )(xv . So, we define 0~,0~21 gg in (26) for the initial

approximations 00 , hh vu .

Based on Eqs. (25), we state the absolute and relative errors for Dx :

)(),(,)(),(max),,( xvxvxuxuNxerr hNhNabs

),(),(,),(),(max),,( 1,1, xvxvxuxuNxerr NhhNNhhNrel

),(,),(max xvxu hNhN . (28)

In computation it is set the auxiliary function 1)( xH . Table 1 contains the

values of DxNxerrabs ),,(supmin

for two grids and different orders of

approximations N.

Table 1

The absolute errors for the two grids

(m,n) order of approximation

N

control convergence

parameter abs

Dx

err

sup

(10,10)

5 –1 610203240.3

10 –1 710521049.4

15 –0.85 710469028.4

(20,20)

5 –1 610552858.1

10 –1 710063960.1

15 –1.15 710016957.1

Figures 1, 2 are referring to the graphs of the functions

}/),,(sup{

}/),,(sup{

DxNxerr

DxNxerr

rel

abs

(29)


for 10,10 nmN and respectively, 20,15 nmN . The graphic represen-

tations reveal very good accordance between the convergence regions for absolute

and relative errors:

3.2. APPLICATION 2

We examine Burgers’ equations (1) and (2) in the circular crown sector

],[],,[,sin,cos/),( 110212

21 trrrrtryxtrxxxxD R ,

where:

0,0,||

arcsin, 01

02201 ay

r

yayr .

The boundary conditions (2) will be defined by the exact solution. Taking

into account the shape of the domain, we consider grids of triangles, which cover

the problem’s domain:

hTK

h KDD

.

The construction of the approximation spaces is established by the relations:

})(,)(/)({ )()(3

1

0h

Ki

Kih

i

Khhhh TKauuDCuW

;

}0/{ hhhhh uWuV ;

h is the boundary of hD ;

Fig. 1 – -curve of abserr (solid line) and

relerr (dotted line), 10,10 nmN .


relerr (dotted line), 20,15 nmN .


iKia )(

)( are the nodes of interpolation in the triangle K;

iKi )(

)( is the local polynomial basis determined by the system:

1),( ),,(),( 21

3

1

21)(

21

3

1

xxxxOaxx j

j

Kjj

j

.

Figures 3 and 4 contain the involved grids of triangles for the domain D.

Fig. 3 – Grid 1 of problem’s domain.

Fig. 4 – Grid 2 of problem’s domain.

Our purpose is to compare the numerical solution and the exact solution

obtained in (27) with

)2cos()10(2cosh20205410),( 21212121 xxxxxxxx .

We set 0Re 500, 5, 1, 0.1y a r and consider 0~,0~21 gg for

the initial approximations 00 , hh vu in (26). The numerical solutions of the problem

are given by Eq. (25). Table 2 presents the absolute errors for the two grids from

Figs. 3 and 4.


Table 2

The absolute errors for the two grids

order of approximation

N

control convergence

parameter abs

Dx

err

sup

Grid 1

5 –1 610742891.1

10 –1 710095101.2

15 –1 710094876.2

Grid 2

5 –1 710339230.1

10 –1 710202674.1

15 –1 710202498.1

The Figures 5, 6 deal with the graphs of the functions from (29). The

computations with 10N for grid 1 and 15N for grid 2 are considered. Again,

the convergence regions of the absolute and relative errors are in excellent

agreement.

4. CONCLUSIONS

In the present paper it is investigated a new finite element technique based

on homotopy analysis to determine the numerical solution of the coupled two-

dimensional Burgers’ equation. The method allows to consider a complex shape of

the problem’s domain. The applications prove the proposed method works very

well.


relerr (dotted line), N = 10, grid 1.


relerr (dotted line), N = 15, grid 2.


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new finite element algorithm for the coupled · finite element method and the finite differences...

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