Orthogonal Collocation Method of the Two dimensional Burgers Equations

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2018 3rd International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2018) ISBN: 978-1-60595-035-8

Orthogonal Collocation Method of

the Two-dimensional

Burgers Equations

Ning MA

*

and Wen-liang BIAN

College of Science, China University of Petroleum, Beijing, China 102249

*Corresponding author

Keywords: Burgers equations, Orthogonal collocation method, Error estimate.

Abstract. In this paper, the semi-discrete orthogonal collocation method for two-dimensional Burgers equations is considered. And the existence and uniqueness of orthogonal collocation solution is proved and the error estimate is obtained.

Introduction

Burgers equation is one of the most important nonlinear wave equations in physics and mathematics. It is an equation that can find the analytical solutions[1]. Many numerical methods have been proposed, such as the finite difference method[2], the finite element method[2] and the mixed element method[3].

The collocation method at Gauss points[4] has high convergence order and does not need to calculate numerical integration so that the calculation is simple. So it is widely used in many kinds of equations, such as the quasilinear parabolic equations[5], the parabolic equation[6], the two dimensional heat conduction equation[7], the convection diffusion problem[8] and the one dimensional Burgers equation[9]. Now we consider the Orthogonal Collocation Method of the two-dimensional Burgers equations.

We consider the two-dimensional Burgers equations as follows:

0 0

, ( , ) , 0 ,

0, ( , ) , 0 ,

( , ). t

u u u

u u u x y t T

t x y

u x y t T

u u x y







  

 _ _ _{ } _{  }

   



 _ _{  }



 



(1)

In the equations,( , )x y   , [0,1] [0,1], xI I_x, _x [0,1],yI I_y, _y [0,1], is the border of

, 1

Re

  is the viscosity coefficient. Re is the Reynolds number, which is a positive constant.

Then we discrete the spatial region  into grids by points

( ,x y_i _j),i0,1, 2, ,M, j0,1, 2, ,N and ,x y_i _j are satisfied

0 1

0 1 ,

M N

x x x

y y y

    

Let

1 1

[ , ] [ , ] , 0,1, , , 0,1, ,

ij xi xi yj yj i M j N

    

1 1

, max , , max , max{ , },

i i i j

x i i x x y j j y y x y

i M j N

h x x_ h h h x x _ h h h h h

   

      



1



X ( ) is a Bi-cubic Hermite polynomial 0,1, , , 0,1, , ,

ij

(2)





0

X_h  v X_h v|_ =0 .

The four Gauss points (x_ik,y_jl); ,k l1, 2 in _ij are collocation points as follows:

1 1

( ) ( 1) , ( ) ( 1)

2 2 3 2 2 3

j

i y

x

k l

ik i i jl j j

h h

x  x_ x   y  y _  y   .

Now we set the following notations[7]:

2

1 1 , 1

1

, ( , ) ( , ) ,

4 M N

xi yj ik jl ik jl

i j k l

u v h h u x y v x y

  







, x, x y, y ,

u v u v u v

   

2

1 1 1

1

, , ( ) ( ) ,

2

M M

xi ik ik

x ix

i i k

u v u v h u x v x

  







 

2

1 1 1

1

, , ( ) ( ) ,

2

N N

yj jl jl

y jy

j j k

u v u v h u y v y

  







 

(3)

1 1 1 1

( , ) ( , ) ,

ij

M N M N

ij

i j i j

u v u v uvdxdy



   









1

1 1

( , ) ( , ) i ,

i

M M _x

x ix _x

i i

u v u v  uvdx

 









1

1 1

( , ) ( , ) j ,

j

N N _y

y jy _y

j j

u v u v  uvdy

 









2

,

 u u u

1

2 ₁ 2 ₁

2 2 2

( ) ₀ ₀

1 1 1 1

1 1

( , ) ( , ).

2 2

o

M N

xi y ik yj x jl

H

i k j l

u _ h u x t h u y t

   













Lemma1.1: If X_h0  



X_h: _ 0



, then c 0

1 1

0 0

2 2 ₀

( ) , ( ), h

H  c H  X

      

where c1, c2 are positive constant s [7].

Lemma1.2: Assume that the anti-hypothesis is tenable in Xh, then[7]

0

h

X

 ,  c 0





1 1

0

2 2

( ) , ( ) .

H  c H 

     

(3)

1

( ) ( ) 2(4 ) 2

1 4

, ( ) , 1, 2,

i

x M

l l l

x x

x

i _x

e e ch D dx l

     

 



  1 6 2 1 5 , ( ) , i i x M

xx xx _x x x

i x

e e ch D dx

      

 



 1 9 2 1 4

,1 ( ) ,

i

x M

x x x x

i _x

e ch D dx

     

 



 1 9 2 1 6

,1 ( ) .

i

x M

xx _x x x

i x

e ch D dx





 

 

 





There are similar conclusions along the y direction[7].

Lemma1.3:If u is sufficiently smooth,with a constant c0,

2 2

1 2

2 4 (4) ₂

( ) ₍ ₎ , ( ) ,      



 ij L _L i j H ch 2 2 1 2

2 4 (4) ₂

( ) ₍ ₎ , ( ) .      



 ij

t t L t _L

i j

H ch

Semi-Discrete Collocation Method

The orthogonal collocation scheme as follows can be established. Seek U:[0, ]T X_hX_h,such that

0 0

( , , ) 0, 0

0, (0, ], 0, ( , ) , 0 ,

( , ).

x y ik jl

t

U

UU UU U x y t t T

t

U t T u x y t T

U u x y

       _ _ _{ }  _ _{ }        _ _ _ _{  }   _  (3)

Next, we are going to prove the existence and uniqueness of collocation solution and obtain the error estimate.

Consider the following Galerkin scheme

, 0.

x y h

U

UU UU U z z X

t 

 _ _ _{ } _{  }

 (4)

Theorem 2.1: The solutions of (3) and (4) are equivalent, existent and unique.

Proof: From the Equation (2), according to the definition of , , it is clear that the solution of (3) must be the solution of (4).

Let



_p: p1, 2, , 4MN

 

 (x_ik,y_jl), i0,1, ,M; j0,1, ,N k l; , 1, 2



, and

 

4

1

MN p _p

z

 be a

group base of X_h0. Thereupon U x y( , )X_h0 can be expressed as

4 1 ( , ) ( , ) MN p p p

U x y  z x y





. So (3)

and (4) can be written in the form as follows

(4)

 

4 4

( _ij) _MN _MN, _ij _j _i ,

F  F _ F z 

4 4

( _ij) _MN _MN, _ij _j, _j ,

C  C _ C  z z

where R,S, are both vectors of 4MN.

Obviously the solution of equation F 0 must be satisfied the equation C 0, where  is a vector of 4MN. So F is nonsingular if C is nonsingular. Then the solution of (3) and (4) are unique. To get the existence and uniqueness, we just need to prove that C is nonsingular when t0.

Proof by contradiction, assume that C is singular, therefore there is  _{which is a vector of 4}MN, and  0, so that C 0.

Let

4

4 1

( , ) ( , ) , ( )

MN

p p p MN

p

W x y  z x y  





 ,

thus W W, 0.

Then

( _p) 0 , 1, , 4

W   p MN,

We can get W x y( , )0, so  0. We can get a contradiction, so C is nonsingular.

In summary, The solutions of (3) and (4) are equivalent, and the solution is unique when t0.

Error Estimate

DefineW Tu, which is Hermite polynomial,   W U ,   u W , where u is the accurate solution, and U is the semi-discrete collocation solution.

LetU_t₀ Tu x y0( , ) , then  t0 0. We can get the error equation

, , ,

t t

U U u u

z z U U u u z

x y x y

                 

   

Theorem 3.1: If u x y( , ) is the accurate solution of (1), U x y( , ) is the solution of the orthogonal collocation method, and u x y( , ) satisfied the condition 6

(0, , ( ))

uL T H  ，ut L (0, ,T H6( ))



  ,

then there is the error estimate as follows

1

3 (0, , ( )) ( )

L T H

u U  _   h

Proof:

Let z_t, we can get

, , , .

t t t t t

U U u u

x y x y

                    

   

It is easily calculated to see that

, , , , , .

t t t t t t t

U U u u

x y x y

                    

   

Define an inductive hypothesis: _when_t_{(0, ]}_T _{, the solution of the orthogonal collocation method}

( , , )

U x y t M , which has been proved in [9].

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

2 2



2 2 2 2

1 2

, t t x y x y ,

U U u u

U U u u c v c v v v

x y x y    

 _  _  _  _ _ _ _ _ _ _

   

where c1, c2 are positive constants which are irrelevant to the time step and the space step.

According to the definition of  , , it is easy to get[7]

1

, , ,

2

t

d dt

   

    

and[7]

1 1

, , ,

2 2 t t 2

d

dt  



     

 _,

, t , t, ,

d dt

       

In the above formula, let  1. Accoding to the Cauchy inequality

2 2

,

-  _t _t K _t  _t

The error equation turns into



2 2



2 2 2 2

1 2

1 1 1

, , , ,

2 t t 2 2 2 t t t

t x y x y

d d d

dt dt dt

c v c v v v



                  

    

      

- ， ≤ - ， - ，

Integrate over the time interval (0, )T . Then





2 2 2 2 2

0 0 0

2 2

2 2 2 2 2

0 0

1 1 1

2 2 2 2

T T T

t t t

T T

t t x y x y

dt dt K dt

dt dt

     

 

  _       _

 



- ， ≤ ，

- ， C



          

         

The results

2

2 ₈ 2 ₆ ₆

( ) ,h _x ( ) ,h _y ( ) ,h

        

can be obtained from the lemma 1.3.

Lemma 1.1, lemma1.2 and Gronwall lemma imply that

1

2 2 ₆

(0, , ( ))

0 ( )

T

t dt L T H    h



 

. Then the following error estimates can be obtained

1 1

2 2 ₆

(0, , ( )) (0, , ( )) ( )

L T H L T H

u U  _     _   h .

So the inductive hypothesis is reasonable, thus the theorem is proved.

Acknowledgement

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