Startseite Technik Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
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Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines

  • Fabio Silva Botelho EMAIL logo
Veröffentlicht/Copyright: 22. März 2024
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 13 Heft 1

Abstract

This article develops approximate numerical solutions through the generalized method of lines for the time-independent, incompressible Navier–Stokes system in fluid mechanics. More specifically, we highlight the main objective of this article is the development of new approximate procedures for solving numerically the equation systems originated from a domain discretization related to a finite differences scheme. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines (or more generally in curves), and the concerning solutions are written on these lines as functions of the boundary conditions and the domain boundary shape. Finally, it is worth emphasizing that in this text, we have presented softwares and results for a concerning approximate proximal approach, as well as results based on the original conception of the generalized method of lines.

Keywords: generalized method of lines; Navier–Stokes system; equivalent elliptic system
MSC 2010: 65N40; 35Q30

1 Introduction

In this article, we develop approximate solutions for the time-independent incompressible Navier–Stokes system through the generalized method of lines. We recall again that, for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and the domain boundary shape. It is worth highlighting that the main aim of this article is the development of a new approximate procedure for solving numerically the equation system originated from a domain discretization concerning standard finite difference schemes to be specified for each case addressed. We also emphasize the first article part concerns the application and extension of an approximate proximal approach published in the study by Botelho [1]. We develop an analogous algorithm as those presented by Botelho [1] but now for a Navier–Stokes system, which is more complex than the systems of partial differential equations previously addressed. In this first step, we present an algorithm and respective software in MAT-LAB.

Furthermore, we have developed and presented related softwares in MATHEMATICA for a simpler type of domain and also concerning the mentioned proximal approach. Finally, in the last section, we present a software and related line expressions through the original conception of the generalized method of lines, so that in such related numerical examples, the main results are established through applications of the Banach fixed point theorem.

Remark 1.1

We also highlight that the next two paragraphs in this article (a relatively small part) overlap with the Chapter 28, starting page 526, in the book by Botelho, [2], published in 2020, by CRC Taylor and Francis. However, we emphasize that the present article includes substantial new parts, including a concerning software not included in the previous version of 2020. Another novelty in the present version is the establishment of appropriate boundary conditions for an equivalent elliptic system to the original Navier–Stokes one. Such new boundary conditions and concerning results are indicated in Section 2.

At this point, we describe the system in question.

Consider Ω ⊂ R 2 an open, bounded, and connected set with a regular (Lipschitzian) internal boundary denoted by Γ 0 , and a regular external one denoted by Γ 1 . For a two-dimensional motion of a fluid on Ω , we denote by u : Ω → R the velocity field in the direction x of the Cartesian system ( x , y ) , by v : Ω → R the velocity field in the direction y , and by p : Ω → R the pressure one. Moreover, ρ denotes the fluid density, μ is the viscosity coefficient, and g denotes the gravity field.

Under such notation and statements, the time-independent incompressible Navier–Stokes system of partial differential equations stands for

(1) μ ∇ 2 u − ρ u u x − ρ v u y − p x + ρ g x = 0 , in Ω , μ ∇ 2 v − ρ u v x − ρ v v y − p y + ρ g y = 0 , in Ω , u x + v y = 0 , in Ω ,

(2) u = v = 0 , on Γ 0 , u = u ∞ , v = 0 , p = p ∞ , on Γ 1 .

At first, we look for solutions ( u , v , P ) ∈ W 1 , 2 ( Ω ) × W 1 , 2 ( Ω ) × W 1 , 2 ( Ω ) . We emphasize that details about such Sobolev spaces may be found in the study by Adams and Fournier [3].

About the references, we emphasize that related existence of numerical and theoretical results for similar systems may be found in [4–9] and [10], respectively. In particular Temam [10] addresses extensively both theoretical and numerical methods and an interesting interplay between them. Moreover, related finite difference schemes are addressed in the study by Strikwerda [11].

Finally, it is worth mentioning that this article has been published as a preprint (see the concerning reference [12] for details).

2 Details about an equivalent elliptic system

Defining now P = p ∕ ρ and ν = μ ∕ ρ , consider again the Navier–Stokes system in the following format:

(3) ν ∇ 2 u − u ∂ x u − v ∂ y u − ∂ x P + g x = 0 , in Ω , ν ∇ 2 v − u ∂ x v − v ∂ y v − ∂ y P + g y = 0 , in Ω , ∂ x u + ∂ y v = 0 , in Ω ,

(4) u = v = 0 , on Γ 0 , u = u ∞ , v = 0 , P = P ∞ , on Γ 1 .

As previously mentioned, at first, we look for solutions ( u , v , P ) ∈ W 1 , 2 ( Ω ) × W 1 , 2 ( Ω ) × W 1 , 2 ( Ω ) in a distributional sense.

We are going to obtain an equivalent elliptic system with appropriate boundary conditions.

Our main result is summarized by the following theorem.

Theorem 2.1

Let Ω ⊂ R 2 be an open, bounded, connected set with a regular (Lipschitzian) boundary.

Assume u , v , P ∈ W 2 , 2 ( Ω ) are such that

(5) ν ∇ 2 u − u u x − v u y − P x + g x = 0 , in Ω , ν ∇ 2 v − u v x − v v y − P y + g y = 0 , in Ω , ∇ 2 P + u x 2 + v y 2 + 2 u y v x − div g = 0 , in Ω ,

(6) u = u 0 , v = v 0 , on ∂ Ω , u x + v y = 0 , on ∂ Ω .

Suppose also, for such u, v fixed, the unique solution of equation in w

ν ∇ 2 w − u w x − v w y = 0 , in Ω

with the boundary conditions

w = 0 , on ∂ Ω ,

is

w = 0 , in Ω .

Under such hypotheses, u , v , and P solve the following Navier–Stokes system:

(7) ν ∇ 2 u − u u x − v u y − P x + g x = 0 , in Ω , ν ∇ 2 v − u v x − v v y − P y + g y = 0 , in Ω , u x + v y = 0 , in Ω ,

(8) u = u 0 , v = v 0 , on ∂ Ω , u x + v y = 0 , on ∂ Ω .

Proof

In Eq. (5), taking the derivative in x of the first equation and adding with the derivative in y of the second equation, we obtain

(9) ν ∇ 2 ( u x + v y ) − u ( u x + v y ) x − v ( u x + v y ) y − ∇ 2 P − u x 2 − v y 2 − 2 u y v x + div g = 0 , in Ω .

From the hypotheses, u , v , and P are such that

∇ 2 P + u x 2 + v y 2 + 2 u y v x − div g = 0 , in Ω .

From this and Eq. (9), we obtain

(10) ν ∇ 2 ( u x + v y ) − u ( u x + v y ) x − v ( u x + v y ) y = 0 , in Ω .

Denoting w = u x + v y , from this last equation, we obtain

ν ∇ 2 w − u w x − v w y = 0 , in Ω .

From the hypotheses, the unique solution of this last equation with the boundary conditions w = 0 , on ∂ Ω , is w ≡ 0 .

From this and Eq. (10), we have

u x + v y = 0 ,

in Ω with the boundary conditions

u x + v y = 0 , on ∂ Ω .

The proof is complete.

Remark 2.2

The process of obtaining such a system with a Laplace operator in P in the third equation is a standard and well-known one.

The novelty here is the identification of the correct related boundary conditions obtained through an appropriate solution of Eq. (10).

3 An approximate proximal approach

In this section, we develop an approximate proximal numerical procedure for the model in question.

Such results are extensions of previous ones published in the study by Botelho [1], now for the Navier–Stokes system context.

More specifically, neglecting the gravity field, we solve the following system of equations:

(11) ν ∇ 2 u − u ∂ x u − v ∂ y u − ∂ x P = 0 , in Ω , ν ∇ 2 v − u ∂ x v − v ∂ y v − ∂ y P = 0 , in Ω , ∇ 2 P + ( ∂ x u ) 2 + ( ∂ y v ) 2 + 2 ( ∂ y u ) ( ∂ x v ) = 0 , in Ω .

We present a software similar to those presented in the study by Botelho [1], with ν = 0.0094 for

Ω = [ 0 , 1 ] × [ 0 , 1 ]

and with the boundary conditions

u = u 0 = 0.55 y ( 1 − y ) , v = v 0 = 0 , P = p 0 = 0.15 , on [ 0 , y ] , ∀ y ∈ [ 0 , 1 ] ,

u = v = P y = 0 , on [ x , 0 ] and [ x , 1 ] , ∀ x ∈ [ 0 , 1 ] ,

u x = v x = 0 , and P x = 0 , on [ 1 , y ] , ∀ y ∈ [ 0 , 1 ] .

Eq. (11), in an appropriate partial finite difference scheme, stands for

(12) ν u n + 1 − 2 u n + u n − 1 d 2 + ∂ 2 u n ∂ y 2 − u n ( u n − u n − 1 ) d − v n ∂ u n ∂ y − P n − P n − 1 d = 0 ,

(13) ν v n + 1 − 2 v n + v n − 1 d 2 + ∂ 2 v n ∂ y 2 − u n ( v n − v n − 1 ) d − v n ∂ v n ∂ y − ∂ P n ∂ y = 0 ,

(14) P n + 1 − 2 P n + P n − 1 d 2 + ∂ 2 P n ∂ y 2 + ( u n − u n − 1 ) ( u n − u n − 1 ) d 2 + ∂ v n ∂ y 2 + 2 ∂ u n ∂ y v n − v n − 1 d = 0 .

After linearizing such a system about U 0 , V 0 , and P 0 and introducing the proximal formulation, for an appropriate nonnegative real constant K , we obtain

(15) ν u n + 1 − 2 u n + u n − 1 d 2 + ∂ 2 u n ∂ y 2 − ( U 0 ) n ( u n − ( U 0 ) n − 1 ) d − ( V 0 ) n ∂ u n ∂ y − ( P 0 ) n − ( P 0 ) n − 1 d − K u n + K ( U 0 ) n = 0 ,

(16) ν v n + 1 − 2 v n + v n − 1 d 2 + ∂ 2 v n ∂ y 2 − ( U 0 ) n ( v n − ( V 0 ) n − 1 ) d − ( V 0 ) n ∂ v n ∂ y − ∂ ( P 0 ) n ∂ y − K v n + K ( V 0 ) n = 0 ,

(17) P n + 1 − 2 P n + P n − 1 d 2 + ∂ 2 P n ∂ y 2 + ( u n + 1 − ( U 0 ) n ) × ( ( u ) n + 1 − ( U 0 ) n ) d 2 + ∂ ( V 0 ) n ∂ y 2 + 2 ∂ ( U 0 ) n ∂ y × v n + 1 − ( V 0 ) n d − K P n + K ( P 0 ) n = 0 .

At this point, denoting ν = e 1 , we define

( T 1 ) n = − ( U 0 ) n ( u n − ( U 0 ) n − 1 ) d − ( V 0 ) n ∂ u n ∂ y − ( P 0 ) n − ( P 0 ) n − 1 d d 2 e 1 + ∂ 2 u n ∂ y 2 d 2 ,

( T 2 ) n = − ( U 0 ) n ( v n − ( V 0 ) n − 1 ) d − ( V 0 ) n ∂ v n ∂ y − ∂ ( P 0 ) n ∂ y d 2 e 1 + ∂ 2 v n ∂ y 2 d 2 ,

and

(18) ( T 3 ) n = ( u n + 1 − ( U 0 ) n ) ( ( u ) n + 1 − ( U 0 ) n ) d 2 d 2 + ∂ ( V 0 ) n ∂ y 2 d 2 + 2 ∂ ( U 0 ) n ∂ y v n + 1 − ( V 0 ) n d d 2 + ∂ 2 P n ∂ y 2 d 2 .

Therefore, we may write

u n + 1 − 2 u n + u n − 1 − K u n d 2 e 1 + ( T 1 ) n + ( f 1 ) n = 0 ,

where

( f 1 ) n = K ( U 0 ) n d 2 e 1

∀ n ∈ { 1 , … , N − 1 } .

In particular, for n = 1 , we obtain

u 2 − 2 u 1 + u 0 − K u 1 d 2 e 1 + ( T 1 ) 1 + ( f 1 ) 1 = 0 ,

so that

u 1 = a 1 u 2 + b 1 u 0 + c 1 ( T 1 ) 1 + ( h 1 ) 1 + ( E r ) 1 ,

where

a 1 = 2 + K d 2 e 1 − 1 , b 1 = a 1 , c 1 = a 1 , ( h 1 ) 1 = a 1 ( f 1 ) 1 , ( E r ) 1 = 0 .

Similarly, for n = 2 , we obtain

u 3 − 2 u 2 + u 1 − K u 2 d 2 e 2 + ( T 1 ) 2 + ( f 1 ) 2 = 0 ,

so that

u 2 = a 2 u 3 + b 2 u 0 + c 2 ( T 1 ) 2 + ( h 1 ) 2 + ( E r ) 2 ,

where

a 2 = 2 + K d 2 e 1 − a 1 − 1 , b 2 = a 2 b 1 , c 2 = a 2 ( c 1 + 1 ) , ( h 1 ) 2 = a 2 ( ( h 1 ) 1 + ( f 1 ) 2 ) , ( E 1 ) 2 = a 2 ( c 1 ( ( T 1 ) 1 − ( T 1 ) 2 ) ) .

Reasoning inductively, having

u n − 1 = a n − 1 u n + b n − 1 u 0 + c n − 1 ( T 1 ) n − 1 + ( h 1 ) n − 1 + ( E r ) n − 1 ,

we obtain

(19) u n = a n u n + 1 + b n u 0 + c n ( T 1 ) n + ( h 1 ) n + ( E r ) n ,

where

a n = 2 + K d 2 e 1 − a n − 1 − 1 , b n = a n b n − 1 , c n = a n ( c n − 1 + 1 ) , ( h 1 ) n = a n ( ( h 1 ) n − 1 + ( f 1 ) n ) , ( E r ) = a n ( ( E r ) n − 1 + c n − 1 ( ( T 1 ) n − 1 − ( T 1 ) n ) ) , ∀ n ∈ { 1 , … , N − 1 } .

Observe now that n = N − 1 , we have u N − 1 = u N , so that

(20) u N − 1 ≈ a N − 1 u N − 1 + b N − 1 u 0 + c N − 1 ( T 1 ) N − 1 + ( h 1 ) N − 1 × a N − 1 u N − 1 + b N − 1 u 0 + c N − 1 ∂ 2 u N − 1 ∂ y 2 d 2 + c N − 1 − ( U 0 ) n ( u n − ( U 0 ) n − 1 ) d − ( V 0 ) n ∂ u n ∂ y − ( P 0 ) n − ( P 0 ) n − 1 d d 2 e 1 + ( h 1 ) N − 1 .

This last equation is a second-order ODE in u N − 1 , which must be solved with the boundary conditions

u N − 1 ( 0 ) = u N − 1 ( 1 ) = 0 .

Summarizing, we have obtained u N − 1 .

Similarly, we may obtain v N − 1 and P N − 1 .

Having u N − 1 , we may obtain u N − 2 with n = N − 2 in Eq. (19) (neglecting ( E r ) N − 2 ).

Similarly, we may obtain v N − 2 and P N − 2 .

Having u N − 2 , we may obtain u N − 3 with n = N − 3 in Eq. (19) (neglecting ( E r ) N − 3 ).

Similarly, we may obtain v N − 3 and P N − 3 .

And so on, up to obtaining u 1 , v 1 and P 1 .

The next step is to replace { ( U 0 ) n , ( V 0 ) n , ( P 0 ) n } by { u n , v n , P n } and repeat the process until an appropriate convergence criterion is satisfied.

Here, we present a concerning software in MAT-LAB based in this last algorithm (with small changes and differences where we have set K = 155 and ν = 0.0094 ).

*******************************

  1. clear all

  2. m8=500;

  3. d=1/m8;

  4. m9=140;

  5. d1=1/m9;

  6. e1=0.01;

  7. K=155.0;

  8. m2=zeros(m9-1,m9-1);

  9. for i=2:m9-2

  10. m2(i,i)=-2.0;

  11. m2(i,i+1)=1.0;

  12. m2(i,i-1)=1.0;

  13. end;

  14. m2(1,1)=-1.0;

  15. m2(1,2)=1.0;

  16. m2(m9-1,m9-1)=-1.0;

  17. m2(m9-1,m9-2)=1.0;

  18. m22=zeros(m9-1,m9-1);

  19. for i=2:m9-2

  20. m22(i,i)=-2.0;

  21. m22(i,i+1)=1.0;

  22. m22(i,i-1)=1.0;

  23. end;

  • m22(1,1)=-2.0;

  • m22(1,2)=1.0;

  • m22(m9-1,m9-1)=-2.0;

  • m22(m9-1,m9-2)=1.0;

  • m1a=zeros(m9-1,m9-1);

  • m1b=zeros(m9-1,m9-1);

  • for i=1:m9-2

  • m1a(i,i)=-1.0;

  • m1a(i,i+1)=1.0;

  • end;

  • m1a(m9-1,m9-1)=-1.0;

  • for i=2:m9-1

  • m1b(i,i)=1.0;

  • m1b(i,i-1)=-1.0;

  • end;

  • m1b(1,1)=1.0;

  • m1=(m1a+m1b)/2;

  • Id=eye(m9-1);

  • a(1)=1/(2+K* d 2 /e1);

  • b(1)=1/(2+K* d 2 /e1);

  • c(1)=1/(2+K* d 2 /e1);

  • for i=2:m8-1

  • a(i)=1/(2-a(i-1)+K* d 2 /e1);

  • b(i)=a(i)*b(i-1);

  • c(i)=(c(i-1)+1)*a(i);

  • end;

  • for i=1:m9-1

  • u5(i,1)=0.55*i*d1*(1-i*d1);

  • end;

  • uo=u5;

  • o=zeros(m9-1,1);

  • po=0.15*ones(m9-1,1);

  • for i=1:m8-1

  • Uo(:,i)=0.25*ones(m9-1,1);

  • Vo(:,i)=0.05*ones(m9-1,1);

  • Po(:,i)=0.05*ones(m9-1,1);

  • end;

  • for i=1:m8-1

  • U1(:,i)=Uo(:,i);

  • V1(:,i)=Vo(:,i);

  • P1(:,i)=Po(:,i);

  • end;

  • for k7=1:1

  • e1=e1*.94;

  • b14=1.0;

  • k1=1;

  • k1max=3000;

  • while ( b 14 > 1 0 − 3.5 ) and ( k 1 < k 1 m a x )

  • k1=k1+1;

  • a(1)=1/(2+K* d 2 /e1);

  • b(1)=a(1);

  • c1(:,1)=a(1)*K*Uo(:,1)* d 2 /e1;

  • c2(:,1)=a(1)*K*Vo(:,1)* d 2 /e1;

  • c3(:,1)=a(1)*(K*Po(:,1)* d 2 /e1);

  • for i=2:m8-1

  • a(i)=1/(2+K* d 2 /e1-a(i-1));

  • b(i)=a(i)*(b(i-1));

  • c1(:,i)=a(i)*(c1(:,i-1)+K*Uo(:,i)* d 2 /e1);

  • c2(:,i)=a(i)*(c2(:,i-1)+K*Vo(:,i)* d 2 /e1);

  • c3(:,i)=a(i)*(c3(:,i-1)+K*Po(:,i)* d 2 /e1);

  • end;

  • i=1;

  • M50=(Id-a(m8-1)*Id-c(m8-1)*m22/ d 1 2 * d 2 );

  • z1=b(m8-1)*uo+c1(:,m8-i)+c(m8-1)*(-Vo(:,m8-i).*(m1*Uo(:,m8-i)))/d1* d 2 /e1;

  • M60=(Id-a(m8-1)*Id-c(m8-1)*m22/ d 1 2 * d 2 );

  • z2=b(m8-1)*vo+c2(:,m8-i)+c(m8-1)*(-Vo(:,m8-i)).*(m1*Vo(:,m8-i))/d1* d 2 /e1;

  • M70=(Id-a(m8-1)*Id-c(m8-1)*m2/ d 1 2 * d 2 );

  • z3=b(m8-1)*po+c(m8-1)*((m1/d1*Vo(:,m8-i)).*(m1/d1*Vo(:,m8-i))* d 2 )+c3(:,m8-i);

  • U(:,m8-1)=inv(M50)*z1;

  • V(:,m8-1)=inv(M60)*z2;

  • P(:,m8-1)=inv(M70)*z3;

  • for i=2:m8-1

  • M50=(Id-c(m8-i)*m22/ d 1 2 * d 2 );

  • z1=b(m8-i)*uo+a(m8-i)*U(:,m8-i+1);

  • z1=z1+c(m8-i)*(-U(:,m8-i+1).*(Uo(:,m8-i+1)-Uo(:,m8-i))*d/e1;

  • z1=z1-(Po(:,m8-i+1)-Po(:,m8-i))*d/e1);

  • z1=z1+c1(:,m8-i);

  • z1=z1+c(m8-i)*(-V(:,m8-i+1).*(m1*Uo(:,m8-i))/d1* d 2 )/e1;

  • M60=(Id-c(m8-i)*m22/ d 1 2 * d 2 );

  • z2=b(m8-i)*vo+a(m8-i)*V(:,m8-i+1);

  • z2=z2+c(m8-i)*(-(U(:,m8-i+1).*(Vo(:,m8-i+1)-Vo(:,m8-i))*d/e1;

  • z2=z2+V(:,m8-i+1)).*(m1*Vo(:,m8-i)/d1* d 2 )/e1);

  • z2=z2-c(m8-i)*(m1*Po(:,m8-i)/d1* d 2 )/e1;

  • z2=z2+c2(:,m8-i);

  • M70=(Id-c(m8-i)*m2/ d 1 2 * d 2 );

  • z3=b(m8-i)*po+a(m8-i)*P(:,m8-i+1);

  • z3=z3+c(m8-i)*((Uo(:,m8-i+1)-Uo(:,m8-i)).*(Uo(:,m8-i+1)-Uo(:,m8-i));

  • z3=z3+(m1/d1*Vo(:,m8-i)).*(m1/d1*Vo(:,m8-i))* d 2 );

  • z3=z3+2*(m1/d1*Uo(:,m8-i)).*(Vo(:,m8-i+1)-Vo(:,m8-i))*d;

  • z3=z3+c3(:,m8-i);

  • U(:,m8-i)=inv(M50)*z1;

  • V(:,m8-i)=inv(M60)*z2;

  • P(:,m8-i)=inv(M70)*z3;

  • end;

  • b14=max(max(abs(U-Uo)));

  • b14

  • Uo=U;

  • Vo=V;

  • Po=P;

  • k1

  • U(m9/2,10)

  • end;

  • k7

  • end;

  • for i=1:m9-1

  • y(i)=i*d1;

  • end;

  • for i=1:m8-1

  • x(i)=i*d;

  • end;

  • mesh(x,y,U);

**********************************

For the field of velocities U and V and the pressure field P , please see Figures 1, 2, and 3, respectively.

Figure 1 Solution U ( x , y ) U\left(x,y) for the case ν = 0.0094 \nu =0.0094 .
Figure 1

Solution U ( x , y ) for the case ν = 0.0094 .

Figure 2 Solution V ( x , y ) V\left(x,y) for the case ν = 0.0094 \nu =0.0094 .
Figure 2

Solution V ( x , y ) for the case ν = 0.0094 .

Figure 3 Solution P ( x , y ) P\left(x,y) for the case ν = 0.0094 \nu =0.0094 .
Figure 3

Solution P ( x , y ) for the case ν = 0.0094 .

4 A software in MATHEMATICA related to the previous algorithm

In this section, we develop the solution for the Navier–Stokes system through the generalized method of lines, similar to the results presented in the study by Botelho [1], but now in a Navier–Stokes system context.

We present a software in MATHEMATICA for N = 10 lines for the case in which

(21) ν ∇ 2 u − u ∂ x u − v ∂ y u − ∂ x P = 0 , in Ω , ν ∇ 2 v − u ∂ x v − v ∂ y v − ∂ y P = 0 , in Ω , ∇ 2 P + ( ∂ x u ) 2 + ( ∂ y v ) 2 + 2 ( ∂ y u ) ( ∂ x v ) = 0 , in Ω .

We consider it in polar coordinates, with ν = e 1 = 0.1 , and with

Ω = { ( r , θ ) ∈ R 2 : 1 ≤ r ≤ 2 , 0 ≤ θ ≤ 2 π } ,

∂ Ω 1 = { ( 1 , θ ) ∈ R 2 : 0 ≤ θ ≤ 2 π }

and

∂ Ω 2 = { ( 2 , θ ) ∈ R 2 : 0 ≤ θ ≤ 2 π } .

The boundary conditions are

u = v = 0 , P = 0.15 , on ∂ Ω 1 ,

u = u f [ x ] , v = 0 , P = 0.12 , on ∂ Ω 2 .

From now and on, x stands for θ .

We remark that some changes have been made, concerning the original conception, in order to make it suitable through the software MATHEMATICA for such a Navier–Stokes system.

We highlight that the approximation error for the computation between two adjacent lines is proportional to 1 ∕ K , similarly as indicated in previous section. Therefore, as K > 0 is larger, the related approximation is of a better quality. However, if K > 0 is too much large, the converging process obtains slower.

At this point, we provide some details about the numerical procedure.

For d = 1 ∕ N and r n = 1 + n d , in a partial finite differences scheme and in polar coordinates, we generically denote

L n ( u ) = u n + 1 − 2 u n + u n − 1 d 2 + 1 r n u n − u n − 1 d + 1 r n 2 ∂ 2 u n ∂ x 2 ,

( d 1 ) n = ( u n − u n − 1 ) d f 1 ( x ) − ∂ u n ∂ x f 2 ( x ) r n ,

( d 2 ) n = ( u n − u n − 1 ) d f 2 ( x ) + ∂ u n ∂ x f 1 ( x ) r n ,

where f 1 ( x ) = cos ( x ) and f 2 ( x ) = sin ( x ) .

Thus, the concerning approximate Navier–Stokes system stands for

ν L n ( u ) − u n ( d 1 ) n ( u ) − v n ( d 2 ) n ( u ) − ( d 1 ) n P = 0 ,

ν L n ( v ) − u n ( d 1 ) n ( v ) − v n ( d 2 ) n ( v ) − ( d 2 ) n P = 0 ,

L n P + ( d 1 ) n ( u ) 2 + ( d 2 ) n ( v ) 2 + 2 ( d 2 ) n ( u ) ( d 1 ) n = 0

∀ n ∈ { 1 , … , N − 1 } , for an appropriate small real parameter ν > 0 to be specified.

Similarly as in the previous section, we may obtain

u n ≈ a n u n + 1 + b n ( T 1 ) n ( u n + 1 , u n ) + ( c 1 ) n , v n ≈ a n v n + 1 + b n ( T 2 ) n ( u n + 1 , u n ) + ( c 2 ) n , P n ≈ a n P n + 1 + b n ( T 3 ) n ( u n + 1 , u n ) + ( c 3 ) n + ( b 11 ) n P 0 ,

where the process to obtain { a n , b n , ( c 1 ) n , ( c 2 ) n , ( c 3 ) n , ( b 11 ) n } and { ( T 1 ) n , ( T 2 ) n , ( T 3 ) n } is indicated in the software presented in the next lines.

Observe that for n = N − 1 , we obtain u N = u f [ x ] , v N = v f [ x ] , and P N = P f [ x ] , so that generically denoting u = ( u , v , P ) , we set

u N − 1 ≈ a N − 1 u N + b N − 1 ( T 1 ) N − 1 ( u N , ( u 0 ) N − 1 ) + ( c 1 ) N − 1 ,

v N − 1 ≈ a N − 1 v N + b N − 1 ( T 2 ) N − 1 ( u N , ( u 0 ) N − 1 ) + ( c 2 ) N − 1 ,

and

P N − 1 ≈ a N − 1 P N + b N − 1 ( T 3 ) N − 1 ( u N , ( u 0 ) N − 1 ) + ( c 3 ) N − 1 .

Similarly, we may obtain

{ u N − 2 , v N − 2 , P N − 2 }

and so on up to obtaining

{ u 1 , v 1 , P 1 } .

The next step is to replace

{ ( u 0 ) n , ( v 0 ) n , ( P 0 ) n }

by

{ u n , v n , P n }

and then to repeat the process until an appropriate convergence criterion is satisfied.

Here the concerning software.

*************************************************

  1. m 8 = 10 ;

  2. C l e a r [ t 3 , t 4 ] ;

  3. d = 1.0 ∕ m 8 ;

  4. K = 4.0 ;

  5. e 1 = 0.1 ; ( here ν = e 1 )

  6. U o o [ x − ] = 0.0 ;

  7. V o o [ x − ] = 0.0 ;

  8. P o o [ x − ] = 0.15 ;

  9. F o r [ i = 1 , i < m 8 + 1 , i + + ,

  10. u o [ i ] = 0.05 ;

  11. v o [ i ] = 0.05 ;

  12. P o [ i ] = 0.05 ] ;

  1. F o r [ k = 1 , k < 80 , k + + , (here we have fixed the number of iterations)

  2. P r i n t [ k ] ;

  3. a [ 1 ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 ) ;

  4. b [ 1 ] = a [ 1 ] ;

  5. b 11 [ 1 ] = a [ 1 ] ;

  6. c 1 [ 1 ] = a [ 1 ] * ( K * u o [ 1 ] ) * d 2 ∕ e 1 ;

  7. c 2 [ 1 ] = a [ 1 ] * ( K * v o [ 1 ] ) * d 2 ∕ e 1 ;

  8. c 3 [ 1 ] = a [ 1 ] * ( K * P o [ 1 ] + P 1 ) * d 2 ∕ e 1 ;

  1. F o r [ i = 2 , i < m 8 , i + + ,

  2. a [ i ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 − a [ i − 1 ] ) ;

  3. b [ i ] = a [ i ] * ( b [ i − 1 ] + 1 ) ;

  4. b 11 [ i ] = a [ i ] * b 11 [ i − 1 ] ;

  5. c 1 [ i ] = a [ i ] * ( c 1 [ i − 1 ] + ( K * u o [ i ] ) * d 2 ∕ e 1 ) ;

  6. c 2 [ i ] = a [ i ] * ( c 2 [ i − 1 ] + ( K * v o [ i ] ) * d 2 ∕ e 1 ) ;

  7. c 3 [ i ] = a [ i ] * ( c 3 [ i − 1 ] + ( K * P o [ i ] ) * d 2 ∕ e 1 ) ] ;

  8. u [ m 8 ] = u f [ x ] * t 3 ; v [ m 8 ] = v f [ x ] * t 3 ; P [ m 8 ] = 0.12 ; d 1 = 1.0 ;

  1. F o r [ i = 1 , i < m 8 , i + + ,

  2. P r i n t [ i ] ;

  3. t [ m 8 − i ] = 1.0 + ( m 8 − i ) * d ;

  4. D x u = ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 −

  5. D [ u o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  6. D y u = ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 +

  7. D [ u o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  8. D x v = ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 −

  9. D [ v o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  10. D y v = ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 +

  11. D [ v o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  12. D x P = ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 −

  13. D [ P o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  1. D y P = ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 +

  2. D [ P o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  3. T 1 = − ( u [ m 8 − i + 1 ] * D x u + v [ m 8 − i + 1 ] * D y u + D x P ) * d 2 ∕ e 1

  4. + ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2 +

  5. D [ u o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  6. T 2 = − ( u [ m 8 − i + 1 ] * D x v + v [ m 8 − i + 1 ] * D y v + D y P ) *

  7. d 2 ∕ e 1 + ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2

  8. + D [ v o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  9. T 3 = ( D x u 2 + D y v 2 + 2 * D y u * D x v ) * d 2

  10. + ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2 +

  11. D [ P o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  1. A 1 = a [ m 8 − i ] * u [ m 8 − i + 1 ] + b [ m 8 − i ] * T 1 + c 1 [ m 8 − i ] ;

  2. A 1 = E x p a n d [ A 1 ] ;

  3. A 1 = S e r i e s [ A 1 , { t 4 , 0 , 1 } , { t 3 , 0 , 2 } , { u f [ x ] , 0 , 2 } , { u f ′ [ x ] , 0 , 1 } , { u f ′ ′ [ x ] , 0 , 1 } ,

  4. { u f ′ ′ ′ [ x ] , 0 , 0 } , { u f ′ ′ ′ ′ [ x ] , 0 , 0 } , { v f [ x ] , 0 , 1 } , { v f ′ [ x ] , 0 , 1 } , { v f ′ ′ [ x ] , 0 , 1 } ,

  5. { v f ′ ′ ′ [ x ] , 0 , 0 } , { v f ′ ′ ′ ′ [ x ] , 0 , 0 } ,

  6. { f 1 [ x ] , 0 , 1 } , { f 2 [ x ] , 0 , 1 } , { f 1 ′ [ x ] , 0 , 0 } , { f 2 ′ [ x ] , 0 , 0 } , { f 1 ′ ′ [ x ] , 0 , 0 } , { f 2 ′ ′ [ x ] , 0 , 0 } ] ;

  7. A 1 = N o r m a l [ A 1 ] ;

  8. u [ m 8 − i ] = E x p a n d [ A 1 ] ;

  1. A 2 = a [ m 8 − i ] * v [ m 8 − i + 1 ] + b [ m 8 − i ] * T 2 + c 2 [ m 8 − i ] ;

  2. A 2 = E x p a n d [ A 2 ] ;

  3. A 2 = S e r i e s [ A 2 , { t 4 , 0 , 1 } , { t 3 , 0 , 2 } , { u f [ x ] , 0 , 1 } , { u f ′ [ x ] , 0 , 1 } , { u f ′ ′ [ x ] , 0 , 1 } ,

  4. { u f ′ ′ ′ [ x ] , 0 , 0 } , { u f ′ ′ ′ ′ [ x ] , 0 , 0 } , { v f [ x ] , 0 , 2 } , { v f ′ [ x ] , 0 , 1 } , { v f ′ ′ [ x ] , 0 , 1 } ,

  5. { v f ′ ′ ′ [ x ] , 0 , 0 } , { v f ′ ′ ′ ′ [ x ] , 0 , 0 } ,

  6. { f 1 [ x ] , 0 , 1 } , { f 2 [ x ] , 0 , 1 } , { f 1 ′ [ x ] , 0 , 0 } , { f 2 ′ [ x ] , 0 , 0 } ,

  7. { f 1 ′ ′ [ x ] , 0 , 0 } , { f 2 ′ ′ [ x ] , 0 , 0 } ] ;

  8. A 2 = N o r m a l [ A 2 ] ;

  9. v [ m 8 − i ] = E x p a n d [ A 2 ] ;

  1. A 3 = a [ m 8 − i ] * P [ m 8 − i + 1 ] + b [ m 8 − i ] * T 3 + c 3 [ m 8 − i ] + b 11 [ m 8 − i ] * P o o [ x ] ;

  2. A 3 = E x p a n d [ A 3 ] ;

  3. A 3 = S e r i e s [ A 3 , { t 4 , 0 , 2 } , { t 3 , 0 , 2 } , { u f [ x ] , 0 , 1 } , { u f ′ [ x ] , 0 , 1 } , { u f ′ ′ [ x ] , 0 , 0 } , { u f ′ ′ ′ [ x ] , 0 , 0 } ,

  4. { u f ′ ′ ′ ′ [ x ] , 0 , 0 } , { v f [ x ] , 0 , 1 } , { v f ′ [ x ] , 0 , 1 } , { v f ′ ′ [ x ] , 0 , 0 } , { v f ′ ′ ′ [ x ] , 0 , 0 } ,

  5. { v f ′ ′ ′ ′ [ x ] , 0 , 0 } , { f 1 [ x ] , 0 , 1 } , { f 2 [ x ] , 0 , 1 } ,

  6. { f 1 ′ [ x ] , 0 , 0 } , { f 2 ′ [ x ] , 0 , 0 } , { f 1 ′ ′ [ x ] , 0 , 0 } , { f 2 ′ ′ [ x ] , 0 , 0 } ] ;

  7. A 3 = N o r m a l [ A 3 ] ;

  8. P [ m 8 − i ] = E x p a n d [ A 3 ] ] ;

  1. F o r [ i = 1 , i < m 8 + 1 , i + + ,

  2. u o [ i ] = u [ i ] ;

  3. v o [ i ] = v [ i ] ;

  4. P o [ i ] = P [ i ] ] ; d 1 = 1.0 ;

  5. P r i n t [ E x p a n d [ U [ m 8 ∕ 2 ] ] ] ]

  6. F o r [ i = 1 , i < m 8 , i + + ,

  7. P r i n t [ “ u [ ′ ′ , i , ′ ′ ] = ” , u [ i ] [ x ] ] ]

Here we present the related line expressions obtained for the lines n = 1 , n = 5 and n = 9 of a total of N = 10 lines.

1. L i n e n = 1

(22) u [ 1 ] = 1.58658 * 1 0 − 9 + 0.019216 f 1 [ x ] + 3.68259 * 1 0 − 11 f 2 [ x ] + 0.132814 u f [ x ] + 2.28545 * 1 0 − 8 f 1 [ x ] u f [ x ] − 1.69037 * 1 0 − 8 f 2 [ x ] u f [ x ] − 0.263288 f 1 [ x ] u f [ x ] 2 − 6.02845 * 1 0 − 9 f 1 [ x ] ( u f ′ ) [ x ] + 6.02845 * 1 0 − 9 f 2 [ x ] ( u f ′ ) [ x ] + 0.104284 f 2 [ x ] u f [ x ] ( u f ′ ) [ x ] + 0.0127544 ( u f ″ ) [ x ] + 9.39885 * 1 0 − 9 f 1 [ x ] ( u f ″ ) [ x ] − 5.26303 * 1 0 − 9 f 2 [ x ] ( u f ″ ) [ x ] − 0.0340276 f 1 [ x ] u f [ x ] ( u f ″ ) [ x ] + 0.0239544 f 2 [ x ] ( u f ′ ) [ x ] ( u f ″ ) [ x ]

2. L i n e n = 5

(23) u [ 5 ] = 4.25933 * 1 0 − 9 + 0.0436523 f 1 [ x ] + 9.88625 * 1 0 − 11 f 2 [ x ] + 0.572969 u f [ x ] + 6.87985 * 1 0 − 8 f 1 [ x ] u f [ x ] − 4.40534 * 1 0 − 8 f 2 [ x ] u f [ x ] − 0.765222 f 1 [ x ] u f [ x ] 2 − 1.61319 * 1 0 − 8 f 1 [ x ] ( u f ′ ) [ x ] + 1.61319 * 1 0 − 8 f 2 [ x ] ( u f ′ ) [ x ] + 0.363471 f 2 [ x ] u f [ x ] ( u f ′ ) [ x ] + 0.0333685 ( u f ″ ) [ x ] + 2.39576 * 1 0 − 8 f 1 [ x ] ( u f ″ ) [ x ] − 1.27491 * 1 0 − 8 f 2 [ x ] ( u f ″ ) [ x ] − 0.0342544 f 1 [ x ] u f [ x ] ( u f ″ ) [ x ] + 0.0509889 f 2 [ x ] ( u f ′ ) [ x ] ( u f ″ ) [ x ]

3. L i n e n = 9

(24) u [ 9 ] = 1.15848 * 1 0 − 9 + 0.0136828 f 1 [ x ] + 2.68892 * 1 0 − 11 f 2 [ x ] + 0.922534 u f [ x ] + 2.16498 * 1 0 − 8 f 1 [ x ] u f [ x ] − 1.16065 * 1 0 − 8 f 2 [ x ] u f [ x ] − 0.278966 f 1 [ x ] u f [ x ] 2 − 4.25263 * 1 0 − 9 f 1 [ x ] ( u f ′ ) [ x ] + 4.25263 * 1 0 − 9 f 2 [ x ] ( u f ′ ) [ x ] + 0.154642 f 2 [ x ] u f [ x ] ( u f ′ ) [ x ] + 0.0110114 ( u f ″ ) [ x ] + 6.13523 * 1 0 − 9 f 1 [ x ] ( u f ″ ) [ x ] − 3.23081 * 1 0 − 9 f 2 [ x ] ( u f ″ ) [ x ] + 0.0146222 f 1 [ x ] u f [ x ] ( u f ″ ) [ x ] + 0.0090088 f 2 [ x ] ( u f ′ ) [ x ] ( u f ″ ) [ x ]

5 The software and numerical results for a more specific example

In this section, we present numerical results for the same Navier–Stokes system and domain as in the previous one, but now with different boundary conditions.

In this example, we set ν = 0.1 and the boundary conditions are

u = v = 0 , P = 0.15 , on ∂ Ω 1 , u = − 1.0 sin [ x ] , v = 1.0 cos [ x ] , P = 0.12 , on ∂ Ω 2 .

Here the concerning software:

*************************************************

  1. m 8 = 10 ;

  2. C l e a r [ t 3 , t 4 ] ;

  3. d = 1.0 ∕ m 8 ;

  4. K = 4.0 ;

  5. e 1 = 0.1 ;

  6. U o o [ x − ] = 0.0 ;

  7. V o o [ x − ] = 0.0 ;

  8. P o o [ x − ] = 0.15 ;

  1. F o r [ i = 1 , i < m 8 + 1 , i + + ,

  2. u o [ i ] = 0.05 ;

  3. v o [ i ] = 0.05 ;

  4. P o [ i ] = 0.05 ] ;

  5. f 1 [ x − ] = C o s [ x ] ;

  6. f 2 [ x − ] = S i n [ x ] ;

  7. F o r [ k = 1 , k < 80 , k + + , (here we have fixed the number of iterations)

  8. P r i n t [ k ] ;

  9. a [ 1 ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 ) ;

  10. b [ 1 ] = a [ 1 ] ;

  11. b 11 [ 1 ] = a [ 1 ] ;

  12. c 1 [ 1 ] = a [ 1 ] * ( K * u o [ 1 ] ) * d 2 ∕ e 1 ;

  13. c 2 [ 1 ] = a [ 1 ] * ( K * v o [ 1 ] ) * d 2 ∕ e 1 ;

  14. c 3 [ 1 ] = a [ 1 ] * ( K * P o [ 1 ] + P 1 ) * d 2 ∕ e 1 ;

  1. F o r [ i = 2 , i < m 8 , i + + ,

  2. a [ i ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 − a [ i − 1 ] ) ;

  3. b [ i ] = a [ i ] * ( b [ i − 1 ] + 1 ) ;

  4. b 11 [ i ] = a [ i ] * b 11 [ i − 1 ] ;

  5. c 1 [ i ] = a [ i ] * ( c 1 [ i − 1 ] + ( K * u o [ i ] ) * d 2 ∕ e 1 ) ;

  6. c 2 [ i ] = a [ i ] * ( c 2 [ i − 1 ] + ( K * v o [ i ] ) * d 2 ∕ e 1 ) ;

  7. c 3 [ i ] = a [ i ] * ( c 3 [ i − 1 ] + ( K * P o [ i ] ) * d 2 ∕ e 1 ) ] ;

  8. u f [ x − ] = − 1.0 * S i n [ x ] ;

  9. v f [ x ] = 1.0 * C o s [ x ] ;

  10. u [ m 8 ] = u f [ x ] * t 3 ;

  11. v [ m 8 ] = v f [ x ] * t 3 ;

  12. P [ m 8 ] = 0.12 ;

  1. F o r [ i = 1 , i < m 8 , i + + ,

  2. P r i n t [ i ] ;

  3. t [ m 8 − i ] = 1.0 + ( m 8 − i ) * d ;

  4. D x u = ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 − D [ u o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  5. D y u = ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 + D [ u o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  6. D x v = ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 − D [ v o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  7. D y v = ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 + D [ v o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  8. D x P = ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d * f 1 [ x ] * t 4 − D [ P o [ m 8 − i ] , x ] * f 2 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  9. D y P = ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d * f 2 [ x ] * t 4 + D [ P o [ m 8 − i ] , x ] * f 1 [ x ] ∕ t [ m 8 − i ] * t 4 ;

  1. T 1 = − ( u [ m 8 − i + 1 ] * D x u + v [ m 8 − i + 1 ] * D y u + D x P ) * d 2 ∕ e 1 + ( u o [ m 8 − i + 1 ] − u o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2 + D [ u o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  2. T 2 = − ( u [ m 8 − i + 1 ] * D x v + v [ m 8 − i + 1 ] * D y v + D y P ) * d 2 ∕ e 1 + ( v o [ m 8 − i + 1 ] − v o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2 + D [ v o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  3. T 3 = ( D x u 2 + D y v 2 + 2 * D y u * D x v ) * d 2 + ( P o [ m 8 − i + 1 ] − P o [ m 8 − i ] ) ∕ d ∕ t [ m 8 − i ] * d 2 + D [ P o [ m 8 − i + 1 ] , { x , 2 } ] ∕ t [ m 8 − i ] 2 * d 2 ;

  4. A 1 = a [ m 8 − i ] * u [ m 8 − i + 1 ] + b [ m 8 − i ] * T 1 + c 1 [ m 8 − i ] ;

  5. A 1 = E x p a n d [ A 1 ] ;

  6. A 1 = S e r i e s [ A 1 , { t 4 , 0 , 1 } , { t 3 , 0 , 2 } , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  7. A 1 = N o r m a l [ A 1 ] ;

  1. u [ m 8 − i ] = E x p a n d [ A 1 ] ;

  2. A 2 = a [ m 8 − i ] * v [ m 8 − i + 1 ] + b [ m 8 − i ] * T 2 + c 2 [ m 8 − i ] ;

  3. A 2 = E x p a n d [ A 2 ] ;

  4. A 2 = S e r i e s [ A 2 , { t 4 , 0 , 1 } , { t 3 , 0 , 2 } , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  5. A 2 = N o r m a l [ A 2 ] ;

  6. v [ m 8 − i ] = E x p a n d [ A 2 ] ;

  7. A 3 = a [ m 8 − i ] * P [ m 8 − i + 1 ] + b [ m 8 − i ] * T 3 + c 3 [ m 8 − i ] + b 11 [ m 8 − i ] * P o o [ x ] ;

  8. A 3 = E x p a n d [ A 3 ] ;

  9. A 3 = S e r i e s [ A 3 , { t 4 , 0 , 2 } , { t 3 , 0 , 2 } , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  10. A 3 = N o r m a l [ A 3 ] ;

  11. P [ m 8 − i ] = E x p a n d [ A 3 ] ] ;

  1. F o r [ i = 1 , i < m 8 + 1 , i + + ,

  2. u o [ i ] = u [ i ] ;

  3. v o [ i ] = v [ i ] ;

  4. P o [ i ] = P [ i ] ] ;

  5. P r i n t [ E x p a n d [ P [ m 8 ∕ 2 ] ] ] ]

  6. F o r [ i = 1 , i < m 8 , i + + ,

  7. P r i n t [ ” u [ ” , i , ” ] = ” , u [ i ] [ x ] ] ] .

Here the corresponding line expressions for N = 10 lines

1. L i n e n = 1

(25) u [ 1 ] = 1.445 * 1 0 − 10 + 0.0183921 C o s [ x ] + 1.01021 * 1 0 − 9 C o s [ x ] 2 − 0.120676 S i n [ x ] + 3.62358 * 1 0 − 10 C o s [ x ] S i n [ x ] + 1.37257 * 1 0 − 9 S i n [ x ] 2 + 0.0620534 C o s [ x ] S i n [ x ] 2

2. L i n e n = 2

(26) u [ 2 ] = 2.60007 * 1 0 − 10 + 0.0307976 C o s [ x ] + 1.81088 * 1 0 − 9 C o s [ x ] 2 − 0.233061 S i n [ x ] + 6.53242 * 1 0 − 10 C o s [ x ] S i n [ x ] + 2.46412 * 1 0 − 9 S i n [ x ] 2 + 0.123121 C o s [ x ] S i n [ x ] 2

3. L i n e n = 3

(27) u [ 3 ] = 3.40796 * 1 0 − 10 + 0.0384482 C o s [ x ] + 2.36167 * 1 0 − 9 C o s [ x ] 2 − 0.339657 S i n [ x ] + 8.5759 * 1 0 − 10 C o s [ x ] S i n [ x ] + 3.21926 * 1 0 − 9 S i n [ x ] 2 + 0.180891 C o s [ x ] S i n [ x ] 2

4. L i n e n = 4

(28) u [ 4 ] = 3.83612 * 1 0 − 10 + 0.0420843 C o s [ x ] + 2.64262 * 1 0 − 9 C o s [ x ] 2 − 0.441913 S i n [ x ] + 9.66336 * 1 0 − 10 C o s [ x ] S i n [ x ] + 3.60895 * 1 0 − 9 S i n [ x ] 2 + 0.230559 C o s [ x ] S i n [ x ] 2

5. L i n e n = 5

(29) u [ 5 ] = 3.87923 * 1 0 − 10 + 0.0421948 C o s [ x ] + 2.65457 * 1 0 − 9 C o s [ x ] 2 − 0.540729 S i n [ x ] + 9.77606 * 1 0 − 10 C o s [ x ] S i n [ x ] + 3.63217 * 1 0 − 9 S i n [ x ] 2 + 0.266239 C o s [ x ] S i n [ x ] 2

6. L i n e n = 6

(30) u [ 6 ] = 3.56064 * 1 0 − 10 + 0.0391334 C o s [ x ] + 2.419 * 1 0 − 9 C o s [ x ] 2 − 0.636718 S i n [ x ] + 8.97185 * 1 0 − 10 C o s [ x ] S i n [ x ] + 3.31618 * 1 0 − 9 S i n [ x ] 2 + 0.281514 C o s [ x ] S i n [ x ] 2

7. L i n e n = 7

(31) u [ 7 ] = 2.93128 * 1 0 − 10 + 0.033175 C o s [ x ] + 1.97614 * 1 0 − 9 C o s [ x ] 2 − 0.730328 S i n [ x ] + 7.38127 * 1 0 − 10 C o s [ x ] S i n [ x ] + 2.71426 * 1 0 − 9 S i n [ x ] 2 + 0.269642 C o s [ x ] S i n [ x ] 2

8. L i n e n = 8

(32) u [ 8 ] = 2.0656 * 1 0 − 10 + 0.0245445 C o s [ x ] + 1.38127 * 1 0 − 9 C o s [ x ] 2 − 0.821902 S i n [ x ] + 5.19585 * 1 0 − 10 C o s [ x ] S i n [ x ] + 1.90085 * 1 0 − 9 S i n [ x ] 2 + 0.22362 C o s [ x ] S i n [ x ] 2

9. L i n e n = 9

(33) u [ 9 ] = 1.05509 * 1 0 − 10 + 0.0134316 C o s [ x ] + 6.99554 * 1 0 − 10 C o s [ x ] 2 − 0.911718 S i n [ x ] + 2.65019 * 1 0 − 10 C o s [ x ] S i n [ x ] + 9.64573 * 1 0 − 10 S i n [ x ] 2 + 0.13622 C o s [ x ] S i n [ x ] 2

Here, we present the related plots for the lines n = 2 , n = 4 , n = 6 , and n = 8 of a total of N = 10 lines.

For each line, we set N = 500 nodes on the interval [ 0 , 2 π ] , so that the units in x are 2 π ∕ 500 , where again x stands for θ .

For such lines, please see Figures 4, 5, 6, and 7, respectively.

Figure 4 Solution u 2 ( x ) {u}_{2}\left(x) for the line n = 2 n=2 , for the case ν = 0.1 \nu =0.1 .
Figure 4

Solution u 2 ( x ) for the line n = 2 , for the case ν = 0.1 .

Figure 5 Solution u 4 ( x ) {u}_{4}\left(x) for the line n = 4 n=4 , for the case ν = 0.1 \nu =0.1 .
Figure 5

Solution u 4 ( x ) for the line n = 4 , for the case ν = 0.1 .

Figure 6 Solution u 6 ( x ) {u}_{6}\left(x) for the line n = 6 n=6 , for the case ν = 0.1 \nu =0.1 .
Figure 6

Solution u 6 ( x ) for the line n = 6 , for the case ν = 0.1 .

Figure 7 Solution u 8 ( x ) {u}_{8}\left(x) for the line n = 8 n=8 , for the case ν = 0.1 \nu =0.1 .
Figure 7

Solution u 8 ( x ) for the line n = 8 , for the case ν = 0.1 .

6 Numerical results through the original conception of the generalized method of lines for the Navier–Stokes system

In this section, we develop the solution for the Navier–Stokes system through the generalized method of lines, as originally introduced in [13], with further developments in [14].

We present a software in MATHEMATICA for N = 10 lines for the case in which

(34) ν ∇ 2 u − u ∂ x u − v ∂ y u − ∂ x P = 0 , in Ω , ν ∇ 2 v − u ∂ x v − v ∂ y v − ∂ y P = 0 , in Ω , ∇ 2 P + ( ∂ x u ) 2 + ( ∂ y v ) 2 + 2 ( ∂ y u ) ( ∂ x v ) = 0 , in Ω .

Such a software refers to an algorithm presented in Chapter 27, in [2], in polar coordinates, with ν = 1.0 , and with

Ω = { ( r , θ ) ∈ R 2 : 1 ≤ r ≤ 2 , 0 ≤ θ ≤ 2 π } ,

∂ Ω 1 = { ( 1 , θ ) ∈ R 2 : 0 ≤ θ ≤ 2 π } ,

and

∂ Ω 2 = { ( 2 , θ ) ∈ R 2 : 0 ≤ θ ≤ 2 π } .

The boundary conditions are

u = v = 0 , P = 0.15 , on ∂ Ω 1 ,

u = − 1.0 sin ( θ ) , v = 1.0 cos ( θ ) , P = 0.10 , on ∂ Ω 2 .

We remark some changes have been made, concerning the original conception, to make it suitable through the software MATHEMATICA for such a Navier–Stokes system.

We highlight that the nature of this approximation is qualitative.

Here the concerning software in MATHEMATICA.

****************************************

  1. m 8 = 10 ;

  2. C l e a r [ z 1 , z 2 , z 3 , u 1 , u 2 , P , b 1 , b 2 , b 3 , a 1 , a 2 , a 3 ] ;

  3. C l e a r [ P f , t , a 11 , a 12 , a 13 , b 11 , b 12 , b 13 , t 3 ] ;

  4. d = 1.0 ∕ m 8 ;

  5. e 1 = 1.0 ;

  6. a 1 = 0.0 ;

  7. a 2 = 0.0 ;

  8. a 3 = 0.15 ;

  9. F o r [ i = 1 , i < m 8 , i + + ,

  10. P r i n t [ i ] ;

  11. C l e a r [ b 1 , b 2 , b 3 , u 1 , u 2 , P ] ;

  12. b 1 [ x − ] = u 1 [ i + 1 ] [ x ] ;

  13. b 2 [ x − ] = u 2 [ i + 1 ] [ x ] ;

  14. b 3 [ x − ] = P [ i + 1 ] [ x ] ;

  15. t [ i ] = 1 + i * d ;

  16. d u 1 x = C o s [ x ] * ( b 1 [ x ] − a 1 ) ∕ d * t 3 − 1 ∕ t [ i ] * S i n [ x ] * D [ b 1 [ x ] , x ] * t 3 ;

  17. d u 1 y = S i n [ x ] * ( b 1 [ x ] − a 1 ) ∕ d * t 3 + 1 ∕ t [ i ] * C o s [ x ] * D [ b 1 [ x ] , x ] * t 3 ;

  18. d u 2 x = C o s [ x ] * ( b 2 [ x ] − a 2 ) ∕ d * t 3 − 1 ∕ t [ i ] * S i n [ x ] * D [ b 2 [ x ] , x ] * t 3 ;

  19. d u 2 y = S i n [ x ] * ( b 2 [ x ] − a 2 ) ∕ d * t 3 + 1 ∕ t [ i ] * C o s [ x ] * D [ b 2 [ x ] , x ] * t 3 ;

  20. d P x = C o s [ x ] * ( b 3 [ x ] − a 3 ) ∕ d * t 3 − 1 ∕ t [ i ] * S i n [ x ] * D [ b 3 [ x ] , x ] * t 3 ;

  21. d P y = S i n [ x ] * ( b 3 [ x ] − a 3 ) ∕ d * t 3 + 1 ∕ t [ i ] * C o s [ x ] * D [ b 3 [ x ] , x ] * t 3 ;

  1. F o r [ k = 1 , k < 6 , k + + , (in this example, we have fixed a relatively small number of iterations)

  2. P r i n t [ k ] ;

  3. z 1 = ( u 1 [ i + 1 ] [ x ] + b 1 [ x ] + a 1 + 1 ∕ t [ i ] * ( b 1 [ x ] − a 1 ) * d + 1 ∕ t [ i ] 2 * D [ b 1 [ x ] , x , 2 ] * d 2

  4. − ( b 1 [ x ] * d u 1 x + b 2 [ x ] * d u 1 y ) * d 2 ∕ e 1 − d P x * d 2 ∕ e 1 ) ∕ 3.0 ;

  5. z 2 = ( u 2 [ i + 1 ] [ x ] + b 2 [ x ] + a 2 + 1 ∕ t [ i ] * ( b 2 [ x ] − a 2 ) * d + 1 ∕ t [ i ] 2 * D [ b 2 [ x ] , x , 2 ] * d 2

  6. − ( b 1 [ x ] * d u 2 x + b 2 [ x ] * d u 2 y ) * d 2 ∕ e 1 − d P y * d 2 ∕ e 1 ) ∕ 3.0 ;

  7. z 3 = ( P [ i + 1 ] [ x ] + b 3 [ x ] + a 3 + 1 ∕ t [ i ] * ( b 3 [ x ] − a 3 ) * d + 1 ∕ t [ i ] 2 * D [ b 3 [ x ] , x , 2 ] * d 2

  8. + ( d u 1 x * d u 1 x + d u 2 y * d u 2 y + 2.0 * d u 1 y * d u 2 x ) * d 2 ) ∕ 3.0 ;

  1. z 1 = S e r i e s [ z 1 , { u 1 [ i + 1 ] [ x ] , 0 , 2 } , { u 1 [ i + 1 ] ′ [ x ] , 0 , 1 } , { u 1 [ i + 1 ] ′ ′ [ x ] , 0 , 1 } ,

  2. { u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 1 } ,

  3. { u 2 [ i + 1 ] ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,

  4. { u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 0 } ,

  5. { P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,

  6. { S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;

  7. z 1 = N o r m a l [ z 1 ] ;

  8. z 1 = E x p a n d [ z 1 ] ;

  9. P r i n t [ z 1 ] ;

  1. z 2 = S e r i e s [ z 2 , { u 1 [ i + 1 ] [ x ] , 0 , 1 } , { u 1 [ i + 1 ] ′ [ x ] , 0 , 1 } , { u 1 [ i + 1 ] ′ ′ [ x ] , 0 , 1 } ,

  2. { u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 2 } ,

  3. { u 2 [ i + 1 ] ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,

  4. { u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 0 } ,

  5. { P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,

  6. { S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;

  7. z 2 = N o r m a l [ z 2 ] ;

  8. z 2 = E x p a n d [ z 2 ] ;

  9. P r i n t [ z 2 ] ;

  1. z 3 = S e r i e s [ z 3 , { u 1 [ i + 1 ] [ x ] , 0 , 2 } , { u 1 [ i + 1 ] ′ [ x ] , 0 , 1 } , { u 1 [ i + 1 ] ′ ′ [ x ] , 0 , 1 } ,

  2. { u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 2 } ,

  3. { u 2 [ i + 1 ] ′ [ x ] , 0 , 1 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,

  4. { u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 1 } ,

  5. { P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,

  6. { S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;

  7. z 3 = N o r m a l [ z 3 ] ;

  8. z 3 = E x p a n d [ z 3 ] ;

  9. P r i n t [ z 3 ] ;

  1. b 1 [ x − ] = z 1 ;

  2. b 2 [ x − ] = z 2 ;

  3. b 3 [ x − ] = z 3 ;

  4. b 11 = z 1 ;

  5. b 12 = z 2 ;

  6. b 13 = z 3 ; ] ;

  1. a 11 [ i ] = b 11 ;

  2. a 12 [ i ] = b 12 ;

  3. a 13 [ i ] = b 13 ;

  4. P r i n t [ a 11 [ i ] ] ;

  5. C l e a r [ b 1 , b 2 , b 3 ] ;

  6. u 1 [ i + 1 ] [ x − ] = b 1 [ x ] ;

  7. u 2 [ i + 1 ] [ x − ] = b 2 [ x ] ;

  8. P [ i + 1 ] [ x − ] = b 3 [ x ] ;

  9. a 1 = S e r i e s [ b 11 , { t 3 , 0 , 1 } ; { b 1 [ x ] , 0 , 1 } , { b 2 [ x ] , 0 , 1 } , { b 3 [ x ] , 0 , 1 } ,

  10. { b 1 ′ [ x ] , 0 , 0 } , { b 2 ′ [ x ] , 0 , 0 } , { b 3 ′ [ x ] , 0 , 0 } , { b 1 ″ [ x ] , 0 , 0 } , { b 2 ″ [ x ] , 0 , 0 } , { b 3 ″ [ x ] , 0 , 0 } ] ;

  11. a 1 = N o r m a l [ a 1 ] ;

  12. a 1 = E x p a n d [ a 1 ] ;

  13. a 2 = S e r i e s [ b 12 , { t 3 , 0 , 1 } ; { b 1 [ x ] , 0 , 1 } , { b 2 [ x ] , 0 , 1 } , { b 3 [ x ] , 0 , 1 } ,

  14. { b 1 ′ [ x ] , 0 , 0 } , { b 2 ′ [ x ] , 0 , 0 } , { b 3 ′ [ x ] , 0 , 0 } , { b 1 ″ [ x ] , 0 , 0 } , { b 2 ″ [ x ] , 0 , 0 } , { b 3 ″ [ x ] , 0 , 0 } ] ;

  15. a 2 = N o r m a l [ a 2 ] ;

  16. a 2 = E x p a n d [ a 2 ] ;

  17. a 3 = S e r i e s [ b 13 , { t 3 , 0 , 1 } ; { b 1 [ x ] , 0 , 1 } , { b 2 [ x ] , 0 , 1 } , { b 3 [ x ] , 0 , 1 } ,

  18. { b 1 ′ [ x ] , 0 , 0 } , { b 2 ′ [ x ] , 0 , 0 } , { b 3 ′ [ x ] , 0 , 0 } , { b 1 ″ [ x ] , 0 , 0 } , { b 2 ″ [ x ] , 0 , 0 } , { b 3 ″ [ x ] , 0 , 0 } ] ;

  19. a 3 = N o r m a l [ a 3 ] ;

  20. a 3 = E x p a n d [ a 3 ] ;

  1. b 1 [ x ] = − 1.0 * S i n [ x ] ;

  2. b 2 [ x ] = 1.0 * C o s [ x ] ;

  3. b 3 [ x ] = 0.10 ;

  1. F o r [ i = 1 , i < m 8 , i + + ,

  2. A 11 = a 11 [ m 8 − i ] ;

  3. A 11 = S e r i e s [ A 11 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  4. A 11 = N o r m a l [ A 11 ] ;

  5. A 11 = E x p a n d [ A 11 ] ;

  6. A 12 = a 12 [ m 8 − i ] ;

  7. A 12 = S e r i e s [ A 12 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  8. A 12 = N o r m a l [ A 12 ] ;

  9. A 12 = E x p a n d [ A 12 ] ;

  10. A 13 = a 13 [ m 8 − i ] ;

  11. A 13 = S e r i e s [ A 13 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;

  12. A 13 = N o r m a l [ A 13 ] ;

  13. A 13 = E x p a n d [ A 13 ] ;

  1. u 1 [ m 8 − i ] [ x − ] = A 11 ;

  2. u 2 [ m 8 − i ] [ x − ] = A 12 ;

  3. P [ m 8 − i ] [ x − ] = E x p a n d [ A 13 ] ;

  4. t 3 = 1.0 ;

  5. P r i n t [ ” u 1 [ ” , m 8 − i , ” ] = ” , A 11 ] ;

  6. C l e a r [ t 3 ] ;

  7. b 1 [ x ] = A 11 ;

  8. b 2 [ x ] = A 12 ;

  9. b 3 [ x ] = A 13 ; ] ;

  10. t 3 = 1.0 ;

  11. F o r [ i = 1 , i < m 8 , i + + ,

  12. P r i n t [ ” u 1 [ ” , i , ” ] = ” , u 1 [ i ] [ x ] ] ]

***************************************************

Here, the line expressions for the field of velocity u = { u 1 [ n ] ( x ) } , where again we emphasize N = 10 lines and ν = e 1 = 1.0 :

1.

(35) u 1 [ 1 ] ( x ) = 0.0044548 C o s [ x ] − 0.174091 S i n [ x ] + 0.00041254 C o s [ x ] 2 S i n [ x ] + 0.0260471 C o s [ x ] S i n [ x ] 2 − 0.000188598 C o s [ x ] 2 S i n [ x ] 2

2.

(36) u 1 [ 2 ] ( x ) = 0.00680614 C o s [ x ] − 0.331937 S i n [ x ] + 0.000676383 C o s [ x ] 2 S i n [ x ] + 0.0501544 C o s [ x ] S i n [ x ] 2 − 0.000176433 C o s [ x ] 2 S i n [ x ] 2

3.

(37) u 1 [ 3 ] ( x ) = 0.00775103 C o s [ x ] − 0.470361 S i n [ x ] + 0.000863068 C o s [ x ] 2 S i n [ x ] + 0.0682792 C o s [ x ] S i n [ x ] 2 − 0.000121656 C o s [ x ] 2 S i n [ x ] 2

4.

(38) u 1 [ 4 ] ( x ) = 0.00771379 C o s [ x ] − 0.589227 S i n [ x ] + 0.000994973 C o s [ x ] 2 S i n [ x ] + 0.0781784 C o s [ x ] S i n [ x ] 2 − 0.00006958 C o s [ x ] 2 S i n [ x ] 2

5.

(39) u 1 [ 5 ] ( x ) = 0.00701567 C o s [ x ] − 0.690152 S i n [ x ] + 0.00106158 C o s [ x ] 2 S i n [ x ] + 0.0796091 C o s [ x ] S i n [ x ] 2 − 0.0000330485 C o s [ x ] 2 S i n [ x ] 2

6.

(40) u 1 [ 6 ] ( x ) = 0.00589597 C o s [ x ] − 0.775316 S i n [ x ] + 0.00104499 C o s [ x ] 2 S i n [ x ] + 0.0734277 C o s [ x ] S i n [ x ] 2 − 0.0000121648 C o s [ x ] 2 S i n [ x ] 2

7.

(41) u 1 [ 7 ] ( x ) = 0.00452865 C o s [ x ] − 0.846947 S i n [ x ] + 0.000931782 C o s [ x ] 2 S i n [ x ] + 0.0609739 C o s [ x ] S i n [ x ] 2 − 2.74137 * 1 0 − 6 C o s [ x ] 2 S i n [ x ] 2

8.

(42) u 1 [ 8 ] ( x ) = 0.00303746 C o s [ x ] − 0.907103 S i n [ x ] + 0.000716865 C o s [ x ] 2 S i n [ x ] + 0.0437018 C o s [ x ] S i n [ x ] 2

9.

(43) u 1 [ 9 ] ( x ) = 0.00150848 C o s [ x ] − 0.957599 S i n [ x ] + 0.000403216 C o s [ x ] 2 S i n [ x ] + 0.0229802 C o s [ x ] S i n [ x ] 2

7 Conclusion

In this article, we have presented solutions for examples concerning the two-dimensional, time-independent, and incompressible Navier–Stokes system through the generalized method of lines. In particular, we have developed software in MAT-LAB and MATHEMATICA for approximate solutions of the numerical systems originated from a full (for the MAT-LAB software) and partial (for the MATHEMATICA codes) domain discretization concerning appropriate finite difference schemes. It is worth highlighting that, with little adaptations, it is possible to apply the MAT-LAB software for a very large class of domain shapes and related boundary conditions. We also obtain the appropriate boundary conditions for an equivalent elliptic system to the original Navier–Stokes one.

Finally, the extension of such results to R 3 , compressible and time-dependent cases is planned for a future work.

  1. Funding information: The author states that there is no external funding involved.

  2. Author contributions: The author has the entire responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The author declares no conflict of interest concerning this article.

  4. Data availability statement: Details on the software for numerical results available upon request.

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Received: 2023-11-04
Revised: 2024-01-13
Accepted: 2024-01-23
Published Online: 2024-03-22

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
  76. Collaborative control of multi-manipulator systems in intelligent manufacturing based on event-triggered and adaptive strategy
  77. Greenhouse monitoring system integrating NB-IOT technology and a cloud service framework
  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
  79. Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme
  80. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part I
  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
  82. Advanced integration of IoT and AI algorithms for comprehensive smart meter data analysis in smart grids
https://doi.org/10.1515/nleng-2022-0376
Schlagwörter für diesen Artikel
generalized method of lines; Navier–Stokes system; equivalent elliptic system
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
  6. A physically consistent AI-based SPH emulator for computational fluid dynamics
  7. Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
  8. A novel framework for effective structural vulnerability assessment of tubular structures using machine learning algorithms (GA and ANN) for hybrid simulations
  9. Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects
  10. Stability analysis of the corruption dynamics under fractional-order interventions
  11. Solutions of certain initial-boundary value problems via a new extended Laplace transform
  12. Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method
  13. Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system
  14. Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method
  15. Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach
  16. On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations
  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
  18. Influence of damage on large displacement buckling analysis of beams
  19. Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
  20. Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach
  21. A new operational matrix method to solve nonlinear fractional differential equations
  22. New solutions for the generalized q-deformed wave equation with q-translation symmetry
  23. Optimize the corrosion behaviour and mechanical properties of AISI 316 stainless steel under heat treatment and previous cold working
  24. Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena
  25. Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model
  26. Modeling credit risk with mixed fractional Brownian motion: An application to barrier options
  27. Method of feature extraction of abnormal communication signal in network based on nonlinear technology
  28. An innovative binocular vision-based method for displacement measurement in membrane structures
  29. An analysis of exponential kernel fractional difference operator for delta positivity
  30. Novel analytic solutions of strain wave model in micro-structured solids
  31. Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model
  32. Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel
  33. Non-linear influences of track dynamic irregularities on vertical levelling loss of heavy-haul railway track geometry under cyclic loadings
  34. Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm
  35. Validity and error analysis of calculating matrix exponential function and vector product
  36. Optimizing execution time and cost while scheduling scientific workflow in edge data center with fault tolerance awareness
  37. Estimating the dynamics of the drinking epidemic model with control interventions: A sensitivity analysis
  38. Online and offline physical education quality assessment based on mobile edge computing
  39. Discovering optical solutions to a nonlinear Schrödinger equation and its bifurcation and chaos analysis
  40. New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation
  41. Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation
  42. Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input
  43. Error analysis of arbitrarily high-order stepping schemes for fractional integro-differential equations with weakly singular kernels
  44. Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation
  45. An unconditionally stable numerical scheme for solving nonlinear Fisher equation
  46. Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium
  47. Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity
  48. Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation
  49. Experimental and numerical investigations into torsional-flexural behaviours of railway composite sleepers and bearers
  50. Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis
  51. Analysis of the magnetohydrodynamic effects on non-Newtonian fluid flow in an inclined non-uniform channel under long-wavelength, low-Reynolds number conditions
  52. Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems
  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
  76. Collaborative control of multi-manipulator systems in intelligent manufacturing based on event-triggered and adaptive strategy
  77. Greenhouse monitoring system integrating NB-IOT technology and a cloud service framework
  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
  79. Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme
  80. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part I
  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
  82. Advanced integration of IoT and AI algorithms for comprehensive smart meter data analysis in smart grids
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