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

Fabio Silva Botelho

doi:10.1515/nleng-2022-0376

Open Access Published by De Gruyter March 22, 2024

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

Fabio Silva Botelho

From the journal Nonlinear Engineering

https://doi.org/10.1515/nleng-2022-0376

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 ,

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 ).

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

clear all
m8=500;
d=1/m8;
m9=140;
d1=1/m9;
e1=0.01;
K=155.0;
m2=zeros(m9-1,m9-1);
for i=2:m9-2
m2(i,i)=-2.0;
m2(i,i+1)=1.0;
m2(i,i-1)=1.0;
end;
m2(1,1)=-1.0;
m2(1,2)=1.0;
m2(m9-1,m9-1)=-1.0;
m2(m9-1,m9-2)=1.0;
m22=zeros(m9-1,m9-1);
for i=2:m9-2
m22(i,i)=-2.0;
m22(i,i+1)=1.0;
m22(i,i-1)=1.0;
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 }

{ u n , v n , P n }

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

Here the concerning software.

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

m 8 = 10 ;
C l e a r [ t 3 , t 4 ] ;
d = 1.0 ∕ m 8 ;
K = 4.0 ;
e 1 = 0.1 ; ( here ν = e 1 )
U o o [ x − ] = 0.0 ;
V o o [ x − ] = 0.0 ;
P o o [ x − ] = 0.15 ;
F o r [ i = 1 , i < m 8 + 1 , i + + ,
u o [ i ] = 0.05 ;
v o [ i ] = 0.05 ;
P o [ i ] = 0.05 ] ;

F o r [ k = 1 , k < 80 , k + + , (here we have fixed the number of iterations)
P r i n t [ k ] ;
a [ 1 ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 ) ;
b [ 1 ] = a [ 1 ] ;
b 11 [ 1 ] = a [ 1 ] ;
c 1 [ 1 ] = a [ 1 ] * ( K * u o [ 1 ] ) * d 2 ∕ e 1 ;
c 2 [ 1 ] = a [ 1 ] * ( K * v o [ 1 ] ) * d 2 ∕ e 1 ;
c 3 [ 1 ] = a [ 1 ] * ( K * P o [ 1 ] + P 1 ) * d 2 ∕ e 1 ;

F o r [ i = 2 , i < m 8 , i + + ,
a [ i ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 − a [ i − 1 ] ) ;
b [ i ] = a [ i ] * ( b [ i − 1 ] + 1 ) ;
b 11 [ i ] = a [ i ] * b 11 [ i − 1 ] ;
c 1 [ i ] = a [ i ] * ( c 1 [ i − 1 ] + ( K * u o [ i ] ) * d 2 ∕ e 1 ) ;
c 2 [ i ] = a [ i ] * ( c 2 [ i − 1 ] + ( K * v o [ i ] ) * d 2 ∕ e 1 ) ;
c 3 [ i ] = a [ i ] * ( c 3 [ i − 1 ] + ( K * P o [ i ] ) * d 2 ∕ e 1 ) ] ;
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 ;

F o r [ i = 1 , i < m 8 , i + + ,
P r i n t [ i ] ;
t [ m 8 − i ] = 1.0 + ( m 8 − i ) * d ;
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 ;
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 ;
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 ;
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 ;
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 ;

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 ;
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 ;
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 ;
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 ;

A 1 = a [ m 8 − i ] * u [ m 8 − i + 1 ] + b [ m 8 − i ] * T 1 + c 1 [ m 8 − i ] ;
A 1 = E x p a n d [ A 1 ] ;
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 } ,
{ 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 } ,
{ v f ′ ′ ′ [ x ] , 0 , 0 } , { v f ′ ′ ′ ′ [ x ] , 0 , 0 } ,
{ 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 } ] ;
A 1 = N o r m a l [ A 1 ] ;
u [ m 8 − i ] = E x p a n d [ A 1 ] ;

A 2 = a [ m 8 − i ] * v [ m 8 − i + 1 ] + b [ m 8 − i ] * T 2 + c 2 [ m 8 − i ] ;
A 2 = E x p a n d [ A 2 ] ;
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 } ,
{ 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 } ,
{ v f ′ ′ ′ [ x ] , 0 , 0 } , { v f ′ ′ ′ ′ [ x ] , 0 , 0 } ,
{ 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 } ] ;
A 2 = N o r m a l [ A 2 ] ;
v [ m 8 − i ] = E x p a n d [ A 2 ] ;

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 ] ;
A 3 = E x p a n d [ A 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 } ,
{ 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 } ,
{ v f ′ ′ ′ ′ [ x ] , 0 , 0 } , { 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 } ] ;
A 3 = N o r m a l [ A 3 ] ;
P [ m 8 − i ] = E x p a n d [ A 3 ] ] ;

F o r [ i = 1 , i < m 8 + 1 , i + + ,
u o [ i ] = u [ i ] ;
v o [ i ] = v [ i ] ;
P o [ i ] = P [ i ] ] ; d 1 = 1.0 ;
P r i n t [ E x p a n d [ U [ m 8 ∕ 2 ] ] ] ]
F o r [ i = 1 , i < m 8 , i + + ,
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:

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

m 8 = 10 ;
C l e a r [ t 3 , t 4 ] ;
d = 1.0 ∕ m 8 ;
K = 4.0 ;
e 1 = 0.1 ;
U o o [ x − ] = 0.0 ;
V o o [ x − ] = 0.0 ;
P o o [ x − ] = 0.15 ;

F o r [ i = 1 , i < m 8 + 1 , i + + ,
u o [ i ] = 0.05 ;
v o [ i ] = 0.05 ;
P o [ i ] = 0.05 ] ;
f 1 [ x − ] = C o s [ x ] ;
f 2 [ x − ] = S i n [ x ] ;
F o r [ k = 1 , k < 80 , k + + , (here we have fixed the number of iterations)
P r i n t [ k ] ;
a [ 1 ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 ) ;
b [ 1 ] = a [ 1 ] ;
b 11 [ 1 ] = a [ 1 ] ;
c 1 [ 1 ] = a [ 1 ] * ( K * u o [ 1 ] ) * d 2 ∕ e 1 ;
c 2 [ 1 ] = a [ 1 ] * ( K * v o [ 1 ] ) * d 2 ∕ e 1 ;
c 3 [ 1 ] = a [ 1 ] * ( K * P o [ 1 ] + P 1 ) * d 2 ∕ e 1 ;

F o r [ i = 2 , i < m 8 , i + + ,
a [ i ] = 1 ∕ ( 2.0 + K * d 2 ∕ e 1 − a [ i − 1 ] ) ;
b [ i ] = a [ i ] * ( b [ i − 1 ] + 1 ) ;
b 11 [ i ] = a [ i ] * b 11 [ i − 1 ] ;
c 1 [ i ] = a [ i ] * ( c 1 [ i − 1 ] + ( K * u o [ i ] ) * d 2 ∕ e 1 ) ;
c 2 [ i ] = a [ i ] * ( c 2 [ i − 1 ] + ( K * v o [ i ] ) * d 2 ∕ e 1 ) ;
c 3 [ i ] = a [ i ] * ( c 3 [ i − 1 ] + ( K * P o [ i ] ) * d 2 ∕ e 1 ) ] ;
u f [ x − ] = − 1.0 * S i n [ x ] ;
v f [ x ] = 1.0 * C o s [ x ] ;
u [ m 8 ] = u f [ x ] * t 3 ;
v [ m 8 ] = v f [ x ] * t 3 ;
P [ m 8 ] = 0.12 ;

F o r [ i = 1 , i < m 8 , i + + ,
P r i n t [ i ] ;
t [ m 8 − i ] = 1.0 + ( m 8 − i ) * d ;
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 ;
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 ;
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 ;
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 ;
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 ;
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 ;

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 ;
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 ;
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 ;
A 1 = a [ m 8 − i ] * u [ m 8 − i + 1 ] + b [ m 8 − i ] * T 1 + c 1 [ m 8 − i ] ;
A 1 = E x p a n d [ A 1 ] ;
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 } ] ;
A 1 = N o r m a l [ A 1 ] ;

u [ m 8 − i ] = E x p a n d [ A 1 ] ;
A 2 = a [ m 8 − i ] * v [ m 8 − i + 1 ] + b [ m 8 − i ] * T 2 + c 2 [ m 8 − i ] ;
A 2 = E x p a n d [ A 2 ] ;
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 } ] ;
A 2 = N o r m a l [ A 2 ] ;
v [ m 8 − i ] = E x p a n d [ A 2 ] ;
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 ] ;
A 3 = E x p a n d [ A 3 ] ;
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 } ] ;
A 3 = N o r m a l [ A 3 ] ;
P [ m 8 − i ] = E x p a n d [ A 3 ] ] ;

F o r [ i = 1 , i < m 8 + 1 , i + + ,
u o [ i ] = u [ i ] ;
v o [ i ] = v [ i ] ;
P o [ i ] = P [ i ] ] ;
P r i n t [ E x p a n d [ P [ m 8 ∕ 2 ] ] ] ]
F o r [ i = 1 , i < m 8 , i + + ,
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.

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

m 8 = 10 ;
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 ] ;
C l e a r [ P f , t , a 11 , a 12 , a 13 , b 11 , b 12 , b 13 , t 3 ] ;
d = 1.0 ∕ m 8 ;
e 1 = 1.0 ;
a 1 = 0.0 ;
a 2 = 0.0 ;
a 3 = 0.15 ;
F o r [ i = 1 , i < m 8 , i + + ,
P r i n t [ i ] ;
C l e a r [ b 1 , b 2 , b 3 , u 1 , u 2 , P ] ;
b 1 [ x − ] = u 1 [ i + 1 ] [ x ] ;
b 2 [ x − ] = u 2 [ i + 1 ] [ x ] ;
b 3 [ x − ] = P [ i + 1 ] [ x ] ;
t [ i ] = 1 + i * d ;
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 ;
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 ;
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 ;
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 ;
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 ;
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 ;

F o r [ k = 1 , k < 6 , k + + , (in this example, we have fixed a relatively small number of iterations)
P r i n t [ k ] ;
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
− ( 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 ;
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
− ( 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 ;
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
+ ( 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 ;

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 } ,
{ u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 1 } ,
{ u 2 [ i + 1 ] ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,
{ u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 0 } ,
{ P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,
{ S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;
z 1 = N o r m a l [ z 1 ] ;
z 1 = E x p a n d [ z 1 ] ;
P r i n t [ z 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 } ,
{ u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 2 } ,
{ u 2 [ i + 1 ] ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,
{ u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 0 } ,
{ P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,
{ S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;
z 2 = N o r m a l [ z 2 ] ;
z 2 = E x p a n d [ z 2 ] ;
P r i n t [ z 2 ] ;

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 } ,
{ u 1 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { u 1 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] [ x ] , 0 , 2 } ,
{ u 2 [ i + 1 ] ′ [ x ] , 0 , 1 } , { u 2 [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { u 2 [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } ,
{ u 2 [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] [ x ] , 0 , 1 } , { P [ i + 1 ] ′ [ x ] , 0 , 1 } ,
{ P [ i + 1 ] ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ [ x ] , 0 , 0 } , { P [ i + 1 ] ′ ′ ′ ′ [ x ] , 0 , 0 } ,
{ S i n [ x ] , 0 , 1 } , { C o s [ x ] , 0 , 1 } ] ;
z 3 = N o r m a l [ z 3 ] ;
z 3 = E x p a n d [ z 3 ] ;
P r i n t [ z 3 ] ;

b 1 [ x − ] = z 1 ;
b 2 [ x − ] = z 2 ;
b 3 [ x − ] = z 3 ;
b 11 = z 1 ;
b 12 = z 2 ;
b 13 = z 3 ; ] ;

a 11 [ i ] = b 11 ;
a 12 [ i ] = b 12 ;
a 13 [ i ] = b 13 ;
P r i n t [ a 11 [ i ] ] ;
C l e a r [ b 1 , b 2 , b 3 ] ;
u 1 [ i + 1 ] [ x − ] = b 1 [ x ] ;
u 2 [ i + 1 ] [ x − ] = b 2 [ x ] ;
P [ i + 1 ] [ x − ] = b 3 [ x ] ;
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 } ,
{ 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 } ] ;
a 1 = N o r m a l [ a 1 ] ;
a 1 = E x p a n d [ a 1 ] ;
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 } ,
{ 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 } ] ;
a 2 = N o r m a l [ a 2 ] ;
a 2 = E x p a n d [ a 2 ] ;
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 } ,
{ 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 } ] ;
a 3 = N o r m a l [ a 3 ] ;
a 3 = E x p a n d [ a 3 ] ;

b 1 [ x ] = − 1.0 * S i n [ x ] ;
b 2 [ x ] = 1.0 * C o s [ x ] ;
b 3 [ x ] = 0.10 ;

F o r [ i = 1 , i < m 8 , i + + ,
A 11 = a 11 [ m 8 − i ] ;
A 11 = S e r i e s [ A 11 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;
A 11 = N o r m a l [ A 11 ] ;
A 11 = E x p a n d [ A 11 ] ;
A 12 = a 12 [ m 8 − i ] ;
A 12 = S e r i e s [ A 12 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;
A 12 = N o r m a l [ A 12 ] ;
A 12 = E x p a n d [ A 12 ] ;
A 13 = a 13 [ m 8 − i ] ;
A 13 = S e r i e s [ A 13 , { S i n [ x ] , 0 , 2 } , { C o s [ x ] , 0 , 2 } ] ;
A 13 = N o r m a l [ A 13 ] ;
A 13 = E x p a n d [ A 13 ] ;

u 1 [ m 8 − i ] [ x − ] = A 11 ;
u 2 [ m 8 − i ] [ x − ] = A 12 ;
P [ m 8 − i ] [ x − ] = E x p a n d [ A 13 ] ;
t 3 = 1.0 ;
P r i n t [ ” u 1 [ ” , m 8 − i , ” ] = ” , A 11 ] ;
C l e a r [ t 3 ] ;
b 1 [ x ] = A 11 ;
b 2 [ x ] = A 12 ;
b 3 [ x ] = A 13 ; ] ;
t 3 = 1.0 ;
F o r [ i = 1 , i < m 8 , i + + ,
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 :

(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

(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

(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

(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

(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

(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

(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

(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

(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.

Funding information: The author states that there is no external funding involved.
Author contributions: The author has the entire responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author declares no conflict of interest concerning this article.
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

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

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

Abstract

1 Introduction

Remark 1.1

2 Details about an equivalent elliptic system

Theorem 2.1

Proof

Remark 2.2

3 An approximate proximal approach

4 A software in MATHEMATICA related to the previous algorithm

5 The software and numerical results for a more specific example

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

7 Conclusion

References

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