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## Computers and Mathematics with Applications

journal homepage:www.elsevier.com/locate/camwa

## Convergence of equilibria for numerical approximations of a suspension model

### J. Valero

^{a,}

^{∗}

### , A. Giménez

^{a}

### , O.V. Kapustyan

^{b}

### , P.O. Kasyanov

^{c}

### , J.M. Amigó

^{a}

a*Universidad Miguel Hernández de Elche, Centro de Investigación Operativa, 03202 Elche (Alicante), Spain*

b*Taras Shevchenko National University of Kyiv, Kyiv, Ukraine*

c*Institute for Applied System Analysis, National Technical University of Ukraine ‘‘Kyiv Polytechnic Institute’’, Kyiv, Ukraine*

a r t i c l e i n f o

*Article history:*

Received 23 December 2015 Received in revised form 11 April 2016 Accepted 23 May 2016

Available online xxxx

*Keywords:*

Non-Newtonian fluids Suspensions

Numerical approximations Finite-difference schemes Partial differential equations

a b s t r a c t

In this paper we study the numerical approximations of a non-Newtonian model for concentrated suspensions.

First, we prove that the approximative models possess a unique fixed point and study their convergence to a stationary point of the original equation.

Second, we implement an implicit Euler scheme, proving the convergence of these approximations as well.

Finally, numerical simulations are provided.

©2016 Elsevier Ltd. All rights reserved.

**1. Introduction**

Non-Newtonian (or complex) fluids often appear in nature and industry. Good examples of such fluids are toothpaste, ketchup, magma, blood, mucus or emulsions such as mayonnaise among many others. A special type of complex fluids are concentrated suspensions, which can be found, for example, in medicine (blood) or in building industry (cement). The dynamical behaviour of suspensions is still far from being well understood as developing a faithful mathematical model of such processes is not an easy task.

We are interested in an equation modelling suspensions which was proposed in [1]. In the last years, several authors have studied for this equation the existence and uniqueness of solutions [2,3], the asymptotic behaviour [4,5] and numerical approximations [6–8].

In our previous paper [8] we studied a sequence of approximative problems for this model, in which finite-difference
schemes were used to deal with the partial derivative with respect to the spatial variable. The problem was split in three
steps: a partial differential equation with a large diffusion, an infinite system of ordinary differential equations and finally a
finite system of ordinary differential equations. For initial data satisfying suitable assumptions it was proved that the iterate
limit of the solutions of the approximative problems in the space*C*

### ( [

0### ,

^{T}### ] ,

^{L}^{2}

### (

R### ))

is equal to the solution of the original equation.In this paper we extend the results from [8] in two ways.

∗Corresponding author.

*E-mail addresses:*[email protected](J. Valero),[email protected](A. Giménez),[email protected](O.V. Kapustyan),[email protected]
(P.O. Kasyanov),[email protected](J.M. Amigó).

http://dx.doi.org/10.1016/j.camwa.2016.05.034 0898-1221/©2016 Elsevier Ltd. All rights reserved.

First, we study the convergence of the fixed points of the approximative problems. It is well-known [2] that for certain values of the parameters of the equation there exists a unique fixed point of the problem with support not included in the interval

### [−

1### ,

^{1}

### ]

. This equilibrium is asymptotically stable [4] and the numerical simulations in [7] suggest that every solution with initial data with support not included in### [−

1### ,

^{1}

### ]

converges to this fixed point as time goes to### +∞

. We prove that each of the approximative problems possesses a unique fixed point and also that the iterate limit of the equilibria of the approximative problems in the space*L*

^{2}

### (

R### )

is equal to the equilibrium of the original equation with support not included in the interval### [−

1### ,

^{1}

### ]

.Second, we complete the sequence of approximations of the problem by implementing an implicit Euler scheme for
the discretization of the time derivative. We prove that the solution of the resulting system converges in the space
*C*

### ( [

0### ,

^{T}### ] ,

^{L}^{2}

### (

R### ))

to the solution of the finite system of ordinary differential equations approximating the original equation.Finally, some numerical simulations are provided in the last section.

**2. Previous results**

In the previous paper [8] the authors considered the convergence of finite-difference approximations of the problem

### ∂

^{p}### ∂

^{t}### −

*D*

### (

^{p}### (

^{t}### )) ∂

^{2}

^{p}### ∂σ

^{2}

### +

^{1}

*T*_{0}

### χ

R\[−1,1]### (σ)

^{p}### =

^{D}### (

^{p}### (

^{t}### ))

### α δ

0### (σ) ,

^{(1)}

*p*

### ≥

0### ,

^{p}### (

^{0}

### , σ) =

*p*

^{0}

### (σ) ,

^{(2)}

where*p*

### =

*p*

### (

^{t}### , σ) ,

^{t}### ∈ [

0### ,

^{T}### ] , σ ∈

_{R},

*T*

_{0}and

### α

are positive constants.Here,

### δ

0is the Dirac### δ

-function with support in the origin,*D*

### (

^{p}### (

^{t}### )) = α

*T*_{0}

###

|σ|>1

*p*

### (

^{t}### , σ)

^{d}### σ

and

### χ

*I*is the indicator function in the interval

*I.*

The function*p*

### (

^{t}### , σ )

is a probability density at time*t*, so for any

*t*

### ∈ [

0### ,

^{T}### ]

,###

R

*p*

### (

^{t}### , σ )

^{d}### σ =

1### ,

^{(3)}

*p*

### (

^{t}### , σ ) ≥

0### ,

^{for a.a.}

### σ ∈

_{R}

### .

It is well-known [2] that for any*p*^{0}

### ∈

*L*

^{1}

### (

R### ) ∩

*L*

^{∞}

### (

R### )

^{such that}

^{p}^{0}

### ≥

0 a.e.,###

R*p*^{0}

### (σ)

^{d}### σ =

1,###

R

### | σ |

*p*

^{0}

### (σ)

^{d}### σ < ∞

and*D*

### (

*0*

^{p}### ) >

0 there exists a unique solution*p*

### =

*p*

### (

^{t}### , σ)

^{of problem}(1)–(2), which satisfies(3).

We consider as a first step the approximative problem

### ∂

*t*

*p*

^{c}### −

###

*D*

###

*p*^{c}

### (

^{t}### ) +

^{1}

*c*

###

### ∂

_{σ σ}

^{2}

^{p}### +

^{1}

*T*_{0}

### χ

R\[−1,1]### (σ )

^{p}

^{c}### =

^{D}### (

^{p}

^{c}### (

^{t}### ))

### α δ

*c*

### (σ ) ,

^{(4)}

*p*^{c}

### ≥

0### ,

^{p}

^{c}### (

^{0}

### , σ) =

*p*

^{0}

_{c}### (σ ) ,

^{(5)}

where*p*^{c}

### =

*p*

^{c}### (

^{t}### , σ )

^{,}

^{c}### >

0 is a large parameter and the### δ

^{-function}

### δ

0is replaced by the step continuous from the right function### δ

*c*

### (σ) =

###

###

###

###

###

###

###

0

### ,

^{if}

### σ < −

^{1}2c

### ,

*c*

### ,

^{if}

### −

^{1}

2c

### ≤ σ <

^{1}2c

### ,

0### ,

^{if}

### σ ≥

^{1}

2c

### .

We would like to highlight the fact that the new term^{1}_{c}

### ∂

_{σ σ}

^{2}

*is an artificial diffusion which helps us to prove the convergence of the approximative solutions. Such a trick is very common in the numerical approximations of problems in Physics. Also,*

^{p}### [−

^{1}

2c

### ,

_{2c}

^{1}

### ]

is the support of the map### δ

*c*, which approximates the

### δ

^{-function}

### δ

0. Therefore, when*c*

### → +∞

, the artificial diffusion and the support of### δ

*c*converge to 0 in unison.

Let*p*^{0}* _{c}*be such that

*p*^{0}_{c}

### ∈

*C*

_{0}

^{∞}

### (

R### ),

^{p}^{0}

*c*

### ≥

0 a.e.,###

R

*p*^{0}_{c}

### (σ )

^{d}### σ =

1### ,

^{(6)}

*p*^{0}_{c}

### →

*p*

^{0}in

*L*

^{2}

### (

R### ), σ

^{p}^{0}

*c*

### → σ

^{p}^{0}

^{in}

^{L}^{1}

### (

R### ),

^{as}

^{c}### → +∞ .

^{(7)}

It is proved in [8, Theorem 3] that such approximation exists and that the unique solution*p** ^{c}*to problem(4)–(5)converges
to the unique solution

*p*to problem(1)–(2)in the space

*C*

### ( [

0### ,

^{T}### ] ,

^{X}### )

^{, where}

*X*

### =

###

*p*

### ∈

*L*

^{2}

### (

R### ) :

###

R

### | σ | |

*p*

### |

*d*

### σ < +∞

### ,

endowed with the norm### ∥

*p*

### ∥

_{X}### = ∥

*p*

### ∥

*2(R)*

_{L}### +

R

### | σ | |

*p*

### |

*d*

### σ

. It is important to remark that for all*t*

### ∈ [

0### ,

^{T}### ]

the solution satisfies###

R*p*^{c}

### (

^{t}### , σ)

^{d}### σ =

1 and*p*

^{c}### (

^{t}### , σ) ≥

0, for a.a.### σ ∈

_{R}, since it is a probability density.

Further, we shall consider the following approximating infinite system of ordinary differential equations:

*dp*^{c}_{i}^{,}^{h}*dt*

### −

###

*D*_{h}

### (

^{p}

^{c}*h*

### (

^{t}### )) +

^{1}

*c*

###

_{p}*c*,

*h*

*i*+1

### (

^{t}### ) −

2p

^{c}

_{i}^{,}

^{h}### (

^{t}### ) +

*p*

^{c}

_{i}_{−}

^{,}

^{h}_{1}

### (

^{t}### )

*h*^{2}

### +

^{1}

*T*_{0}

### χ

Z\[−2n1,2n1]### (

^{i}### )

^{p}

^{c}*i*

^{,}

^{h}### (

^{t}### ) =

^{D}

^{h}###

*p*

^{c}

_{h}### (

^{t}### )

### α δ

*c*

^{i}### ,

^{(8)}

*p*^{c}_{i}^{,}^{h}

### (

^{0}

### ) =

*p*

^{0}

_{c}_{,}

_{h}_{,}

_{i}### ,

^{i}### ∈

_{Z}

### ,

^{(9)}

where*h*

### >

^{0}

### σ

*i*

### =

*ih,i*

### ∈

_{Z},

^{1}

_{h}### =

2n_{1}, with

*n*

_{1}

### ∈

_{N},

### χ

Z\[−2n1,2n1]### (

^{i}### ) =

###

1### ,

^{if}

^{i}### ̸∈

[### −

2n_{1}

### ,

^{2n}1]

### ,

0### ,

^{otherwise}

### .

Here,*p*

^{c}

_{h}### (

^{t}### ) =

###

*p*

^{c}

_{i}^{,}

^{h}### (

^{t}### )

*i*∈_{Z}denotes a sequence satisfying
*p*^{c}_{i}^{,}^{h}

### (

^{t}### ) ≃

*p*

^{c}### (

^{t}### , σ

*i*

### ) ,

and we made the following approximations
*D*

###

*p*^{c}

### (

^{t}### )

### = α

*T*

_{0}

###

|σ|>1

*p*^{c}

### (

^{t}### , σ)

^{d}### σ ≃

*D*

_{h}###

*p*

^{c}

_{h}### (

^{t}### )

### = α

^{h}*T*

_{0}

###

|* _{i}*|>2n1

*p*^{c}_{i}^{,}^{h}

### (

^{t}### ),

^{(10)}

### δ

^{i}*c*

### = δ

*c*

### (σ

*i*

### ) =

###

_{0}

### ,

^{if}

^{i}### < −

*n*

_{h}_{,}

_{c}### ,

*c*

### ,

^{if}

### −

*n*

_{h}_{,}

_{c}### ≤

*i*

### <

^{n}*h*,

*c*

### ,

0

### ,

^{if}

^{i}### ≥

*n*

_{h}_{,}

_{c}### .

^{(11)}

Also,*c*is taken such that_{2ch}^{1}

### =

*n*

_{h}_{,}

_{c}### ∈

_{N}and

*n*

_{h}_{,}

_{c}### <

^{2n}1.

We observe that the parameter*h*is the length of the intervals in the finite-difference approximation of the second
derivative

### ∂

_{σ σ}

^{2}

*p, which is a diffusion term. The approximation is getting better ash*goes to 0. Also,

*n*

_{h}_{,}

*is the number of subintervals of length*

_{c}*h*of the interval

### [

0### ,

_{2c}

^{1}

### ]

, the support of the function### δ

*c*in the positive semi-axis. The condition

*n*

_{h}_{,}

_{c}### <

^{2n}1is equivalent to say that

_{2c}

^{1}

### <

1, that is, the support of### δ

*c*is strictly included in the interval

### [−

1### ,

^{1}

### ]

.We define the following partition of the real line:

Ω*h*

### = { σ

*i*

### =

*ih*

### }

_{i}_{∈}

_{Z}

### ,

^{I}*i*

^{h}### = [ σ

*i*

### , σ

*i*

### +

*h*

### ).

For*p*^{0}* _{c}*from(6)we define the step function

###

^{p}0

*c*,^{h}

### (σ) =

*i*∈_{Z}

*p*^{0}_{c}

### (

^{ih}### ) χ

*I*

_{i}

^{h}### (σ )

and normalize it by setting*p*^{0}_{c}_{,}_{h}

### (σ) =

^{p}0
*c*,^{h}

### (σ )

###

###

^{p}^{0}

_{c}_{,}

_{h}###

###

*L*

^{1}(R)

### .

^{(12)}

It holds that*p*^{0}_{c}_{,}_{h}

### →

*p*

^{0}

*in*

_{c}*L*

^{2}

### (

R### ) ∩

*L*

^{1}

### (

R### )

^{as}

^{h}### →

0.Further, we fix*c*

### ∈

_{N}and take a sequence

*h*

_{n}### →

0 such that_{2ch}

^{1}

*n*

### =

*n*

_{h}

_{n}_{,}

_{c}### ∈

_{N}. Then conditions

_{h}^{1}

*n*

### =

2n_{1}

### ,

*1*

^{n}### ∈

_{N},

*n*

_{h}

_{n}_{,}

_{c}### <

^{2n}1are satisfied. We take

*p*

^{0}

_{c}_{,}

_{h}*n*,*i*

### =

*p*

^{0}

_{c}_{,}

_{h}*n*

### (

^{ih}*n*

### )

as the initial data in problem(8)and define the step functions*p*

^{c}

_{h}

_{n}### (

^{t}### , σ ) =

*i*∈Z

*p*^{c}_{i}^{,}^{h}^{n}

### (

^{t}### ) χ

*I*

_{i}

^{h}### (σ ),

^{(13)}

where*p*^{c}_{h}_{n}

### (

^{t}### ) = {

*p*

^{c}

_{i}^{,}

^{h}

^{n}### (

^{t}### ) }

_{i}_{∈}

_{Z}is the unique solution to problem(8)–(9). It is proved in [8, Theorem 2] that

*p*^{c}_{h}_{n}

### →

*p*

*strongly in*

^{c}*C*

### ( [

0### ,

^{T}### ];

*L*

^{2}

### (

R### )),

^{(14)}

where*p** ^{c}*is the solution to problem(4)–(5)with initial data

*p*

^{0}

*. As before, the solutions*

_{c}*p*

^{c}

_{h}*satisfy that*

_{n}###

R

*p*^{c}_{h}_{n}

### (

^{t}### , σ)

^{d}### σ =

*i*∈_{Z}

*p*^{c}_{i}^{,}^{h}^{n}

### (

^{t}### )

^{h}*n*

### =

1 and*p*

^{c}

_{i}^{,}

^{h}

^{n}### (

^{t}### ) ≥

0### ,

^{for any}

^{t}### ∈ [

0### ,

^{T}### ] ,

^{i}### ∈

_{Z}

### .

Let us consider now finite-dimensional approximations. We define the operator*A*^{N}_{h}

### :

_{R}

^{2N}

^{+}

^{1}

### →

_{R}

^{2N}

^{+}

^{1}by

*A*^{N}_{h}

### :=

^{1}

*h*

^{2}

###

###

###

###

###

###

###

###

###

###

###

###

1

### −

1 0### · · ·

0 0 0### −

1 2### −

1 0### · · ·

0 0 0### −

1 2### −

1### ... · · ·

0### ...

^{0}

### ... ... ...

^{0}

### ...

0

### · · · ... −

1 2### −

1 0 0 0### · · ·

0### −

1 2### −

1 0 0 0### · · ·

0### −

1 1###

###

###

###

###

###

###

###

###

###

###

###

(2N+1)×(2N+1)

### .

^{(15)}

Then we consider the finite-dimensional system

###

###

###

###

###

*dp*

^{c}

_{h}^{,}

_{,}

^{N}

_{i}*dt*

### = −

###

*D*

^{N}

_{h}###

*p*

^{c}

_{h}^{,}

^{N}### (

^{t}### )

### +

^{1}

*c*

###

*A*

^{N}

_{h}*p*

^{c}

_{h}^{,}

^{N}###

*i*

### −

^{1}

*T*_{0}

### χ

Z\[−2n1,2n1]### (

^{i}### )

^{p}

^{c}*h*

^{,},

^{N}*i*

### +

*D*

^{N}

_{h}###

*p*

^{c}

_{h}^{,}

^{N}### (

^{t}### )

### α δ

^{i}*c*

### ,

*p*

^{c}

_{h}^{,}

_{,}

^{N}

_{i}### (

^{0}

### ) =

*p*

^{N}

_{c}_{,}

^{,}

_{h}^{0}

_{,}

_{i}### , −

*N*

### ≤

*i*

### ≤

*N*

### ,

(16)

where*N*

### >

^{2n}1,

^{1}

_{h}### =

2n_{1}

### ,

*1*

^{n}### ∈

_{N},

_{2ch}

^{1}

### =

*n*

_{h}_{,}

_{c}### ∈

_{N}

### ,

^{n}*h*,

*c*

### <

^{2n}1and

*D*

^{N}

_{h}###

*p*

^{c}

_{h}^{,}

^{N}###

### = α

^{h}*T*

_{0}

###

2n1<|*i*|≤*N*

*p*^{c}_{h}^{,}_{,}^{N}_{i}

### .

What we have done is to cut the tails of the system(8)off in order to work with a finite number of equations. The condition
*N*

### >

^{2n}1implies that we solve the problem for

### σ

in an interval containing### [−

1### ,

^{1}

### ]

. It is obvious that*N*has to be large to get good approximations.

For the initial data*p*^{0}_{c}_{,}* _{h}*from(12)we consider the approximations

*p*

^{N}

_{c}_{,}

^{,}

_{h}^{0}given by

*p*^{N}_{c}_{,}^{,}_{h}^{0}_{,}_{i}

### =

^{p}0
*c*,*h*,*i*

###

|* _{i}*|≤

_{N}*hp*^{0}_{c}_{,}_{h}_{,}_{i}

### ,

^{for}

### |

*i*

### | ≤

*N*

### ,

^{(17)}

where*p*^{0}_{c}_{,}_{h}_{,}_{i}

### =

*p*

^{0}

_{c}_{,}

_{h}### (

^{ih}### )

. Then we define the step functions*p*

^{c}

_{h}^{,}

^{N}### (

^{t}### , σ ) =

*i*∈Z

*p*^{c}_{h}^{,}_{,}^{N}_{i}

### (

^{t}### ) χ

*I*

*i*

### (σ ),

^{(18)}

where

###

*p*

^{c}

_{h}^{,}

_{,}

^{N}

_{i}### ( · )

|* _{i}*|≤

*is the unique solution to problem(16)with initial data(17)and*

_{N}*p*

^{c}

_{h}^{,}

_{,}

^{N}

_{i}### (

^{t}### ) =

0 if### |

*i*

### | >

*N. It is proved*in [8, Section 5] that

*p*^{c}_{h}^{,}^{N}

### →

*p*

^{c}*in*

_{h}*C*

### ( [

0### ,

^{T}### ] ,

^{L}^{2}

### (

R### ))

^{as}

^{N}### → ∞ ,

where*p*^{c}* _{h}*is the function defined in(13)with initial data(12). Again, the property of being a probability density is satisfied:

###

R

*p*^{c}_{h}^{,}^{N}

### (

^{t}### , σ)

^{d}### σ =

|*i*|≤*N*

*p*^{c}_{h}^{,}_{,}^{N}_{i}

### (

^{t}### )

^{h}### =

1 and*p*

^{c}

_{h}^{,}

_{,}

^{N}

_{i}### (

^{t}### ) ≥

0### ,

^{for any}

^{t}### ≥

0### , |

*i*

### | ≤

*N*

### .

**3. Fixed points of approximations**

Our aim in this section is to study the fixed points of the approximative problems and their convergence to the fixed
points of the original problem(1). For simplicity, we shall consider the particular case where*T*_{0}

### =

1.First, recall that for Eq.(1)with*T*_{0}

### =

1 the fixed points, given by the solutions of*D*

### (

^{p}### ) ∂

^{2}

^{p}### ∂σ

^{2}

### − χ

R\[−1,1]### (σ)

^{p}### +

^{D}### (

^{p}### )

### α δ

0### (σ) =

0### ,

^{(19)}

are the following [2]:

### •

Any probability density*p*

### ( · )

with support in### [−

1### ,

^{1}

### ]

solves(19). We note that all these solutions satisfy*D*

### (

^{p}### ) =

0;### •

If### α ≤

^{1}

2, there are no more equilibria. If

### α >

^{1}

_{2}, then there exists a unique fixed point

*p*with positive value of

*D*

### (

^{p}### )

^{,}which is given by

*p*

### (σ ) =

###

###

###

###

###

###

###

###

###

###

###

###

###

### √

*D*

^{∗}2

### α

^{e}^{(}

^{1}

+σ )/√
*D*∗

### ,

^{if}

### σ ≤ −

1### ,

### √

*D*

^{∗}

### +

12

### α +

^{1}

2

### α σ,

^{if}

### −

1### ≤ σ ≤

0### ,

### √

*D*

^{∗}

### +

12

### α −

^{1}

2

### α σ,

^{if 0}

### ≤ σ ≤

1### ,

### √

*D*

^{∗}2

### α

^{e}^{(}

^{1}

−σ )/√
*D*∗

### ,

^{if}

### σ ≥

1### ,

(20)

where

*D*^{∗}

### =

###

### −

^{1}2

### +

### √

4### α −

12

###

2(21)

and*z*

### =

### √

*D*^{∗}is the unique positive solution of the equation
*h*

### (

^{z}### ) =

*z*

^{2}

### +

*z*

### − α +

^{1}

2

### =

0### .

We observe that when

### α >

^{1}

_{2}the stationary point

*p*is asymptotically stable [4]. Moreover, the numerical simulations in [7] suggest that every solution with initial data satisfying

*D*

###

*p*^{0}

###

### >

0 converges to this fixed point as time goes to### +∞

_{.}We shall prove that for

### α >

^{1}

_{2}the approximative problems possess a unique fixed point converging to(20).

*3.1. Equation with large diffusion*

Let us consider now the fixed points of problem(4). In order to find them we fix first*D*

### >

0 and solve first the following ordinary differential equation:###

*D*

### +

^{1}

*c*

###

*d*

^{2}

*p*

^{c}*d*

### σ

^{2}

### − χ

R\[−1,^{1]}

### (σ)

^{p}

^{c}### +

^{D}### α δ

*c*

### (σ) =

0### .

We note that in this case, unlike problem(1), there is no stationary solutions with*D*

### (

^{p}### ) =

0.Taking into account the condition*p*^{c}

### (σ) →

0, as### σ → ±∞

, it is not difficult to check that this equation possesses a unique solution defined by*p*^{c}_{D}

### (σ ) =

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

*D*2

### α

*D*

### +

^{1}

*c*

*e*^{(}^{1}^{+}^{σ )/}

*D*+^{1}_{c}

### ,

^{if}

### σ ≤ −

1### ,

*D*
2

### α

1

### +

###

*D*

### +

^{1}

*c*

*D*

### +

^{1}

*c*

### +

^{D}2

### α

*D*

### +

^{1}

*c*

### σ,

^{if}

### −

1### ≤ σ ≤ −

^{1}2c

### ,

*D*

2

### α

1### +

###

*D*

### +

^{1}

*c*

*D*

### +

^{1}

*c*

### −

^{1}8

*D*

### α

*D*

### +

^{1}

*c*

###

*c*

### −

*2*

^{Dc}### α

*D*

### +

^{1}

*c*

### σ

^{2}

### ,

^{if}

### −

^{1}

2c

### ≤ σ ≤

^{1}2c

### ,

*D*

2

### α

1### +

###

*D*

### +

^{1}

*c*

*D*

### +

^{1}

*c*

### −

^{D}2

### α

*D*

### +

^{1}

*c*

### σ,

^{if}

^{1}

2c

### ≤ σ ≤

1### ,

*D*

2

### α

*D*

### +

^{1}

*c*

*e*^{(}^{1}^{−}^{σ )/}

*D*+^{1}_{c}

### ,

^{if}

### σ ≥

1### .

Since
*D*

###

*p*^{c}_{D}

###

### = α

###

|σ|>1

*p*^{c}_{D}

### (σ)

^{d}### σ =

*D*

### ,

^{for any}

^{D}### >

^{0}

### ,

in order to obtain a fixed point it remains to find a positive value of*D*such that

###

R*p*^{c}_{D}

### (σ )

^{d}### σ =

1. Calculating the integral we obtain1
48c^{2}

### α

2D
*D*

### +

^{1}

*c*

###

24c

### +

24c^{2}

*D*

### +

24c^{2}

###

*D*

### +

^{1}

*C*

### +

12c^{2}

### −

1###

### =

1### ,

and after the change of variable*z*

### =

###

*D*

### +

^{1}

*c* we finally have the equation
*g*^{c}

### (

^{z}### ) =

*z*

^{4}

### +

*z*

^{3}

### −

###

1 24c^{2}

### +

^{1}

*c*

### −

^{1}2

### + α

###

*z*

^{2}

### −

^{1}

*cz*

### −

^{1}2c

### +

^{1}

24c^{3}

### =

0### .

^{(22)}

It follows from the Descarte’s rule of signs that for

### α >

^{0}

### .

^{5 and}

*large enough this polynomial possesses a unique positive root*

^{c}*z*

*. More precisely,*

^{c}*c*has to satisfy

*c*

### >

^{√}

^{1}

_{12}. We will take

*c*

### ≥

1. Such condition is compatible with the meaning of the term^{1}

_{c}### ∂

_{σ σ}

^{2}

^{p}^{in}(4), as this is an artificial diffusion that has to be small in order to approximate the original system properly, which means that we need

*c*to be large.

If we pass to the limit as*c*

### → ∞

the polynomial*g*

^{c}### (

^{z}### )

^{tends to}

*g*

### (

^{z}### ) =

*z*

^{2}

###

*z*^{2}

### +

*z*

### +

^{1}2

### − α

### .

By continuity, the root*z** ^{c}*converges to the unique positive root of

*h*

### (

^{z}### )

, which is equal to*z*

^{∗}

### =

### √

*D*^{∗}. Therefore,
*D*^{c}

### =

*z*^{c}

###

2### −

^{1}

*c*

### →

*D*

^{∗}

### >

^{0}

### ,

where*D*^{∗}is given in(21). Hence,*D*^{c}

### >

^{0, for}

*large enough, and thus there is a unique stationary point*

^{c}*p*

^{c}### (σ) =

*p*

^{c}

_{D}*c*

### (σ )

^{.}Moreover, it is easy to see using

*D*

^{c}### →

*D*

^{∗}that

*p*^{c}

### →

*p*in

*X*

### .

Therefore, we have proved the following result.

**Theorem 1.** *Let*

### α >

^{0}

### .

^{5. Problem}

^{(4)}

*possesses a unique fixed point p*

^{c}*for c*

### ≥

1*and*

*p*

^{c}### →

*p*

*in X*

### ,

^{as c}### → ∞ ,

*where p is the unique fixed point of problem*(1)*such that D*

### (

^{p}### ) >

^{0}

^{defined in}^{(20).}

*3.2. Lattice dynamical system*

Further, we will study the fixed points of Eq.(8)with_{2ch}^{1}

### =

*n*

_{h}_{,}

_{c}### ∈

_{N}and

*n*

_{h}_{,}

_{c}### <

^{2n}1

### =

^{1}

*h*. As before, we fix first*D*

### >

^{0}and solve the following equation in differences

###

###

###

###

###

###

###

###

###

###

###

###

###

### −

^{D}### +

^{1}

*c*

*h*^{2}

### (

^{p}*i*+1

### −

2p

_{i}### +

*p*

*−1*

_{i}### ) +

*p*

_{i}### =

0### ,

^{if}

^{i}### < −

2n_{1}

### ,

*p*

*+1*

_{i}### −

2p

_{i}### +

*p*

*−1*

_{i}### =

0### ,

^{if}

### −

2n_{1}

### ≤

*i*

### < −

*n*

_{h}_{,}

_{c}### ,

### −

^{D}### +

^{1}

*c*

*h*^{2}

### (

^{p}*i*+1

### −

2p

_{i}### +

*p*

*−1*

_{i}### ) =

^{D}### α

^{c}### ,

^{if}

### −

*n*

_{h}_{,}

_{c}### ≤

*i*

### <

^{n}*h*,

*c*

### ,

*p*

*+1*

_{i}### −

2p

_{i}### +

*p*

*−1*

_{i}### =

0### ,

^{if}

^{n}*h*,

^{c}### ≤

*i*

### ≤

2n_{1}

### ,

### −

^{D}### +

^{1}

*c*

*h*^{2}

### (

^{p}*i*+1

### −

2p

_{i}### +

*p*

*−1*

_{i}### ) +

*p*

_{i}### =

0### ,

^{if}

^{i}### >

^{2n}1

### ,

(23)

whose solution, taking into account that*p*_{i}

### →

*i*→±∞0, is given by

*p*^{c}_{i}_{,}^{,}_{D}^{h}

### =

###

###

###

###

###

###

###

###

###

*C*_{1}

### λ

*1*

^{i}### ,

^{if}

^{i}### < −

2n_{1}

### ,

*A*

### +

*Bi*

### ,

^{if}

### −

2n_{1}

### ≤

*i*

### < −

*n*

_{h}_{,}

_{c}### ,

*E*

### +

*Fi*

### −

^{ch}2*D*
2

###

*D*

### +

^{1}

*c*

### α

^{i}2

### ,

^{if}

### −

*n*

_{h}_{,}

_{c}### ≤

*i*

### <

^{n}*h*,

*c*

### ,

*A*

### +

*Bi*

### ,

^{if}

^{n}*h*,

*c*

### ≤

*i*

### ≤

2n_{1}

### ,

*C*_{2}

### λ

^{−}1

^{i}### ,

^{if}

^{i}### >

^{2n}1

### ,

(24)

provided that

### λ

1### = λ

*1*

^{c}^{,},

^{D}*h*

### =

1### +

^{h}2

2

###

*D*

### +

^{1}

*c*

### +

^{h}###

*D*

### +

^{1}

*c*

###

1

### +

^{h}2

4

###

*D*

### +

^{1}

*c*

### , λ

2### = λ

*2*

^{c}^{,},

^{D}*h*

### =

^{1}

### λ

*1*

^{c}^{,},

^{D}*h*

### ,

^{(25)}

and the constants*C*_{1}

### ,

^{A}### ,

^{B}### ,

^{E}### ,

^{F}### ,

^{A}### ,

^{B}### ,

*2satisfy the compatibility conditions*

^{C}###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

*A*

### −

^{1}

*hB*

### − λ

^{−}1

^{1}

^{h}*C*

_{1}

### =

0### ,

*A*

### −

^{h}### +

1*h* *B*

### − λ

^{−}1

^{h}^{+}

^{h}^{1}

*C*

_{1}

### =

0### ,

*A*

### −

^{1}

2ch*B*

### −

*E*

### +

^{1}

2ch*F*

### = −

^{D}8c

###

*D*

### +

^{1}

*c*

### α ,

*A*

### −

^{2ch}

### +

12ch *B*

### −

*E*

### +

^{2ch}

### +

12ch *F*

### = −

^{D}### (

^{1}

### +

2ch### )

^{2}8c

###

*D*

### +

^{1}

*c*

### α ,

*A*

### +

^{1}

### −

2ch2ch *B*

### −

*E*

### −

^{1}

### −

2ch2ch *F*

### = −

^{D}### (

^{1}

### −

2ch### )

^{2}8c

###

*D*

### +

^{1}

*c*

### α ,

*A*

### +

^{1}

2ch*B*

### −

*E*

### −

^{1}

2ch*F*

### = −

^{D}8c

###

*D*

### +

^{1}

*c*

### α ,

*A*

### +

^{h}### +

1*h* *B*

### − λ

^{−}1

^{h}^{+}

^{h}^{1}

*C*

_{2}

### =

0### ,

*A*

### +

^{1}

*hB*

### − λ

^{−}1

^{1}

^{h}*C*

_{2}

### =

0### .

(26)

Solving this system we obtain:

*C*_{1}

### = λ

1^{1}

^{h}*D*

###

*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

###

*h*

### (

^{2}

### +

*h*

### ) −

2h^{2}4

###

### −

1### + λ

^{−}1

^{1}

###

1### +

*h*

### − λ

^{−}1

^{1}

### ,

*2*

^{C}### = λ

1^{1}

^{h}*D*

###

*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

###

*h*

### (

^{2}

### −

*h*

### ) −

2h^{2}4

###

### −

1### + λ

^{−}1

^{1}

###

1### +

*h*

### − λ

^{−}1

^{1}

### ,

^{(27)}

*A*

### =

^{D}*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

### (

^{2}

### +

*h*

### ) −

2h 4###

### −

1### + λ

^{−}1

^{1}

### ,

^{A}### =

^{D}*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

### (

^{2}

### −

*h*

### ) −

2h 4###

### −

1### + λ

^{−}1

^{1}

### ,

*B*

### = −

*D*

###

*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

###

*h*

### (

^{2}

### +

*h*

### ) −

2h^{2}4

###

1

### +

*h*

### − λ

^{−}1

^{1}

### ,

^{B}### =

^{D}*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

###

*h*

### (

^{2}

### −

*h*

### ) −

2h^{2}4

###

1

### +

*h*

### − λ

^{−}1

^{1}

### ,

*F*

### = −

*2*

^{Dh}###

*D*

### +

^{1}

*c*

### α (

^{1}

### +

*ch*

### ) −

^{D}###

*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

###

*h*

### (

^{2}

### +

*h*

### ) −

2h^{2}4

###

1

### +

*h*

### − λ

^{−}1

^{1}

### ,

*E*

### = −

^{D}8c

###

*D*

### +

^{1}

*c*

### α (

^{1}

### +

2ch### ) +

^{D}###

*D*

### +

^{1}

*c*

### α

### −

1### + λ

^{−}1

^{1}

### (

^{2}

### +

*h*

### ) −

2h 4###

### −

1### + λ

^{−}1

^{1}

### .

For simplicity of notation here and throughout the paper, if no confusion is possible, sometimes we omit the indexes
*c*

### ,

^{h}### ,

*and write just*

^{D}### λ

1.We need to check first that

### α

^{h}###

|*i*|>2n1*p*^{c}_{i}_{,}^{,}_{D}^{h}

### =

*D. Indeed, we can easily compute that*

### α

^{h}###

|*i*|>2n1

*p*^{c}_{i}_{,}^{,}_{D}^{h}

### =

^{Dh}2

###

*D*

### +

^{1}

*c*

### λ

1### (λ

1### −

1### )

^{2}

### =

*D*4

###

*D*

### +

^{1}

*c*

### +

2h^{2}

### +

2h###

4###

*D*

### +

^{1}

*c*

### +

*h*

^{2}

###

*h*

### +

###

4###

*D*

### +

^{1}

*c*

### +

*h*

^{2}

###

2### =

*D*for any

*D*

### >

^{0}

### .

We need to find*D*

^{c}

_{h}### >

0 such that*S*

_{h}

^{c}### =

*i*∈_{Z}*p*^{c}_{i}_{,}^{,}_{D}^{h}*c*
*h*

*h*

### =

1. Using mathematical software we obtain*S*^{c}_{h}

### (

^{D}### ) =

*i*∈_{Z}

*p*^{c}_{i}_{,}^{,}_{D}^{h}*h*

### =

^{b}*c*
*h*

### (

^{D}### )

### w

*h*

^{c}### (

^{D}### ) ,

^{(28)}