The Difference Splitting Scheme for Hyperbolic Systems with Variable Coefficients

(1)

The Difference Splitting Scheme for Hyperbolic Systems

with Variable Coefficients

Aloev

R.

D.

1,∗

,

Eshkuvatov

Z.

K.

2

,

Khudoyberganov

M.

U.

1

,

Nematova

D.

E.

1

1_Faculty_of_Mathematics,_National_University_of_Uzbekistan_(NUUz),_Tashkent,_Uzbekistan 2_Faculty_of_Science_and_Technology,_Universiti_Sains_Islam_Malaysia_(USIM),_Negeri_Sembilan,_Malaysia

Abstract

In thepaper, we proposeasystematicapproach todesign andinvestigate the adequacy of the computational models for a mixed dissipative boundary value problem posedforthesymmetrict-hyperbolicsystems. Weconsidera two-dimensionallinearhyperbolicsystemwithvariable coef-ficientsandwiththelowerordertermindissipativeboundary conditions. Weconstructthe difference splittingscheme for the numericalcalculationof stablesolutions for thissystem. AdiscreteanalogueoftheLyapunov’sfunctionisconstructed forthe numericalverificationo fs tabilityo fs olutionsf orthe considered problem. A priori estimate is obtained for the discrete analogueof the Lyapunov’sfunction. Thisestimate allowsus toassertthe exponentialstability of thenumerical solution.Atheoremontheexponentialstabilityofthesolution of the boundary value problemfor linear hyperbolic system andonstabilityofdifferencesplittingschemeinthe Sobolev spaces was proved. These stability theorems give us the opportunity to prove the convergence of the numerical solution.

Keywords

Difference Scheme, Lyapunov Function, Mixed Problem, Stability

1 Introduction

We consider the mixed dissipative boundary value problem for a two-dimensional linear hyperbolic system with variable co-efficients and lower order term [1]. For this problem, we con-struct and investigate the difference splitting scheme in order to obtain stable solutions. A discrete analogue of the Lyapunov function is constructed and an a-priori estimate is obtained for it. The obtained a-priori estimate allows us to assert the expo-nential stability of the numerical solution.

It should be noted that numerous problems have been de-voted to the solution of such problems (see [3]-[10] and ref-erences in them). Studying the stability of solutions for one-dimensional hyperbolic systems is the subject of [2]. Authors of the paper investigate the stability of the solution by con-structing the Lyapunov function and using a priori estimates

for solution in various functional spaces. However, stability of the difference schemes, constructed in all these papers, was investigated using the technique of constructing dissipative en-ergy integrals. The a-priori estimates are obtained in these pa-pers. But they are not enough to get exponential stability of the numerical solution.

2 Differential statement of the problem

In the domainG={(t, x, y) : 0< t≤T,0 < x < l,−∞< y <+∞},we considered a symmetric hyperbolic system in a special canonical form

∂v

∂t +K ∂v

∂x +C ∂v

∂y +Mv=0 (1)

with boundary conditions forx= 0 :

vI =svII (2) forx=l:

vII =rvI (3) and with initial data att= 0

vi(0, x, y) =ϕi(x, y), i= 1, . . . , n, 0≤x≤l, −∞ ≤y≤+∞ (4)

where vI ₌ ₍_v

1, v2, . . . , vm)T, vII =

(vm+1, vm+2, . . . , vn)T, K is a diagonal matrix, C is a

positive definite matrix,Mis an-th order square real matrix

v=

vI vII

, K=

K+ 0

0 −K−

,

K+₌



   

k1 0 · · · 0

0 k2 · · · 0

..

. ... . .. ...

0 · · · 0 km 

   

,

K− =



   

km+1 0 · · · 0

0 km+2 · · · 0

..

. ... . .. ...

0 · · · 0 kn 

   

(2)

C = kcij(x)k,M =kmij(x)k, i, j = 1, . . . , n,0 ≤ x≤ l,

sis a matrix of order (n−m)×m,r is a matrix of order

m×(n−m).The initial functions are assumed to vanish for

|y|> 1

2Y . The similar problem is considered in [1].

The assumption of positive definiteness of the matrixCis not mandatory [1]. It is introduced to simplify the construction of the difference scheme. In fact, if the matrixC does not satisfy this assumption, then by introducing new coordinates

x0, y0, t0

x0=x, y0=y+ωt, t0 =t

we can rewrite the equation in the form of

∂v

∂t0 +K

∂v

∂x0 + (C+ωE)

∂v

∂y0 +Mv= 0,

where the symmetric matrixC+ωEof the coefficients at _∂y∂v0

will be positively definite for a sufficiently largeω. HereEis the identity matrix.

Suppose that the initial data ϕ = (ϕ1, ϕ2, . . . , ϕn)T ∈

W1

2((0, l),(−∞,+∞),Rn) satisfy the compatibility

condi-tions:

ϕI₌_s_ϕII_{, x}_{= 0}_{, t}_{= 0}_,

ϕII ₌_r_ϕI_{, x}₌_{l, t}_{= 0}_.. (5)

HereW1

2((0, l),(−∞,+∞),Rn)is the Sobolev space.

Definition 2(exponential stability [2]). The system (1) with boundary conditions (2)-(3) is said to beexponentially stable in the L2₋_{norm if there exists such}_{ν >}₀_and_{c >}₀_{that for}

any initial conditionϕ∈L2₍₍₀_{, l}₎_,₍_−∞_,₊_∞₎_,_Rn₎_,_the_L2

-solution of the mixed problem (1)-(4) satisfies the inequality

kv(t,·)kL2₍₍₀_,l₎_,_(−∞_,_+∞);_Rn₎≤ ≤ce−νtkϕkL2₍₍₀_,l₎_,_(−∞_,_+∞);_Rn₎, t≥0

(6)

We construct the Lyapunov’s function in the form:

L(t) =

+∞

Z

−∞

l Z

0

(µ(x)v,v)dxdy=

=

+∞

Z

−∞

l Z

0

( m X

i=1

µie−νx[vi(t, x, y)]

2

+

n X

i=m+1

µieνx[vi(t, x, y)]

2

)

dxdy

(7)

whereµi>0, i= 1, . . . , n, µ+= (µ1, ..., µm)T,

µ−= (µm+1, ..., µn)T,

µ(x) =

e−νxµ+ 0 0 eνx_µ−

.

Theorem(exponential stability). The system (1) with the boundary conditions (2)-(3) is exponentially stable in theL2

-norm if there existsν >0andµi >0, i= 1, . . . , nsuch that

the following matrices

K+(l) 0 0 K−₍₀₎

e−νlµ+ 0 0 µ−

−

0 r s 0

×

K+(0) 0 0 K−₍_l₎

µ+ ₀

0 eνlµ−

0 s r 0

(8) and

ν|K(x)|µ(x)−K0(x)µ(x) +µ(x)MT(x)µ(x)+ +µ(x)M(x), x∈(0, l) (9)

are positive definite.

Proof.We calculate the derivative of the Lyapunov function:

L0(t) =

+∞

R

−∞

l R

0

∂t(µ(x)v,v)dxdy=

=

+∞

R

−∞

l R

0

(µ(x)∂tv,v)dxdy+

+∞

R

−∞

l R

0

(µ(x)v, ∂tv)dxdy=

=

+∞

R

−∞

l R

0

(µ(x) [−K(x)∂xv−C(x)∂yv−M(x)v],v)dxdy+

+

+∞

R

−∞

l R

0

(µ(x)v,[−K(x)∂xv−C(x)∂yv−M(x)v])dxdy =

=−

+∞

R

−∞

l R

0

[(µ(x)K(x)∂xv,v) + (µ(x)v,K(x)∂xv) +

+ (µ(x)C(x)∂yv,v) + (µ(x)v,C(x)∂yv)]dxdy− −

+∞

R

−∞

l R

0

[(µ(x)M(x)v,v) + (µ(x)v,M(x)v)]dxdy.

Since

(µ(x)K(x)∂xv,v) = (∂x[µ(x)K(x)v],v)− −(µ0(x)K(x)v,v)−(µ(x)K0(x)v,v)

and

µ0(x)K(x) =−ν|K(x)|µ(x),

we have the following identity:

L0(t) =−

+∞

R

−∞

l R

0

[∂x(K(x)µ(x)v,v) +∂y(C(x)µ(x)v,v)]dxdy−

−

+∞

R

−∞

l R

0

ν|K(x)|µ(x)−K0(x)µ(x) +MT₍_x₎_µ₍_x₎₊

+µ(x)M(x)]v,v)dxdy=−

+∞

R

−∞

(K(x)µ(x)v,v)

l

0

dy−

− l R

0

(C(x)µ(x)v,v)

+∞

−∞

dx−

+∞

R

−∞

l R

0

([ν|K(x)|µ(x)−

K0(x)µ(x) +MT₍_x₎_µ₍_x_{) +}_µ₍_x₎_M₍_x₎

v,v

dxdy= =−

+∞

R

−∞

(K(x)µ(x)v,v)

l

0

dy−

+∞

R

−∞

l R

0

([ν|K(x)|µ(x)−

K0₍_x₎_µ₍_x_{) +}_MT₍_x₎_µ₍_x_{) +}_µ₍_x₎_M₍_x₎

v,v

dxdy.

Transform separately the each term of the obtained identity:

1)−(K(x)µ(x)v,v)

l

0

=−[(K(l)µ(l)v(t, l, y),v(t, l, y))−

(K(0)µ(0)v(t,0, y),v(t,0, y))] = =−

K+₍_l₎ ₀

0 −K−(l)

e−νl_µ+ ₀

0 eνl_µ−

×

vI₍_{t, l, y}₎ vII₍_{t, l, y}₎

,

vI₍_{t, l, y}₎ vII₍_{t, l, y}₎

(3)

+

K+₍₀₎ ₀

0 −K−(0)

µ+ ₀

0 µ−

×

vI₍_t,₀_{, y}₎ vII₍_t,₀_{, y}₎

,

vI₍_t,₀_{, y}₎ vII₍_t,₀_{, y}₎

=

=−

K+₍_l₎ ₀

0 K−(0)

e−νlµ+ 0 0 µ−

×

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

,

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

+ +

K+₍₀₎ ₀

0 K−(l)

µ+ ₀

0 eνl_µ−

×

vI₍_t,₀_{, y}₎ vII₍_{t, l, y}₎

,

vI₍_t,₀_{, y}₎ vII₍_{t, l, y}₎

= =−

K+₍_l₎ ₀

0 K−(0)

e−νl_µ+ ₀

0 µ−

×

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

,

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

+ +

K+₍₀₎ ₀

0 K−(l)

µ+ ₀

0 eνl_µ−

×

0 s r 0

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

,

0 s r 0

×

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

=−

K+₍_l₎ ₀

0 K−(0)

×

e−νl_µ+ ₀

0 µ−

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

,

vI₍_{t, l, y}₎ vII₍_t,₀_{, y}₎

+

0 r s 0

×

K+₍₀₎ ₀

0 K−(l)

µ+ 0

0 eνl_µ−

0 s r 0

×

vI(t, l, y)

vII(t,0, y)

,

vI(t, l, y)

vII(t,0, y)

<0.

According to (9) we have

2)− l R

0

([ν|K(x)|µ(x)−K0(x)µ+ +MT(x)µ(x) +µ(x)M(x)v,vdx <0.

Taking into account these transformations, we obtain

L0(t) =−

+∞

R

−∞

(K(x)µ(x)v,v)

l

0

dy−

+∞

R

−∞

l R

0

([ν|K(x)| × ×µ(x)−K0(x)µ(x) +MT(x)µ(x)+

+µ(x)M(x)]v,v)dxdy <0.

Since the matrices (9) and ν|K(x)|µ(x) are posi-tive definite for any x,we get following inequality

ν|K(x)|µ(x)−K0(x)µ(x) +MT(x)µ(x)+

+µ(x)M(x)]v,v)> ν(|K(x)|µ(x)v,v)> να(µ(x)v,v), α=ν min

1≤i≤n

0≤x≤l

ki(x)

for any vectorv∈Rn_{. Then we have}

L0(t)<−να

Z +∞

−∞

Z l

0

(µ(x)v,v)dxdy=−ηL(t), η=να

Hence,

L(t)≤e−ηtL(0), t >0.

However, since there is such a constantγ >0such that

1

γkv(t,·)k

2

L2₍₍₀_,l₎_,_(−∞_,_+∞)_,Rn₎≤L(t)≤ ≤γkv(t,·)k2_L2₍₍₀_,l₎_,_(−∞_,_+∞)_,Rn₎, kv(t,·)k2_L2₍₍₀_,l₎_,_(−∞_,_+∞)_,Rn₎=

R+∞

−∞

Rl

0(v,v)dxdy

we obtain

kv(t,·)k_L2₍₍₀_,l₎_,_(−∞_,_+∞)_,Rn₎≤ ≤γe−νt/2_k_Φ_k

L2₍₍₀_,l₎_,_(−∞_,_+∞)_,Rn₎, t∈[0,+∞).

3 The difference scheme

In the domainG, construct the difference grid

Gh ={(tκ, xj, yq) : 0 ≤tκ ≤T, 0≤xj ≤l, −∞<

yq <+∞}wheretκ=κ∆t, κ= 0, . . . , N;N∆t=T,

xj= j+1₂

∆x; J∆x=l; j= 0, . . . , J−1;

yq = q+1₂

∆y; q=−∞, . . . ,+∞;

and denote the values of the numerical solution at the nodal points by

vκjq=

1 ∆x

1 ∆y

y_q_{+ 1}

2

Z

y_q₋1 2

x_j_{+ 1}

2

Z

x_j₋1 2

v(tκ, x, y)dxdy, j= 0, . . . , J−1.

For the numerical solution of the mixed problem (1)-(4), we suggest a difference splitting scheme

"

wIκ jq

wIIκ

jq #

=

"

vIκ jq

vIIκ

jq #

− ∆t

∆x×

×

K+_j₋₁ 0 0 K−_j₊₁

" _vIκ

jq− v

Iκ

j−1,q

vIIκ

jq− v

IIκ j+1,q

#

,

j= 0, . . . , J−1; κ= 0, . . . , N−1;

q=−∞, . . . ,+∞;

(10)

uκ

jq=wκjq−

∆t

∆yCj

wκ

jq−wκjq−1

, j= 0, . . . , J−1; κ= 0, . . . , N−1;

q=−∞, . . . ,+∞;

(11)

v_jqκ+1=uκjq−∆tMjuκ_jq, j = 0, . . . , J−1;

κ= 0, . . . , N−1. (12)

The initial conditions (4) are approximated as follows:

v_jq0 = 1 ∆x

1 ∆y

y q+ 1₂

Z

y q−1

2

x j+ 1₂

Z

x j−1

2

Φ(x, y)dxdy,

j = 0, . . . , J−1;q=−∞, . . .+∞

(13)

The boundary conditions are approximated in this way:

vIκ+1

−1q

vIIκ+1 J q

!

=

0 s r 0

_vIκ+1 J−1,q

vIIκ+1

0q !

,

κ= 0, . . . , N−1;q=−∞, . . .+∞

(14)

Suppose that the condition of the CFL

∆t

∆x1≤maxi≤n

0≤j≤J−1

|ki,j| ≤1,

∆t

∆y 1≤maxi≤n

0≤j≤J−1

(4)

holds. Hereλi(Cj)are eigenvalues of the matrixCj.

Now we study the question on the exponential stability of the solution of the difference problem (10)-(14).

Definition 2.The difference scheme (10)-(12) with a differ-ence boundary condition (14) exponentially stable if there exist constantsη > 0andc >0such that for any initial condition

v0

jq ∈ L2((xj−1

2, xj+ 1 2), (yq−

1 2, yq+

1 2),R

n₎_{, the solution of}

the difference boundary value problem (10)-(14) satisfies the equality

∆x∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq,vκjq

≤ce−ηtκ_×

×∆x∆y

+∞

P

q=−∞

J−1

P

j=0

v0

jq,vjq0

, κ= 1, . . . , N.

Here, we formally write out the sums of an infinite number of terms. However this is done only formally, since only a finite number of them are nonzero. It is due to the fact that the difference solution is non-zero only at a finite number of points.

Consider the difference boundary-value problem (10) - (14) with the stationary solution

vκ_jq= 0, κ= 0, . . . , N−1; j= 0, . . . , J−1;

q=−∞, . . . ,+∞.

In order to prove the stability of the difference boundary-value problem (10) - (14), we propose the following function as the discrete Lyapunov function

L(vκ) =Lκ= ∆x∆y

+∞

X

q=−∞

J−1

X

j=0

vκjq, µjvκjq

,

µj=µ(xj), j= 1, . . . , J−1

(15)

Hereµj =

e−νxj_µ+ ₀

0 eνxj_µ−

.

Theorem 2.LetT >0and the discrete Lyapunov function be defined by (15). If the condition of CFL _∆∆_xt max

1≤i≤n

0≤j≤J−1

|ki,j| ≤

1, _∆∆_yt max

1≤i≤n

0≤j≤J−1

|λi(Cj)| ≤ 1, holds, and there exist real

numbers ν > 0 and µi > 0, i = 1, . . . , n such that

0 < η ≡ ναe−ν∆x −β < 1,whereα = min

1≤i≤n

0≤j≤J−1

|ki,j|,

β = max

1≤i≤n

0≤j≤J−1

k0_i,j

,MjTµj+µjMj−∆tMjTµjMj, j=

0, . . . , J −1are non-negative definite matrices and

µ+_e−νxJ_K+

J−1 0

0 µ−eνx−1_K− 0

−

0 r s 0

×

µ+_e−νx0_K+

−1 0

0 µ−eνxJ−1_K−

J

0 s r 0

is a positive definite matrix, then the numerical solutionvκ

jqof

the difference boundary value problem (10)-(14) converges to the stationary solutionv_jq∗ = 0for the normL2_.

Proof. Using the Lyapunov discrete function, we calculate the derivative of the Lyapunov function (15) as follows

L(vκ+1₎₋_L₍_vκ₎

∆t =

L(vκ+1₎₋_L₍_uκ₎

∆t +

L(uκ₎₋_L₍_wκ₎

∆t +

+L(wκ)−_∆_tL(vκ),

whereL(vκ_{) = ∆}_x_∆_y +∞P

q=−∞

J−1

P

j=0

(vκ

jq, µjvκjq), L(wκ) =

= ∆x∆y

+∞

P

q=−∞

J−1

P

j=0

(wκ

jq, µjwjqκ),

L(uκ) = ∆x∆y

+∞

X

q=−∞

J−1

X

j=0

(uκ_jq, µjuκjq), κ= 0, . . . , N.

Now we prove that this quadratic form is negatively defined. For this, it suffices to show that all three quadratic forms on the right-hand side of the previous equation are negatively defined:

L(vκ+1)−L(uκ)

∆t =

∆x∆y

∆t ×

×

+∞

P

q=−∞

J−1

P

j=0

(vκ_jq+1, µjvκjq+1)−(uκjq, µjuκjq)

,

L(uκ)−L(wκ)

∆t =

∆x∆y

∆t ×

×

+∞

P

q=−∞

J−1

P

j=0

(uκ

jq, µjuκjq)−(wκjq, µjwκjq)

,

L(wκ)−L(vκ)

∆t =

∆x∆y

∆t ×

× +∞P

q=−∞

J−1

P

j=0

(wκ

jq, µjwκjq)−(vκjq, µjvκjq)

.

Lemma 1. Let the conditions of Theorem 2 be satisfied. Then the quadratic form

L(vκ+1)−L(uκ) ∆t ≤0

is not positively defined.

Proof. Using the difference scheme (12), calculate the quadratic form L(vκ+1_∆)−_tL(uκ).

L(vκ+1)−L(uκ)

∆t =

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

hh

uκ_jq−∆tMjuκ_jq

i

,

µj h

uκ_jq−∆tMjuκ_jq

i

−(uκ_jq, µjuκjq) i

=∆x_∆∆_ty× × +∞P

q=−∞

J−1

P

j=0

(uκ

jq, µjuκjq)−(uκjq, µjuκjq)

−∆x∆y×

×

+∞

P

q=−∞

J−1

P

j=0

h

Mjuκ_jq, µjuκjq

+uκ_jq, µjMjuκ_jq

−

−∆t(Mju_jqκ, µjMjuκ_jq)

i

=−∆x∆y× ×

+∞

P

q=−∞

J−1

P

j=0

h

uκ

jq,MTjµjuκjq

+uκ

jq, µjMjuκjq

−

−∆t(uκ

jq,MTjµjMjuκ_jq)

i

=−∆x∆y× ×

+∞

P

q=−∞

J−1

P

j=0

uκ_jq,hMjTµj+µjMj−∆tMjTµjMj i

uκ_jq.

By virtue of Theorem 2. MT

jµj + µjMj −

∆tMT

jµjMj, j = 0, ..., J − 1 are nonnegative definite

matrix. Hence,

∆t ≤0.

Lemma 1 is proved.

Lemma 2. Let the conditions of Theorem 2 be satisfied. Then the quadratic form

(5)

is not positively defined.

Proof. Introduce the notation Oj = _∆∆_ytCj. Using

the difference scheme (11), calculate the quadratic form

∆t .

L(uκ)−L(wκ)

∆t =

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

uκ_jq, µjuκjq

− wκ_jq, µjwjqκ

=

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

wκ_jq−Oj

wκ_jq−wκ_jq₋₁ , µj

wκ_jq−

−Ojwκjq−wjqκ−1 − wκjq, µjwκjq

∆x∆y

∆t +∞ P q=−∞ × × J−1 P j=0

wκ_jq, µjwκjq

− wκ_jq, µjwκjq

−2 µjwjqκ,Oj×

×

wκ_jq−wκ_jq₋₁ +∆x_∆∆_ty

+∞ P q=−∞ J−1 P j=0

µjOj

w_jqκ −wκ_jq₋₁,

Oj

wκ_jq−wκ_jq₋₁= ∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0

µjOj

w_jqκ −w_jqκ₋₁,

Oj

wκ

jq−wκjq−1

−2 µjwκjq,Oj

wκ

jq−wκjq−1 =

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

µjOjwjqκ,Ojwκjq

−2 µjOjwκjq,

Ojwκjq−1

+ µjOjwjqκ−1,Ojwκjq−1 +

+∆x_∆∆_ty

+∞ P q=−∞ J−1 P j=0

−2 µjwκjq,Ojwκjq

+ 2 µjwκjq,

Ojwκjq−1 = ∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

µjOjwjqκ,Ojwκjq

+ +2 µj[E−Oj]wκjq,Ojwκjq−1

+ µjOjwjqκ−1, Ojwκjq−1

−2 µjwκjq,Ojwjqκ .

HereE is the unit matrix. By virtue of the condition of the CFL, the matrix (E−Oj) ≥ 0. Suppose that the matrix

(E−Oj)µjOjis a positive definite matrix. Then we obtain

2µj[E−Oj]wκjq,Ojwκ_jq₋₁

≤ ≤ µj[E−Oj]wκjq,Ojwκjq

+ + µj[E−Oj]wκjq−1,Ojwκjq−1

=

µjwκjq,Ojwκjq

− µjOjwκjq,Ojwκjq

+ + µjwκjq−1,Ojwκjq−1

− µjOjwκjq−1,Ojwκjq−1

.

Taking into account this inequality, we have

∆t =

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0         

µjOjwκjq,Ojwjqκ

+

+2 µj[E−Oj]wjqκ,Ojwκjq−1

+ + µjOjwκjq−1,Ow

κ

jq−1

− −2µjwjqκ,Ojwκ_jq

         ≤

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0 h

µjOjwκ_jq,Ojwκ_jq

+ +µjwκjq,Ojwκjq

−µjOjwκjq,Ojwκjq i

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0             

µjwκjq−1,Ojwκjq−1 −

−µjOjw_jqκ₋₁,Ojwκ_jq₋₁

+ +µjOjw_jqκ₋₁,Ojwκ_jq₋₁

−

−2µjwjqκ,Ojwκ_jq             

= ∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0 h

µjwκjq−1,Ojwκ_jq₋₁

−

−µjwκjq,Ojwκ_jq i

.

Since the initial functions are assumed to be zero at|y| >

1

2Y, then (see [1]) for t = ∆t,the difference solution

ob-tained by the described scheme will be obviously equal to zero if|y|> 1₂Y + ∆yatt= 2∆t, if|y|> 1₂Y + 2∆y, etc. The difference solution will be equal to zero on the last grid layer

t=T if|y|> 1₂Y +_∆T_t∆y. Thus, the grid functionwκ

jqwill

be nonzero on any considered grid layer that only at a finite number of points. Iftis not too large,wκ

jq = 0if|y| > Y

. Taking into account this property of difference solutions, we have:

∆t ≤

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

µjwκjq−1,

Ojwκ_jq₋₁

−µjwκjq,Ojwκ_jq i

= 0.

Therefore

∆t ≤0.

Lemma 2 is proved.

Lemma 3. Let the conditions of Theorem 2 be satisfied. Then the following inequality

L(wκ)−L(vκ)

∆t <−ηL(v

κ₎

holds.

Proof.Introduce the following notation

Wκ_jq=

"

vIκ j−1,q

vIIκ j+1,q

#

, |K|_j =

K+_j₋₁ 0 0 K−_j₊₁

,

Dj= _∆∆_xt |K|j

Taking into account these notations, using the difference scheme (10), calculate the quadratic form L(wκ)−_∆_tL(vκ).Then we obtain:

L(wκ₎₋_L₍_vκ₎

∆t =

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

(wκ

jq, µjwjqκ)− −(vκ_jq, µjvκjq)

== ∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0

v_jqκ −Dj

vκ_jq−

−Wκ

jq , µjvκjq−Djvjqκ −Wjqκ −(vκjq, µjvκjq)

= = ∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0

(vκ

jq, µjvjqκ)−(vκjq, µjvκjq)− −2 µjvκjq,Dj

vκ_jq−Wκ_jq+ ∆_∆x∆_ty× × +∞ P q=−∞ J−1 P j=0

µjDj

vκ_jq−W_jqκ,Dj

vκ_jq−W_jqκ=

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0

µjDjvjqκ −Wκjq

,Djvjqκ −Wκjq

−

−2 µjvκjq,Dj

vκ

jq−Wκjq

= ∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0 h

µjDjvκ_jq,Dvκjq

−2µjDvκjq,DjWκ_jq

+ +µjDjWκ_jq,DjWκ_jq

i

+∆x_∆∆_ty× × +∞ P q=−∞ J−1 P j=0 h

−2µjvjqκ,Djvκ_jq

+ 2µjvκjq,DjWκ_jq i

=

∆x∆y

∆t +∞ P q=−∞ J−1 P j=0 h

µjDjvκ_jq,Djvκ_jq

+ +2µj(E−Dj)vκjq,DjWκ_jq

i

+ +∆_∆x∆_ty

+∞ P q=−∞ J−1 P j=0 h

µjDjWκ_jq,DjWκ_jq

−2µjvκjq,Djvκ_jq i

(6)

By virtue of the CFL condition, matrices

(E−Dj) ≥ 0 and (E−Dj)µjDj are

posi-tive definite diagonal matrices. Hence, we have

2 µj(E−Dj)vκjq,DjWκjq

≤ µj(E−Dj)vκjq,Djvjqκ

+ + µj(E−Dj)Wκjq,DjWκjq

= µjvκjq,Djvjqκ

− −µjDjvκjq,Djvκ_jq

+ µjWκjq,DjWκjq

− µjDWκjq,DWκjq

,

Taking into account this inequality, we obtain

L(wκ)−L(vκ)

∆t =

= ∆_∆x∆_ty

+∞

P

q=−∞

J−1

P

j=0

µjDjvκjq,Djvjqκ

+ +2 µj(E−Dj)vκjq,DjWκjq

+ +∆x_∆∆_ty

+∞

P

q=−∞

J−1

P

j=0

µjDjWκjq,DjWκjq

−

−2 µjvκjq,Djvκjq

≤∆_∆x∆_ty

+∞

P

q=−∞

J−1

P

j=0

µjDjvjqκ,Djvκjq

+ +µjvjqκ,Djvκ_jq

−µjDjvκjq,Djvκ_jq i

+

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

µjWjqκ,DjWκjq

− µjDjWκjq,DjWκjq

+

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

µjDjWκjq,DjWκjq

−2 µjvjqκ,Djvκjq

= ∆_∆x∆_ty

+∞

P

q=−∞

J−1

P

j=0

µjWκjq,DjWjqκ

− µjvκjq,Djvκjq

or L(wκ)−_∆_tL(vκ)≤ ∆_∆x∆_ty

+∞

P

q=−∞

J−1

P

j=0

µjWκjq,DjWjqκ

−

− µjvκjq,Djvκjq

.

Transform separately the first quadratic form in the right side of this inequality:

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

µjWκjq,DjWκjq

=

= ∆y

+∞

P

q=−∞

J−1

P

j=0 Wκ

jq,

µ+e−νxj_K+

j−1 0

0 µ−eνxj_K−

j+1

Wκ_jq

=

e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

Wκ jq,

_µ+_e−νxj−1_K+

j−1 0

0 µ−eνxj+1_K−

j+1

Wκ jq

=

=e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

vIκ j−1,q, µ

+_e−νxj−1_×

× K+_j₋₁ vIκ j−1,q

= =e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

vIIκ

j+1,q, µ

−_eνxj+1_×

×K−_j₊₁ vIIκ j+1,q

=

e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

vIκ

jq, µ

+_e−νxj_K+

j vI

κ jq

e−ν∆x∆y

+∞

P

q=−∞

vIκ₋₁_,q, µ+e−νx−1_K+ −1 vI

κ

−1,q

−

vIκ

J−1,q, µ

+_e−νxJ−1_K+

J−1 vI

κ J−1,q

e−ν∆x∆y

+∞

P

q=−∞

J−1

P

j=0

vIIκ_jq, µ−eνxj_K−

j v

IIκ

jq

e−ν∆x∆y

+∞

P

q=−∞

− vIIκ₀_q, µ−eνx0_K−

0 v

IIκ 0q

+

vIIκ

J q, µ

−_eνxJ_K−

J vII

κ J q

=

e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

(vκ

jq, µjKjvκjq) +

+e−ν∆x_∆_y +∞P

q=−∞

vIκ

−1,q, µ

+_e−νx0_K+ −1 vI

κ

−1,q

−

vIκ_J₋₁_,q, µ+e−νxJ_K+

J−1 v

Iκ

J−1,q

−e−ν∆x∆y

+∞

P

q=−∞

vIIκ₀_q, µ−eνx0_K−

0 v

IIκ 0q

+

vIIκ

J q, µ

−_eνxJ_K−

J v

IIκ J q

= =e−ν∆x∆y

+∞

P

q=−∞

J−1

P

j=0

(vκ_jq, µjKjvκjq) +

+e−ν∆x_∆_y +∞P

q=−∞

"

vIκ₋₁_,q vIIκ

J q #

,

µ+_e−νx0_K+

−1 0

0 µ−eνxJ−1_K−

J

" _vIκ

−1,q

vIIκ J q

#!

−

−e−ν∆x_∆_y +∞P

q=−∞

"

vIκ J−1,q

vIIκ

0q #

,

×

µ+e−νxJ_K+

J−1 0

0 µ−eνx−1_K− 0

" _vIκ J−1,q

vIIκ₀_q

#!

Here

|K|_j=

_K+

j 0

0 K−_j

.

Taking into account the boundary conditions

vIκ₋₁+1_,q

vIIκ+1 J q

!

=

0 s r 0

_vIκ+1 J−1,q

vIIκ+1

0q !

,

(7)

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

µjWjqκ,DjWκ_jq

=e−ν∆x_∆_y_×

×

+∞

P

q=−∞

J−1

P

j=0

(vκ

jq, µjKjvjqκ)+∆y

+∞

P

q=−∞

0 s r 0

×

× v

Iκ+1 J−1,q

vIIκ+1

0q !

,

µ+_e−νx0_K

−1+ 0

0 µ−eνxJ−1_K

J−

×

0 s r 0

_vIκ+1

J−1,q

vIIκ₀_q+1

!

−

−∆y

+∞

P

q=−∞

"

vIκ J−1,q

vIIκ₀_q

#

,

×

µ+_e−νxJ_K

J−1+ 0

0 µ−eνx−1_K0−

" _vIκ

J−1,q

vIIκ

0q #!

Suppose that the dissipativity condition

µ+_e−νxJ_K+

J−1 0

0 µ−_eνx−1_K− 0

−

0 r s 0

×

µ+_e−νx0_K+

−1 0

0 µ−_eνxJ−1_K−

J

0 s r 0

> 0 holds. Then we obtain

∆x∆y

∆t

+∞

P

q=−∞

J−1

P

j=0

µjWκjq,DjWκjq

≤

≤e−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0

(vκ

jq, µjKjvκjq)and, hence, L(wκ)−L(vκ)

∆t e

−ν∆x_∆_y +∞P

q=−∞

J−1

P

j=0 vκ

jq, µjKjvκjq

−

−∆y

+∞

P

q=−∞

J−1

P

j=0

vκ

jq, µj|K|_jvκjq

=

= ∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq, µj

h

e−ν∆x_K

j− |K|_j

i

vκ jq

=

= ∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq, µj

h

e−ν∆x_K

j−Kj+ Kj− |K|_j

i

vκ jq

=

= ∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq, µj

e−ν∆x_K

j−Kj

vκ jq

+

+ ∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq, µj

h

Kj− |K|j i

vκ jq

=

= (e−ν∆x₋_1)∆_y +∞P

q=−∞

J−1

P

j=0 vκ

jq, µjKjvκjq

+

∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq, µj

_K+

j −K

+

j−1 0

0 K−_j −K−_j₊₁

vκ jq

=

= −ν∆x∆y

+∞

P

q=−∞

J−1

P

j=0

vκ_jq, µjKjvκjq

+

∆x∆y

+∞

P

q=−∞

J−1

P

j=0

vκ_jq, µjK0jvκjq

.

Since according to the condition of Theorem 2, there are constants

α= min

1≤i≤n

0≤j≤J−1

|ki,j|, β= max

1≤i≤n

0≤j≤J−1

k0i,j

such that0< ναe−ν∆x₋_{β <}₁_,_{we get}

L(wκ₎₋_L₍_vκ₎

∆t <−ηL(v

κ₎_.

Here

η=ναe−ν∆x−β.

Lemma 3 is proved.

As a result, lemmas 1-3 imply the following inequalities:

∆t ≤0,

L(wκ)−L(vκ)

∆t <−ηL(v

κ₎_.

Summing up these inequalities, we have the following for the quadratic form

∆t :

∆t =

∆t +

+L(uκ)−_∆_tL(wκ) + L(wκ)−_∆_tL(vκ) < −ηL(vκ₎_orLκ+1₋_Lκ

∆t <

−ηLκ _or Lκ+1−Lκ

∆t < −ηL

κ_. _{Recursively applying this}

inequality, we obtain

Lκ+1 _< ₍₁ ₋ _∆_tη₎κ+1_L0 _≤ _e−η∆t(κ+1)_L0 ₌

e−ηtκ+1_L0_{, κ}_{= 0}_{, . . . , N}₋₁_.

Denote

C1 = min

1≤i≤n

0≤j≤J−1

{$ij :|µj−$ijE|= 0}, C2 =

max

1≤i≤n

0≤j≤J−1

{$ij:|µj−$ijE|= 0},

Then

C1E≤µj≤C2E, j= 0, . . . , J−1.

It follows from here that

C1∆x∆y +∞

P

q=−∞

J−1

P

j=0 vκ

jq,vκjq

≤Lκ_≤

≤C2e−ηtκ∆x∆y +∞

P

q=−∞

J−1

P

j=0 v0

jq,v0jq

, κ= 0, . . . , N,

∆x∆y

+∞

P

q=−∞

J−1

P

j=0

vκ jq,vκjq

≤Ce−ηtκ_∆_x_∆_y_×

×

+∞

P

q=−∞

J−1

P

j=0 v0

jq,v0jq

, κ= 0, . . . , N; C=C2/C1.

Hence, the numerical solutionv_jqn of the mixed problem is exponentially stable in theL2-norm.

Theorem 2 is proved.

4 Conclusions

(8)

Acknowledgements

This work was supported by Universiti Sains Islam Malaysia (USIM) under RMC Research Grant Scheme (FRGS, 2018). Project code is USIM/FRGS/FST/055002/51118

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