COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.7(2007), No.2, pp.178–190c© 2007 Institute of Mathematics of the National Academy of Sciences of Belarus

EXACT DIFFERENCE SCHEMES FOR

MULTIDIMENSIONAL HEAT CONDUCTION EQUATIONS

M.LAPINSKA-CHRZCZONOWICZ1

Abstract — The initial boundary-value problem for the two-dimensional heat con-duction equation

∂u

∂t=

∂

∂x1

(

k1(u)∂u

∂x1

)

+∂

∂x2

(

k2(u)∂u

∂x2

)

is considered. A new difference scheme approximating the above equation is con-structed. The error of approximation of the considered scheme is O((σ − 0.5)(h1 +h2 + τ) + h2

1+ h2

2+ τ2), where σ is a weight. The main difference between the method

proposed in the present paper and the other difference schemes is that for the travelingwave solutions

u(x, t) = U

(

1

2ax1 +

1

2bx2 − t

)

, 0 6 x1 6 l1, 0 6 x2 6 l2, 0 6 t 6 T,

the considered scheme is exact if the grid steps satisfy definite conditions γ1 = aτ/h1 =1/2, γ2 = bτ/h2 = 1/2. The iteration method is used to solve a nonlinear differenceequation.

2000 Mathematics Subject Classification: 65M06; 65M12.

Keywords: heat conduction equation, exact difference scheme.

1. Introduction

Various schemes have been constructed to approximate the boundary-value problem for theheat conduction equation [8, 24]. Let us consider the one-dimensional parabolic equation

∂u

∂t=∂2u

∂x2+ f(x, t).

The simplest difference scheme approximating the above problem is explicit (backward)difference scheme involving four points of the grid

yn+1i − yn

i

τ=yn

i−1 − 2yni + yn

i+1

h2+ ϕn

i .

The other basic schemes approximating this equation are the Richardsons scheme, theCrank — Nicolsons scheme, the Saulevs schemes or the scheme with weight [1–3, 8, 24].

1Department of Mathematics, The John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20–950Lublin, Poland. E-mail: lapkam@kul.lublin.pl

Exact difference schemes for multidimensional heat conduction equations 179

In investigating difference schemes, one of the most important problem is the approxi-mation order, which is desired to be as high as possible. For example, the scheme with aweight

yn+1i − yn

i

τ= σ

yn+1i−1 − 2yn+1

i + yn+1i+1

h2+ (1 − σ)

yni−1 − 2yn

i + yni+1

h2+ ϕn

i

has the order of approximation O((σ − 0.5)τ + τ 2 + h2), where σ is a weight. For σ =

0.5 − h2/(12τ) and ϕni = (5/6)f

n+1/2i + (1/12)(f

n+1/2i−1 + f

n+1/2i ), if u ∈ C6

3 , the error can bereduced to O(τ 2 + h4). In this case the scheme with a weight is usually termed a higher-accuracy scheme [24].

Difference schemes of a high order of approximation arouse interest among authors [17,21,26,27]. In the article of Mohanty [18], two-level implicit difference methods of O(τ 2 +h4)using 19-spatial grid points for the solving the three space dimensional heat conductionequation is proposed. Radwan in [22] gives a comparison of the numerical solutions obtainedfrom higher-order accurate difference schemes for the two-dimensional Burgers equation.

Definition 1.1. A difference scheme is exact if the truncation error is equal to zero ory = u at the grid nodes.

It is widely known that there exist exact difference schemes approximating transportequations or some ordinary differential equations. For example, for the Cauchy problem

du/dt = f1(t)f2(u), 0 < t 6 T, u(0) = u0,

the difference scheme

yn+1 − yn

τ=

1

τ

tn+1∫

tn

f1(t) dt

(

1

yn+1 − yn

yn+1∫

yn

du

f2(u)

)−1

, t ∈ ωτ , y0 = u0,

where f2(u) 6= 0, is exact [9]. In [24], the exact difference scheme is constructed for theequation

(k(x)u′(x))′ − q(x)u(x) = −f(x), 0 < x < L, u(0) = µ1, u(L) = µ2. (1.1)

It turned out that it is possible to construct other exact difference schemes for someordinary or partial differential equations .

In the last few years, such schemes have been considered in papers [5 – 7, 9, 10, 23]. Itis worth to mention here paper [7], in which under natural conditions the authors provedthe existence of a two-point exact difference scheme for systems of first-order boundaryvalue problems, or papers of Mickens [12 – 16], in which a nonstandard finite differenceschemes were introduced. Mickens [11] gives certain rules for constructing nonstandard finitedifference schemes and emphasizes that an important feature of nonstandard schemes is thatthey often can provide numerical integration techniques, for which elementary numericalinstabilities do not occur. Nowadays, other authors are applying techniques initiated byMickens to obtain exact numerical solutions or numerical solutions of a highorder of accuracyto a wide class of differential equations.

Let us introduce on the domain QT = [0, l1] × [0, l2] × [0, T ] with boundary Γ a uniformgrid

ω = ωh1× ωh2

× ωτ = {(x1,i, x2,j, tn) : x1,i = ih1, i = 0, N1, hN1 = l1,

x2,j = jh2, j = 0, N2, hN2 = l2, tn = nτ, n = 0, N0, τN0 = T}, (1.2)

180 M.Lapinska-Chrzczonowicz

Γh = {(x1i, x2j , tn+1) ∈ Γ : x1i = ih1, i = 0, N1, hN1 = l1,

x2j = jh2, j = 0, N2, hN2 = l2, tn = nτ, n = 0, N0, τN0 = T}. (1.3)

In this paper, we will use the following notation: yij = ynij = y(x1i, x2j , tn), y = yn+1,

y = yn+1/2, yt = (yi,j − yi,j)/τ , yx1= (yi,j − yi−1,j)/h1, yx2

= (yi,j − yi,j−1)/h2, yx1=

(yi+1,j − yi,j)/h1, yx2= (yi,j+1 − yi,j)/h1.

The main goal of this paper is to construct a new difference scheme approximating thetwo-dimensional heat conduction equation

∂u

∂t=

∂

∂x1

(

k1(u)∂u

∂x1

)

+∂

∂x2

(

k2(u)∂u

∂x2

)

, 0<x1<l1, 0<x2<l2, 0<t 6 T. (1.4)

The order of approximation of the scheme, which will be introduced in this paper, isO((σ − 0.5)(h1 + h2 + τ) + h2

1 + h22 + τ 2). The essential difference between the suggested

method and the other difference schemes is that for traveling wave solutions with a constantvelocity u(x1, x2, t) = U(x1/(2a) + x2/(2b)− t) satisfying the conditions stated in paper andwith specific steps h1, h2, τ , the considered scheme is exact.

The paper is organized as follows. In Section 2, the new difference scheme for theboundary-value problem (1.4) is constructed and the error of approximation of this scheme isinvestigated. The iteration method is used to solve the difference equation. In Section 3, thestability of the linear approach of the nonlinear difference scheme (1.4) is proved. In Section4, the results of the numerical experiments are presented to confirm the theoretical resultsinvestigated in the paper and show that an approach can be applied to different problems.

2. Exact difference schemes for the two-dimensional quasilinear

parabolic equation

In this Section, a new difference scheme for boundary-value problem for parabolic equation isconstructed. Primary consideration is given to the investigation of the error of approximationof this scheme. The iteration method is used to to solve the nonlinear difference equation.

Let us consider in the domain QT the boundary-value problem for the two-dimensionalquasi-linear parabolic equation

∂u

∂t=

∂

∂x1

(

k1(u)∂u

∂x1

)

+∂

∂x2

(

k2(u)∂u

∂x2

)

, 0 < x1 < l1, 0 < x2 < l2, 0 < t 6 T, (2.1)

u(x1, x2, 0) = u0(x1, x2), 0 6 x1 6 l1, 0 6 x2 6 l2, (2.2)

u|Γ = µ(x1, x2, t), (x1, x2, t) ∈ Γ, (2.3)

where 0 < c1 6 k1(u), k2(u) 6 c2, u = u(x1, x2, t), (x1, x2, t) ∈ QT , c1, c2 = const. Weassume that u ∈ C4(QT ) and 0 < m1 6 u(x1, x2, t) 6 m2, (x1, x2, t) ∈ QT , m1, m2 = const.

Then, equation (2.1) can be written in the equivalent form

∂u

∂t=

∂

∂x1

(

u∂ϕ1(u)

∂x1

)

+∂

∂x2

(

u∂ϕ2(u)

∂x2

)

, ϕ1(u)=

u∫

u0

k1(ξ)

ξdξ, ϕ2(u)=

u∫

u0

k2(ξ)

ξdξ. (2.4)

Exact difference schemes for multidimensional heat conduction equations 181

On the uniform grid (1.2), problem (2.1) – (2.3) is approximated by the difference scheme

yt =σ(y(ϕ1(y))x1)x1

+ σ (y(ϕ2(y))x2)x2

+ (1− σ)(y(ϕ1(y))x1)x1

+ (1− σ)(y(ϕ2(y))x2)x2, (2.5)

y0ij =u0(x1i, x2j), x1i∈ω1h, x2j ∈ω2h, yn+1

ij =µ(x1i, x2j, tn+1), (x1i, x2j , tn+1)∈Γh.(2.6)

Let us investigate the error of approximation of scheme (2.5), (2.6). We assume thatϕ1, ϕ2 ∈ C4[m1, m2]. Then the error of approximation satisfies the chain of equalities

ψn+1ij =−ut+σ(u(ϕ1(u))x1

)x1+σ(u(ϕ2(u))x2

)x2+(1−σ)(u(ϕ1(u))x1

)x1+(1−σ)(u(ϕ2(u))x2

)x2=

= −ut + σ∂u

∂t+ (1 − σ)

∂u

∂t+ σ(u(ϕ1(u))x1

)x1− σ

∂

∂x1

(

u∂ϕ1(u)

∂x1

)

+ (1 − σ)(u(ϕ1(u))x1)x1

−

−(1 − σ)∂

∂x1

(

u∂ϕ1(u)

∂x1

)

+ σ(u(ϕ2(u))x2)x2

− σ∂

∂x2

(

u∂ϕ2(u)

∂x2

)

+ (1 − σ)(u(ϕ2(u))x2)x2

−

−(1 − σ)∂

∂x2

(

u∂ϕ2(u)

∂x2

)

= ψ1 + ψ2 + ψ3.

We may introduce ψ1 in the form

ψ1 = −ut + σ∂u

∂t+ (1 − σ)

∂u

∂t= (σ − 0.5)τ

∂2u

∂t2+O(τ 2). (2.7)

Substituting the expansions

(ϕ1(u))x1,ij =∂

∂x1(ϕ1(u))ij +

h

2

∂2

∂x21

(ϕ1(u))ij +h2

3!

∂3

∂x31

(ϕ1(u))ij +

+h3

4!

(

∂2

∂x21

(ϕ1(u))

)

, u = u(ξ1, x2j , tn+1), ξ1 ∈ [x1i, x1i+1],

(ϕ1(u))x1,ij =∂

∂x1

(ϕ1(u))ij −h

2

∂2

∂x21

(ϕ1(u))ij +h2

3!

∂3

∂x31

(ϕ1(u))ij −

−h3

4!

(

∂2

∂x21

(

ϕ1(˜u))

)

, ˜u = u(ξ2, x2j , tn+1), ξ2 ∈ [x1i−1, x1i],

into ψ2, it is easy to get the expression

ψ2 =σ(u(ϕ1(u))x1)x1

− σ∂

∂x1

(

u∂ϕ1(u)

∂x1

)

+ (1 − σ)(u(ϕ1(u))x1)x1

− (1 − σ)∂

∂x1

(

u∂ϕ1(u)

∂x1

)

=

= σ

(

ui+1,j − uij

h1−

∂u

∂x1

)

∂

∂x1(ϕ(u)) + σ

(

ui+1,j + uij

2− uij

)

∂2

∂x21

(ϕ1(u))+

+h21

(

1

3!ux1,ij

∂3

∂x31

(ϕ1(u)) +ui+1,j

4

∂4

∂x41

(

ϕ1(ˆu))

+uij

4

∂4

∂x41

(

ϕ1(ˆu)

)

)

+

+(1 − σ)

(

uij − ui−1,j

h1

−∂u

∂x1

)

∂

∂x1

(ϕ(u)) + (1 − σ)

(

uij + ui−1,j

2− uij

)

∂2

∂x21

(ϕ1(u))+

+h21

(

1

3!ux1,ij

∂3

∂x31

(ϕ1(u)) +uij

4

∂4

∂x41

(ϕ1(u)) +ui−1,j

4

∂4

∂x41

(

ϕ1(˜u))

)

.

182 M.Lapinska-Chrzczonowicz

Taking into account the estimates

|ux1| =

∣

∣

∣

∣

∣

∣

1

h1

x1i+1∫

x1i

∂u(ξ, x2j , tn+1)

∂ξdξ

∣

∣

∣

∣

∣

∣

61

h1

x1i+1∫

x1i

∣

∣

∣

∣

∂u(ξ, x2j , tn+1)

∂ξ

∣

∣

∣

∣

dξ 6

61

h1max

(x1,x2,t)∈QT

∣

∣

∣

∣

∂u

∂x1

∣

∣

∣

∣

x1i+1∫

x1i

dξ 6 max(x1,x2,t)∈QT

∣

∣

∣

∣

∂u

∂x1

∣

∣

∣

∣

,

|ux1| =

∣

∣

∣

∣

∣

∣

1

h1

x1i∫

x1i−1

∂u(ξ, x2j , tn)

∂ξdξ

∣

∣

∣

∣

∣

∣

61

h1

x1i∫

x1i−1

∣

∣

∣

∣

∂u(ξ, x2j, tn)

∂ξ

∣

∣

∣

∣

dξ 6

61

h1max

(x1,x2,t)∈QT

∣

∣

∣

∣

∂u

∂x1

∣

∣

∣

∣

x1i∫

x1i−1

dξ 6 max(x1,x2,t)∈QT

∣

∣

∣

∣

∂u

∂x1

∣

∣

∣

∣

,

we arrive at

ψ2 = h1(σ − 0.5)∂2u

∂x21

∂

∂x1

(ϕ1(u)) + h1(σ − 0.5)∂u

∂x1

∂2

∂x21

(ϕ1(u)) +O(τ 2 + h21). (2.8)

Analogously, we write ψ3 in the form

ψ3 = h2(σ − 0.5)∂2u

∂x22

∂

∂x2(ϕ2(u)) + h2(σ − 0.5)

∂u

∂x2

∂2

∂x22

(ϕ2(u)) +O(τ 2 + h22). (2.9)

From equalities (2.7) – (2.9), we find that the difference scheme (2.5), (2.6) has a first orderof approximation O((σ− 0.5)(h1 + h2 + τ) + h2

1 + h22 + τ 2) for σ 6= 0.5 and a second order for

σ = 0.5.

Lemma 2.1. If following conditions are satisfied

∂

∂x1(ϕ1(u)) = −a,

∂

∂x2(ϕ2(u)) = −b, a, b = const > 0, (2.10)

then equality

σ(

u (ϕ1(u))x1

)

x1

+ σ(

u (ϕ2(u))x2

)

x2

+ (1 − σ)(

u (ϕ1(u))x1

)

x1

+ (1 − σ)(

u (ϕ2(u))x2

)

x2

=

= aσux1+ bσux2

+ a(1 − σ)ux1+ b(1 − σ)ux2

(2.11)

is valid.

Proof. Multiplying equations (2.10) by u and differentiating obtained formulas withrespect to x1 and x2 respectively, we get following equations

∂

∂x1

(

u∂ϕ1(u)

∂x1

)

= −a∂u

∂x1,

∂

∂x2

(

u∂ϕ2(u)

∂x2

)

= −b∂u

∂x2.

Integrating equations (2.10) on the intervals [x1,i−1, x1,i], [x1,i, x1,i+1] and [x2,j−1, x2,j ],[x2,j , x2,j+1] respectively, we get equality

σ(

u (ϕ1 (u))x1

)

x1

+ σ(

u (ϕ2 (u))x2

)

x2

+ (1 − σ)(

u (ϕ1 (u))x1

)

x1

+ (1 − σ)(

u (ϕ2 (u))x2

)

x2

=

= −aσux1− bσux2

− a(1 − σ)ux1− b(1 − σ)ux2

. (2.12)

�

Exact difference schemes for multidimensional heat conduction equations 183

Theorem 2.1. If conditions (2.10) of Lemma 2.1 are satisfied and 2aτ/h1 =1, 2bτ/h2 =1,then the difference scheme (2.5), (2.6) is exact for traveling wave solutions of the form

u(x1, x2, tn) = U

(

1

2ax1 +

1

2bx2 − t

)

.

Proof. It is easy to see that

un+1i+1,j = u(x1,i+1, x2,j , tn+1) = U

(

1

2ax1,i+1 +

1

2bx2,j − tn+1

)

=

= U

(

1

2ax1,i +

1

2bx2,j − tn +

1

2ah1 − τ

)

= U

(

1

2ax1,i +

1

2bx2,j − tn

)

= u(x1,i, x2,j , tn) = uni,j,

and similarly un+1i,j+1 = un

i,j. Denote γ1 = aτ/h1 and γ2 = bτ/h2. Then an alternative form ofthe condition 2aτ/h1 = 2bτ/h2 = 1 is γ1 = γ2 = 1/2.

Applying Lemma 2.1, the error of approximation of the difference scheme (2.5), (2.6)satisfies the chain of equalities

ψn+1i,j = −ut + σ

(

u (ϕ1(u))x1

)

x1

+ σ(

u (ϕ2(u))x2

)

x2

+ (1 − σ)(

u (ϕ1(u))x1

)

x1

+

+(1 − σ)(

u (ϕ2(u))x2

)

x2

= −(σ + 1 − σ)ut − aσux1− bσux2

− a(1 − σ)ux1− b(1 − σ)ux2

=

= σ−ut − aτux1

− bτ ux2

τ+ (1 − σ)

−ut − aτux1− bτux2

τ=

= σ−ui,j + ui,j − γ1ui+1,j + γ1ui,j − γ2ui,j+1 + γ2ui,j

τ+

+(1 − σ)−ui,j + ui,j − γ1ui,j + γ1ui−1,j − γ2ui,j + γ2ui,j−1

τ=

= σ(−1 + γ1 + γ2) ui,j + ui,j − γ1ui+1,j − γ2ui,j+1

τ+

+(1 − σ)(1 − γ1 − γ2) ui,j − ui,j + γ1ui−1,j + γ2ui,j−1

τ= σ

ui,j − γ1ui+1,j − γ2ui,j+1

τ+

+(1 − σ)−ui,j + γ1ui−1,j + γ2ui,j−1

τ= σ

(1 − γ1 − γ2) ui,j

τ+ (1 − σ)

(−1 + γ1 + γ2) ui,j

τ= 0.

Thus, the difference scheme (2.5), (2.6) is exact. �

Remark 2.1. The difference scheme (2.5), (2.6) is nonlinear, so there is a need to usethe iteration method, for example,

s+1y − yn

τ= σ

[

sy

(

ϕ1(sy) + ϕ′

1

(

sy)(

s+1y −

sy))

x1

]

x1

+ σ

[

sy

(

ϕ2(sy) + ϕ′

2

(

sy)(

s+1y −

sy))

x2

]

x2

+

+(1 − σ)(

y (ϕ1(y))x1

)

x1

+ (1 − σ)(

y (ϕ2(y))x2

)

x2

, (2.13)

where the initial approximation0y is calculated from the explicit difference scheme

0y − yn

τ=

(

y (ϕ1(y))x1

)

x1

+(

y (ϕ2(y))x2

)

x2

. (2.14)

184 M.Lapinska-Chrzczonowicz

The application of the iteration scheme is connected with checking the condition

‖s+1y −

sy‖C 6 ǫ,

where ǫ is the previously given accuracy and the norm ‖·‖C is defined as follows:

‖v‖C = max0<i<N1,0<j<N2

|vij| .

If this condition is satisfied, then we obtain yn+1i,j =

s+1y i,j, i = 1, N1 − 1, j = 1, N2 − 1.

3. Stability of the linear approach

In the present Section the object of investigation is the stability of the linear approach tothe nonlinear difference scheme (2.5), (2.6).

Let us assume that the coefficients of equation (2.1) depend only on x1, x2 and theboundary conditions

yt = σ(k1yx1)x1

+ σ(k2yx2)x2

+ (1 − σ)(k1yx1)x1

+ (1 − σ)(k2yx2)x2, (3.1)

y0ij = u0(x1i, x2j), x1i ∈ ω1h, x2j ∈ ω2h,

yn+1ij = 0, (x1i, x2j , tn+1) ∈ Γh. (3.2)

are homogeneous. We assume that the coefficients k1 and k2 are positive and bounded:

0 < c1 6 k1(x1, x2), k2(x1, x2) 6 c2, x1 ∈ [0, l1], x2 ∈ [0, l2], (3.3)

where c1, c2 = const.Let us introduce the space H comprising all the functions v defined on the grid ω

and satisfying the condition vnij = 0, (x1i, x2j , tn) ∈ Γh, with the inner product (y, v) =

N1−1∑

i=1

N2−1∑

j=1

yijvijh1h2 and norm ‖v‖ =√

(v, v).

Let us define in the space H the grid operators A, B, A1, A2, A′1, A

′2

(A1v) = −σ(k1vx1)x1, (A2v) = −(1 − σ)(k1vx1

)x1,

(A′1v) = −σ(k2vx2

)x2, (A′

2v) = −(1 − σ)(k2vx2)x2,

A = A1 + A2 + A′1 + A′

2, B = E + τ (A1 + A′1) , (3.4)

where E is the identity operator making it possible to rewrite problem (3.1) - (3.2) in theequivalent operator form

Byt + Ay = 0, y0 = u0. (3.5)

It is easy to see that

Ay = − [σ(k1yx1)x1

+ (1 − σ)(k1yx1)x1

+ σ(k2yx2)x2

+ (1 − σ)(k2yx2)x2

] =

= − [σ(k1yx1)x1

+ (1 − σ)(k1,−1yx1)x1

+ σ(k2yx2)x2

+ (1 − σ)(k2,−1yx2)x2

] =

= −[

(k1(σ)yx1)x1

+ (k2(σ)yx2)x2

]

,

where k1,−1 = k1,i−1,j , k2,−1 = k2,i,j−1, k1(σ) = σk1,i,j+(1−σ)k1,i−1,j , k2(σ) = σk2,i,j+(1−σ)k2,i,j−1.

Let the self-adjoint and positive operator A : H → H be prescribed. The norm ‖v‖A =√

(Av, v) is called the energy norm generated by the operator A. HA denotes a Hilbert spaceconsisting of elements v ∈ H provided with the inner product (y, v)A = (Ay, v) and norm‖v‖A [25].

Exact difference schemes for multidimensional heat conduction equations 185

Lemma 3.1 [25]. Let the operator A be a self-adjoint positive operator independent of

n. The condition

B > 0.5τA, t ∈ ωτ ,

is necessary and sufficient for scheme (3.5) in HA to be stable with respect to the initial data

‖yn‖A 6 ‖u0‖A . (3.6)

It is wellknown that the operator A (independent of n) defined by (3.4) is self-adjointand positive. Thereby, it is sufficient to verify the inequality B > 0.5τA. Let us notice that

B − 0.5τA = E + τ (A1 + A′1) − 0.5τA1 − 0.5τA′

1 − 0.5τA2 − 0.5τA′2 =

= E + 0.5τ(A1 −A2) + 0.5τ(A′1 −A′

2).

Let v ∈ H be an arbitrary function, then

((B − 0.5τA) v, v) = ((E + 0.5τ(A1 − A2) + 0.5τ(A′1 − A′

2)) v, v) =

= ‖v‖2 + 0.5τ ((A1 −A2) v, v) + 0.5τ ((A′1 − A′

2) v, v) = ‖v‖2 − 0.5τσ(

(k1vx1)x1

, v)

+

+0.5τ(1 − σ)(

(k1vx1)x1

, v)

− 0.5τσ(

(k2vx2)x2

, v)

+ 0.5τ(1 − σ)(

(k2vx2)x2

, v)

.

Applying the first Green formula with regard to the operator (kαvxα)xα, α = 1, 2, [24], we

have

((B − 0.5τA) v, v) = ‖v‖2 + 0.5τσ (k1vx1, vx1

]1 − 0.5τ(1 − σ) (k1,−1vx1, vx1

]1 +

+0.5τσ (k2vx2, vx2

]2 − 0.5τ(1 − σ) (k2,−1vx2, vx2

]2 =

= ‖v‖2 + 0.5τ ((σk1 − (1 − σ)k1,−1) vx1, vx1

]1 + 0.5τ ((σk2 − (1 − σ)k2,−1) vx2, vx2

]2 =

= ‖v‖2 + 0.5τ(

σk1 − (1 − σ)k1,−1, v2x1

]

1+ 0.5τ

(

σk2 − (1 − σ)k2,−1, v2x2

]

2,

where

(u, v]1 =N1∑

i=1

N2−1∑

j=1

ui,jvi,jh1h2, ||v]|1 =√

(v, v]1,

(u, v]2 =

N1−1∑

i=1

N2∑

j=1

ui,jvi,jh1h2, ||v]|2 =√

(v, v]2.

Since ‖v‖2> h2

1 ||vx1]|21 /4 and ‖v‖2

> h22 ||vx2

]|22 /4, then

((B−0.5τA)v, v)>

(

h21

8+στk1

2−

(1−σ)τk1,−1

2, v2

x1

]

1

+

(

h22

8+στk2

2−

(1−σ)τk2,−1

2, v2

x2

]

2

. (3.7)

It is easy to see that ((B − 0.5τA)v, v) > 0, if h21/8 + στk1,i,j/2 − (1 − σ)τk1,i−1,j/2 > 0

and h22/8 + στk2,i,j/2 − (1 − σ)τk2,i,j−1/2 > 0 for i = 1, N1, j = 1, N2. These conditions are

equivalent to the following one

σ >3

4−

c14c2

−1

8τc2min

{

h21, h

22

}

. (3.8)

If condition (3.8) is satisfied, then scheme (3.1), (3.2) is stable. By contrast, the well-knownscheme with weight

yt = σ(k1yx1)x1

+ σ(k2yx2)x2

+ (1 − σ)(k1yx1)x1

+ (1 − σ)(k2yx2)x2,

is stable under the condition [24]

σ > 1/2 − (8τc2)−1 min

{

h21, h

22

}

.

186 M.Lapinska-Chrzczonowicz

4. Numerical experiments

In this Section, the results of the numerical experiments presented to show that the approach,presented in Section 2 can be used for different problems and confirm the theoretical resultsinvestigated in the paper.

4.1. Example of the exact difference scheme for the two-dimensional linear

parabolic equation. In the domainQT , let us consider the two-dimensional linear parabolicproblem [19]

∂u

∂t= a2

1

∂2u

∂x21

+ b21∂2u

∂x22

, 0 < x1 < l1, 0 < x2 < l2, 0 < t 6 T, (4.1)

where a1, b1 = const > 0. The exact solution of problem (4.1) given by u(x1, x2, t) =exp{t/2−x1/(2a1)−x2/(2b1)} satisfies conditions (2.10) with constants a = a1/2, b = b1/2.The boundary and initial conditions are consistent with the exact solution.

Problem (4.1) is approximated by the difference scheme

yt,i,j = σ(y[ϕ1(y)]x1)x1,i,j + (1 − σ)(y[ϕ1(y)]x1

)x1,i,j + σ(y[ϕ2(y)]x2)x2,i,j+

+(1 − σ)(y[ϕ2(y)]x2)x2,i,j, x1i ∈ ωh1

, x2j ∈ ωh2, tn ∈ ωτ , (4.2)

where ϕ1(y) = a21 ln y, ϕ2(y) = a2

2 ln y. It is worth to emphasize here that scheme (4.2) isnonlinear as compared to a lot of other schemes approximating the linear problem (4.1).

In all numerical experiments, the accuracy is ǫ = 1.0E − 17 (see Figs. 4.1, 4.2).

F i g. 4.1. Exact solution of problem (4.1) for t = 1,l1 = 5, l2 = 5, T = 1, a1 = 1 and a2 = 1

F i g. 4.2. Solution of the difference scheme (4.2)for σ = 0.5, l1 = 5, l2 = 5, T = 1, a1 = 1, a2 = 1,

N1 = 50, N2 = 50 and N0 = 10

Here ξ is the number of iterations. Let us stress here that the iteration method (2.13)converges after two-three iterations performed. This is due to the calculation of the initial

approximation0yi,j from the exact but unstable explicit scheme (2.14) [9]. Tables 4.1 and

4.2, in which the results of the numerical experiments for different parameter values arepresented, confirm the theoretical results stated in Theorem 2.1.

Exact difference schemes for multidimensional heat conduction equations 187

T ab l e 4.1. σ = 0.5, a1 = 1, b1 = 1, l1 = 1,

l2 = 1, T = 1

h1 h2 τ max06n6N0

‖yn − u(tn)‖C

ξ

0.2 0.2 0.2 8.67E − 19 20.1 0.1 0.1 9.76E − 19 20.05 0.05 0.05 1.95E − 18 20.02 0.02 0.02 6.94E − 18 2

0.01(6) 0.01(6) 0.01(6) 6.94E − 18 2

Ta b l e 4.2. σ = 0.5, a1 = 2, b1 = 4,

l1 = 10, l2 = 10, T = 5

h1 h2 τ max06n6N0

‖yn − u(tn)‖C

ξ

1.0 2.0 0.5 9.98E − 18 30.5 1.0 0.25 1.30E − 17 30.25 0.5 0.125 1.86E − 17 30.2 0.4 0.1 2.13E − 17 3

The results of the numerical experiments presented in Figs. 4.3 and 4.4 show that thenew scheme resolves the boundary-value problem (4.1) retaining the accuracy of the orderO((σ − 0.5)(h1 + h2 + τ) + h2

1 + h22 + τ 2).

F i g. 4.3. Numerical results for σ = 0.5, l1 = 1, l2 = 1, T = 1, a1 = 0.5 and b1 = 4

F i g. 4.4. Numerical results for σ = 0.75, l1 = 1, l2 = 1, T = 1, a1 = 0.5 and b1 = 4

188 M.Lapinska-Chrzczonowicz

4.2. Example of the exact difference scheme for the two-dimensional quasi-

linear heat conduction equation. In the domain QT , let us consider the two-dimensionalquasi-linear heat conduction problem [20]

∂u

∂t=∂u

∂x1

(

k (u)∂u

∂x1

)

+∂u

∂x2

(

k (u)∂u

∂x2

)

, 0 < x1 < l1, 0 < x2 < l2, 0 < t 6 T, (4.3)

where k(u) = ue−u. Function

u(x1, x2, t) =

− ln

[

1 − λ

(

λ

2t−

x1

2−x2

2

)]

, x1 + x2 6 λt < x1 + x2 +2

λ,

0, λt < x1 + x2,

where λ = const > 0, is the solution of problem (4.3) and satisfies condition (2.10) withconstants a = b = λ

2. The boundary and initial conditions are consistent with the exact

solution.Problem (4.3) is approximated by the difference scheme

yt,i,j = σ(

y [ϕ(y)]x1

)

x1,i,j+(1−σ)

(

y [ϕ(y)]x1

)

x1,i,j+σ

(

y [ϕ(y)]x2

)

x2,i,j+(1−σ)

(

y [ϕ(y)]x2

)

x2,i,j,

x1i ∈ ωh1, x2j ∈ ωh2

, tn ∈ ωτ , (4.4)

where ϕ(y) = exp {−y}.In all numerical experiments, the accuracy is ǫ = 1.0E − 17 (see Figs. 4.5, 4.6).Here ξ is the number of iterations. Tables 4.3 and 4.4, in which the results of the

numerical experiments for different parameter values are presented, confirm the theoreticalresults stated in Theorem 2.1.

F i g. 4.5. Exact solution of problem (4.3) for t =0.4, l1 = 0.8, l2 = 0.8, T = 0.4 and λ = 2

F i g. 4.6. Solution of the difference scheme (4.4)for σ = 0.5, t = 0.4, l1 = 0.8, l2 = 0.8, T = 0.4,

λ = 2, N1 = 10, N2 = 10 and N0 = 10

T ab l e 4.3. σ = 0.5, λ = 1, l1 = 1, l2 = 1,

T = 1

h1 h2 τ max06n6N0

‖yn − u(tn)‖C

ξ

0.2 0.2 0.2 1.08E − 19 20.1 0.1 0.1 1.68E − 18 20.05 0.05 0.05 5.10E − 18 20.02 0.02 0.02 3.36E − 18 2

T ab l e 4.4. σ = 0.5, λ = 2, l1 = 0.8, l2 = 0.8,

T = 0.4

h1 h2 τ max06n6N0

‖yn − u(tn)‖C

ξ

0.2 0.2 0.1 1.08E − 19 10.1 0.1 0.05 1.14E − 18 10.05 0.05 0.025 4.23E − 18 20.02 0.02 0.01 7.91E − 18 3

Exact difference schemes for multidimensional heat conduction equations 189

5. Conclusions

In this paper, we have constructed a new difference scheme approximating the two-dimensi-onal heat conduction equation. We have proved that for the traveling wave solutions of theform

u(x, t) = U

(

1

2ax1 +

1

2bx2 − t

)

, 0 6 x1 6 l1, 0 6 x2 6 l2, 0 6 t 6 T,

the considered scheme is exact if the grid steps satisfy definite conditions

γ1 =aτ

h1=

1

2, γ2 =

bτ

h2=

1

2.

Numerical results have been presented to confirm the theoretical results stated in thepaper.

Future research will focus on multidimensional problems for parabolic equations with atraveling wave solution of the form u(x1, x2, t) = U(z1, z2), z1 = k1x1 −λ1t, z2 = k2x2 −λ2t,where λ1/k1 and λ2/k2 are the velocities of wave propagation in the directions x1 and x2,respectively.

References

1. M. Dehghan, Numerical schemes for one-dimensional parabolic equations with nonstandard initial

condition, Applied Mathematics and Computation, 147 (2004), pp. 321–331.2. M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Applied Mathematics

and Computation, 136 (2003), pp. 333–344.3. M. Dehghan, Three-level techniques for one-dimensional parabolic equation with nonlinear initial

condition, Applied Mathematics and Computation, 151 (2004), pp. 567–579.4. L. C. Evans, Partial differential equations, PWN, Warszawa, 2004 (in Polish).5. I. P. Gavrilyuk, Exact difference schemes and difference schemes of arbitrary given degree of accuracy

for generalised one-dimensional third boundary value problem, Z. Anal. Anwend., 12 (1993), pp. 549–566.6. I. P. Gavrilyuk, M. Hermann, M. V. Kutniv, and V. L. Makarov, Difference schemes for nonlinear

BVPs on the half-axis, Computational Methods In Applied Mathematics, 1 (2001), no. 1, pp. 1–30.7. V. L. Makarov, I. P. Gavrilyuk, M. V. Kutniv, and M. Hermann, A two-point difference scheme of an

arbitrary order of accuracy for BVPs for system of first order nonlinear ODEs, Computational Methods InApplied Mathematics, 4 (2004), no. 4, pp. 464–483.

8. T. Malolepszy, Metody badania stabilnosci schematow roznic skonczonych dla jednowymiarowego

rowania przewodnitwa cieplnego, Uniwersytet Zielonog?rski, Zielona Gora, 2004.9. P. Matus, U. Irkhin, and M. Lapinska-Chrzczonowicz, Exact difference schemes for time-dependent

problems, Computational Methods In Applied Mathematics, 5 (2005), no. 4, pp. 422–448.10. P. Matus, U. Irkhin, M. Lapinska-Chrzczonowicz, and S. Lemeshevsky Exact difference schemes for

hyperbolic and parabolic equations, Differents. Uravneniya, 43 (2007), no. 7, (in Russian), (in print).11. R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific Publishing

Co. Pte. Ltd., Singapore, 2000.12. R. E. Mickens, Exact finite difference schemes for two-dimensional advection equations, Journal of

Sound and Vibration, 207 (1997), no. 3, pp. 426–428.13. R. E. Mickens, Suppression of numerically induced chaos with nonstandard finite difference schemes,

Journal of Computational and Applied Mathematics, 106 (1999), pp. 317–324.14. R. E. Mickens and A. B.Gubel Numerical study of a non-standard finite-difference scheme for the

Van Der Pol equation, Journal of Sound and Vibration, 250 (2002), no. 5, pp. 955–963.15. R. E. Mickens, Exact finite difference scheme for a spherical wave equation, Journal of Sound and

Vibration, 182 (1995), no. 2, pp. 342–344.16. R. E. Mickens, A non-standard finite difference scheme for the equations modeling stellar structure,

Journal of Sound and Vibration, 265 (2003), pp. 1116–1120.

190 M.Lapinska-Chrzczonowicz

17. R. K.Mohanty and M. K. Jain, High accuracy difference schemes for the system of two space nonlin-

ear parabolic differential equations with mixed derivatives and variable coefficients, Journal of Computationaland Applied Mathematics, 70 (1996), pp. 15–32.

18. R. K.Mohanty, High accuracy difference schemes for a class of three space dimensional singular

parabolic equations with variable coefficients, Journal of Computational and Applied Mathematics, 89 (1997),pp. 39–51.

19. A. D. Polyanin, Handbook of Linear Mathematical Physics Equations, Fizmatlit, Moscow, 2001, (inRussian).

20. A. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov, Solution methods for nonlinear equations of mathe-

matical physics and mechanics, Fizmatlit, Moscow, 2005, (in Russian).21. Y. Qian, H. Chen, R. Zhang, and S. Chen, A new fourth order finite difference scheme for the heat

equation, Communications in Nonlinear Science & Numerical Solutions, 5 (2000), no. 4, pp. 151–157.22. S. F. Radian, Comparison of higher-order accurate schemes for solving the two-dimensional unsteady

Burgers’ equation, Journal of Computational and Applied Mathematics, 174 (2005), pp. 383–397.23. K. Sakai A new Finite Variable Difference Method with Application to Locally Exact Numerical

Scheme, Journal of Computational Physics, 124 (1996), pp. 301–308.24. A. A. Samarskii, The theory of difference schemes, Marcel Dekker Inc., New York — Basel, 2001.25. A. A. Samarskii, P. P.Matus, and P.N. Vabishchevich, Difference Schemes with Operator Factors,

Kluwer Academic Publishers, Netherlands, 2002.26. J.W. Thomas, Numerical partial differential equations: finite difference methods, Springer-Verlag

New York Inc., New York, 1995.27. D. You, A high order Pade ADI method for unsteady convection-diffusion equations, Journal of

Computational Physics, 214 (2006), pp. 1–11.