General Letters in Mathematics Vol. 7, No. 1, Sep 2019, pp.24-30e-ISSN 2519-9277, p-ISSN 2519-9269Available online at http:// www.refaad.comhttps://doi.org/10.31559/glm2019.7.1.3

Spectral Method for the Heat Equation withAxial Symmetry and a Source

A.Boutaghou

département de mathématiques université Dr: Yahia Fares Médéa,Laboratoire de mécanique, physique et modélisation mathématiques, (LMPMM).

boutaghou a@yahoo.com

Abstract. In this paper, we present a spectral method for solving the heat equation in cylindrical coordinates in a casewhere the data are axisymmetric and independent of the z-coordinate at the same time. The spectral method considered is ofGalerkin type with a Gauss-Radau numerical quadrature formula, it is based on a weighted weak variational formulation of thecontinuous problem. The method considered is discret only in r-variable, the time variable remains continuous. Consequently,the discret problem leads to a system of ordinary differential equations, we solve the system and estimate the error, we alsogive some numerical examples.

Keywords: Orthogonal polynomials, Spectral method, Error estimate, Condition number, Programming.2010 MSC No: 65N35,41A10, 33D45.

1 IntroductionThe heat equation is one of the important partial differential equations. It plays a very important role in all areasof science, such as mathematics, mathematical physics and other sciences. Finding the solution of this equationis therefore a common problem, often difficult or impossible to resonate analytically. In recent years, especiallyafter the emergence of the computer, they have developed many numerical methods to determine an approximatesolution instead of the exact solution to this problem. These methods gave these fruits. In this paper, we present aspectral method for solving the equation of heat in cylindrical coordinates in a case where the data (and therefore thesolution) are axisymmetric and independent of the z-coordinate at the same time. The spectral method consideredis of Galerkin type with a numerical quadrature formula. The study is based on a weak variational formulation ofthe continuous problem. The numerical quadrature formula used is the Gauss-Radau formula with a weight in therange [0,1]. The method considered is discrete only in the r-variable, the time variable remains continuous. As aresult, the discrete problem leads to a system of ordinary differential equations. We solve this system by using thediagonalization of the matrix related to the discretization used. Section 2 is devoted to the theoretical descriptionof the method, section 3 is devoted to the rewriting of the method using the nodes of the polynomial defined ofthe aforementioned Gauss-Radau formula. Section 4 is mainly devoted to analyzing the error, but it also containsnumerical examples.

The concerned problem refer to the equation:ut − α(urr + 1

rur) = f (r, t) in Λ× (0,∞)u(1, t) = 0 t > 0u(r, 0) = u0(r) in Λ

(1)

Spectral Method for the Heat Equation with Axial Symmetry and a Source 25

where Λ =]0, 1] and ∂Λ its boundary, α positive real number, f ∈ L21 (Λ) , where :

L21 ( Λ) =

f : Λ→ R measurable /

∫Λ

(f(r))2rdr <∞

(2)

u0 is smooth on Λ , we denote by ‖.‖L21(Λ) and (., .) the norm and inner product in L2

1 (Λ) and H111 ( Λ) the

subspace is the standard Sobolev space H11 (Λ) satisfying the homogeneous Dirichlet boundary conditions.

In this work we construct approximate solution to the boundary value problem (1) in the form

uN (r, t) =N∑n=0

an(t)ln(r) (3)

but the condition u(1, t) = 0 when t > 0 leads to aN(t) = 0, the approximate solution take the following form

uN (r, t) =N−1∑n=1

an(t)ln(r) (4)

Where the Lagrangian interpolates ln(r), 0 ≤ n ≤ N, are defined at the points rj ∈ Λ = [0, 1] , 0 ≤ j ≤ N . Thegrid made by rj , 0 ≤ j ≤ N, is denoted by

∑N+1 .

The choice of the form (3) for the solution, added to some technics lead to a linear system which can bewritten in a matricial form as Aa + ΓDa = F , where A is a square symmetric positive defined matrix and Γis a diagonal invertible matrix and the operator D = d

dt . We write a = Pv where P is an orthogonal matrix suchthat P−1 (Γ−1A

)P = C is a diagonal matrix, then we obtain a system of N − 1 ordinary differential equations

ddtvj(t) + cjvj(t) = hj(t), j = 1, N − 1 , we can use Lagrange’s method of undetermined parameters to solve foreach component vj(t) of v [9], finally we conclude the expressions of functions an(t) and for which we obtain theapproximation solution.

2 Discretization of the problem2.1 Continuous problem, weak form.the variational formulation of problem (1) it is written :

find u ∈ H11 (Λ), such that u− u1 ∈ H1

11(Λ∀v ∈ H1

11(Λ), (ut, v) + a1(u, v) = (f, v) (5)

where the bilinear form a(., .) is given by:

a(u, v) =∫ 1

0∂ru∂rvrdr (6)

see[7].Where the pivot space of the problem (1) is the space L21 (Λ) , and the variational space is

H11 (Λ) =

v/ v, ∂rv ∈ L2

1 (Λ)

(7)

and the corresponding norm is defined as,‖v‖2L2

1(Λ) =∫

Λv2rdrdt (8)

and the semi norm is defined as,|v|2H1

1 (Λ) =∫

Λ((∂rv)2rdrdt

Also we need the subspace of the variational space with zero Dirichlet trace:

H111 (Λ) =

v ∈ H1

1 (Λ) ; v(1, t) = 0 , t > 0

(9)

26 A.Boutaghou

2.2 Discrete spacesThe approximate spaces is essentially generated by the finite dimensional subspace of L2

1(Λ), IPN (Λ) , IP 1N (Λ) are

the approximate spaces of the spaces H11 (Λ), H1

11(Λ) respectively where

IP 1N (Λ) = pN ∈ PN (Λ) , pN (1) = 0

In this work we consider the quadrature formula and introduce a bilinear form a1N which approach the form a1 andwe approximate (., .)N for (., .) , where N represent in spectral method the degree of polynomials. Here, the discreteproduct is defined for all functions g and h continuous on Λ by

(g, h)N =N∑k=0

g(rk)h(rk)ωk (10)

2.3 Discrete problem, variational formulationthe variational formulation (5) is written :

find uN ∈ IPN (Λ) such that∀v1N ∈ IP 1

N (Λ),(utN , v

1N

)N

+ a1N (uN , v1N ) =

(fN , v

1N

)N

uN0 = IN (u0)(11)

where uN0 is the interpolating polynomial of the initial condition u0 at the Causs-Lobatto nodes and the bilinearform a(., .) is given by:

a1N (uN , v1N ) = (∂ruN , ∂rv1

N )N =N∑k=0

(∂ruN∂rv

1N

)(rk)ωk (12)

The equation is now what are the necessary tools to insure the existence and uniqueness of the approximatesolution which verify the variational formulation (11).

2.4 The spectral methodIn this section we describe the spectral element method applied to subsection 2.3 of the algorithm given in theintroduction. the spectral method is based on a weak formulation of the considered problem. The approximatesolution representation is then given by :

uN (r, t) = u1N (r, t) + uN1,

u1N (r, t) =

N−1∑n=1

an(t)ln(r) ∈ IP 1N (Λ) and uN1(r) = u(1, t) = 0

the Lagrangian interpolates ln(r), 0 ≤ n ≤ N are defined on the interval Λ with r ∈ Λ, by the relationship

ln(r) =N

Πk=0,k 6=n

(r − rkrn − rk

)these interplants satisfy the property ln(rm) = δnm, 1 ≤ n,m ≤ N − 1,where the points rk are the collocation pointson the Gauss-Radau grid, i.e the roots of the polynomial MN (x) = LN+1(x)−LN (x), LN is the Legendre polynomialand x = 2r − 1. There also exists a unique set of positive real numbers ωn, 0 ≤ n ≤ N such that the integral rule∫ 1

0ϕ(r)rdr =

N∑j=0

ϕ (rj)ωj , ωj = rj2

(n+ 1)2L2n(2rj − 1)

is exact for all polynomials ϕ of degree 2N − 1 or less on the interval Λ see [5, 10].

Spectral Method for the Heat Equation with Axial Symmetry and a Source 27

2.5 Existence and uniqueness of solutionProposition 1. The bilinear form a1N (·, ·) satisfies the following properties of continuity:

∀v1N ∈ P 1

N (Λ) ,∀u1N ∈ P 1

N (Λ) ,∣∣a1N

(u1N , v

1N

)∣∣ ≤ ∣∣u1N

∣∣H1

1 (Λ) ·∣∣v1N

∣∣H1

1 (Λ) (13)

and ellipticity∀u1

N ∈ PN (Λ) ,∣∣a1N

(u1N , v

1N

)∣∣ ≥ ∣∣u1N

∣∣2H1

1 (Λ) (14)

Proof. a1N(u1N , v

1N

)=(∂ru

1N , ∂rv

1N

)N

the degree of polynomials ∂ru1N , ∂rv

1N is less than or equal to N − 1 then(

∂ru1N , ∂rv

1N

)N

=∫ 1

0 ∂ru1N∂rv

1Nrdr by the Schwarz inequality we obtain the desired results.

3 Numerical experimentThe variables r and t play different role, to separate these variables we consider the solution u(r, t) and f(r, t) asfunctions of the variable t its values are in the function space defined in Λ , we consider u defined by:

u : [0,∞)→ H11 (Λ)

t → u(t)

then we can not u(r, t) = u(t)(r)the variational formulation can be written as:

find u1N in IPN (Λ), such that

∀v1N ∈ IP 1

N (Λ), ddt

(u1N (t), v1

N

)+ a1N (u1

N (t), vN ) =(fN , v

1N

)N

(15)

the formulation (15) is true for all vN ∈ IP 1N (Λ) then it is true for vm(r) = lm(r), m = 1, N − 1 where (lm(r))0≤m≤N

form a basis to the polynomial space IPN (Λ), the degree of the polynomial uN (t)vN is 2N with respect the variabler, and the degree of the polynomial ∂xu1

N (t)∂xv1N is 2N − 1, then we can write (15) as:

find u1N ∈ IPN (Λ), such that

∀lm ∈ IP 1N (Λ),

N∑k=0

(N−1∑n=1

a′n((t)ln(rk)lm(rk)ωk)

+N∑k=0

(N−1∑n=1

an(t)l′n(rk)l′m(rk)ωk)

=N∑k=0

N−1∑n=0

fN (t, rk)lm(rk))ωk −N∑k=0

u′0(rk)l′m(rk)ωk, m = 1, N − 1

(16)

(16) is equivalent to,am(t)ωm +

N−1∑n=1

(an(t)

N∑k=0

l′n(rk)l′m(rk)ωk)

= fN (rm, t)ωm −N∑k=0

u′0(rk)l′m(rk)ωk

, m = 1, N − 1 in Λ ∩∑N+1

uN (1, t) = 0 , t > 0u(rk, 0) = u0(rk) k = 1, N in Λ

(17)

We obtain a linear system, then we can write this system in a matricial form:

Aa+ ΓDa = F (18)

where A is a symmetric positive defined matrix with order N − 1, its elements have the form:

αmn =N−1∑k=0

l′n(rk)l′m(rk)ωk, n = 0, N − 1,m = 0, N − 1

28 A.Boutaghou

Γ is a diagonal invertible matrix its elements are define as:

γmn =wm , n = m0, n 6= m

, m, n = 0, N − 1

F is a known vector where:

F = (g0(t), g1(t), g2(t), g3(t), ....., gN−2(t), gN−1(t))t

gm(t) = f(rm, t)ωm −N∑k=0

u′0(rk)l′m(rk)ωk, m = 0, N − 1

and the vector a is an unknown vector where

a = (a0(t), a1(t), a2(t), a3(t), ....., aN−1(t), aN−1(t))t

the operator,D = d

dt

multiplying (18) by the invertible matrix Γ−1 of Γ then we find

Γ−1Aa+Da = Γ−1F (19)

the matrix Γ−1A has positive eigenvalues and there exists an orthogonal invertible matrix P such that,

P−1 (Γ−1A)P = C

where C is a diagonal matrix, the elements of the diagonal are the eigenvalues αm,m = 0, N − 1 of the matrix Γ−1A,if we consider the vector v such that

a = Pv

then the system (19) becomes

(Γ−1A)Pv + PDv = Γ−1F (20)

multiplying (20) by the matrix P−1 we obtain,

Cv +Dv = P−1Γ−1F (21)

The matricial form (21) has N − 1 linear equations defined as

v′m(t) + αmvm(t) = hm(t) (22)

where hm(t) =N−1∑j=0

p−1 (m, j) Γ−1 (m, j) gj(t) , 0 ≤ m ≤ N − 1

p−1 (m, j) are the elements of the inverse matrix P−1. To solve the equations (22), we may write the solution in theclosed form :

vm(t) =∫ t

0exp(αm(s− t))hm(s)ds+ dme

−αmt (23)

where dm are constants to be determined, using the boundary conditions then (23) may be written in the followingform:

vm(t) =∫ t

0exp(αm(s− t))hm(s)ds+

N−1∑j=0

p−1 (m, j)u0(rj)

e−αmt (24)

Finally we obtain the functions,

am(t) =N−1∑j=1

pmj

∫ t

0exp(αm(s− t))hm(s)ds+

N−1∑j=1

p−1 (m, j)u0(rj)

e−αmt

Spectral Method for the Heat Equation with Axial Symmetry and a Source 29

where pnj , 1 ≤ n, j ≤ N are the elements of the matrix P and the approximation solution is

uN (r, t) =N−1∑m=0

N−1∑j=0

pmj

∫ t

0exp(αm(s− t))hm(s)ds+

N−1∑j=1

p−1 (m, j)u0(rj)

e−αmt

lm(r).

see[1, 2].

3.1 Condition number

Definition 2. :The condition number of a n× n non-singular matrix A is defined by:

kP (A) = ‖A‖P∥∥A−1∥∥

P(25)

where ‖A‖P is the spectral norm defined by ρ = (AtA) 12

Remark 3. The condition number of a matrix A gives a measure of how sensitive systems of equations, withcoefficients matrix A, are to small perturbations such as those caused by rounding . Then if the condition number ofa matrix is large, the effect of rounding error in the solution process may be serious [9] .To compute the condition number of different order of these matrix we use the spectral norm, and all operations aremade by the Maple V [8], using [6] .

3.2 Figure illustration

The figure 1 present the behavior of the logarithm of the condition number, n vary from 3 to 13, figures 2 and 3present the true and the approximation solution u and uN respectively, these plots occur when N = 14 and thefunction test is : u(r, t) = −exp(−0.002t)(r2 − 1) sin(πr).

The error ‖u− uN‖L21(Ω) , Ω =]0, 1]×]0, 3] is the differance between the exacte and approximate solution, using

30 A.Boutaghou

various degrees for the approximate solution.

N 3 4 5 6 7 8 9Error .4499e− 1 .8387e− 2 .1615e− 2 .1608e− 3 .3144e− 4 .1743e− 5 .3404e− 6

Table 1

Conclusion 4. The main purpose in this work is that we have presented a new approach to the numerical solutionto the heat equation with axial symmetry and a source, the result is seem to converge rapidly as the degree of thepolynomial creases, The method they handled the error study showed that the error is almost deacreses when theparameter of discritisation N increases.

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simulation numerique 4 Janvier 2005.

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[3] A. Boutaghou and F.Z.Nouri,On Finite Spectral Method for Axi-symmetric Elliptic problem, Journal of Analysisand Applications,vol. 4 (2006), No 3, pp.149-168, SAS International Publications.

[4] C. Bernardy and Y-Maday, Approximation spectrale de problemes aux limites elliptiques, 1992, Springer VerlagFrance Paris.

[5] C. Bernardy, M. Dauge and Y. Maday, Numerical algorithms and tests due to Mejdi Azaiez, 1999 Spectal Meth-ods for Axisymmetric Domains, Editions scientifiques et medicales Elsevier France.

[6] D. Richards 2002, Advanced Mathematical Methods with Maple, Cambridge University.

[7] J. de Frutos, R. Munoz-Sola, Chebyshev pseudospectral collocation for parabolic problems with nonconstantcoefficients. Appear in Houston J. Math (Proccedings ICOSAHOM 95.

[8] Maple V, Release 4, Waterloo Maple Inc., 1995.

[9] T. N. Phillips and A. R. Davies, On semi-infinite spectral elements for Poisson problems with re-entrant boundarysingularities, 1988, Journal of Computationl and Apllied Mathematics 21 p173-188.https://doi.org/10.1016/0377-0427(88)90266-x

[10] R. G. Owens and T.N.Phillips , Steady Viscoelastic Flow past a sphere using Spectral Elements . 1996 Int. J.Numer. Meth. Engng., 39: 1517–1534.https://doi.org/10.1002/(sici)1097-0207(19960515)39:9