numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

Cent. Eur. J. Math. • 11(8) • 2013 • 1375-1391DOI: 10.2478/s11533-013-0253-5

Central European Journal of Mathematics

Numerical investigation of a new class of wavesin an open nonlinear heat-conducting medium

Research Article

Milena Dimova1∗, Stea Dimova2† , Daniela Vasileva1‡

1 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8, G. Bonchev Str., 1113, Sofia, Bulgaria

2 Faculty of Mathematics and Informatics, Sofia University “St. Kl. Ohridski”, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

Received 30 June 2012; accepted 4 January 2013

Abstract: The paper contributes to the problem of finding all possible structures and waves, which may arise and preservethemselves in the open nonlinear medium, described by the mathematical model of heat structures. A new classof self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. Aneffective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems –multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.

MSC: 65N30, 65M60, 65P40, 35B44

Keywords: Nonlinear elliptic and parabolic problems • Self-similar solutions • Blow-up • Structural stability • Metastability •Numerical methods© Versita Sp. z o.o.

1. Introduction

The paper is devoted to the numerical study of a new class of solutions of the mathematical model of heat structures [22]ut = N∑

i=1 (ki(u)uxi )xi + Q(u), t > 0, x ∈ RN, (1)∗ E-mail: [email protected]† E-mail: [email protected]‡ E-mail: [email protected]

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Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

whose heat-conductivity coefficients ki(u) ≥ 0 and the self-generating volume source Q(u) ≥ 0 are nonlinear functionsof the temperature u(t, x) ≥ 0. An important case is that of power nonlinearities:ki(u) = uσi, σi > 0, i = 1, . . . , N, Q(u) = uβ, β > 0. (2)

The possible structures and waves, which may arise and preserve themselves in such medium, have been studied forforty years. This field has essentially been developed by the Russian school of Alexandr A. Samarskii and Sergei P.Kurdyumov.In spite of the fact, that the books [2, 22] and many works, beginning with [23], were devoted to the analysis of thecomplexity and the unusual properties of the processes, described by this model, many problems still remain open.The presence of two medium parameters σ and β even in the isotropic case (σi = σ , i = 1, 2, . . . , N), different spacegeometries and dimensions N, pose challenging questions and make the problem (1)–(2) interesting from mathematicaland computational points of view. It is worth mentioning, that in many cases the success in resolving these questionswas achieved by the combination of theoretical investigations and computational experiments.Numerical methods for the solution of (1)–(2) were developed and investigated in several works, including [4, 9, 11, 14,18, 22] and references therein. Anisotropic blow-up self-similar solutions (when σi 6= σj , i 6= j , i, j = 1, 2) of simplestructure and their numerical construction were reported in [3], showing the directed heat diffusion in such medium.In the isotropic two-dimensional case in polar coordinates the problem (1)–(2) readsut = 1

r∂∂r

(ruσ ∂u∂r

)+ 1r2

∂∂φ

(uσ ∂u∂φ

)+ uβ , t > 0, 0 < r <∞, 0 ≤ φ < 2π. (3)It admits the following blow-up self-similar solutions (s.-s.s.) of complex structure:

u(t, r, φ) = g(t)θ(ξ, φ), g(t) = (1− tT0)−1/(β−1)

, ξ = r(1− t

T0)−m

,

m = β − σ − 12(β − 1) , φ = φ + C0β − 1 ln(1− t

T0) (4)

which were found by using the method of invariant-group analysis [3, 12]. Here T0 > 0 is the blow-up time, C0 is aparameter of the family of solutions. For C0 6= 0 from (4) it followsr(t)esφ(t) = r(0)esφ(0) = ξesφ = const, s = β − σ − 12C0 . (5)

The dependence (5) means that the trajectories of the inhomogeneities in the medium (for example, local maxima) would belogarithmic spirals for β 6= σ+1 or circles for β = σ+1. For C0 > 0 the direction of moving of the inhomogeneities is fromthe center along the spiral when β < σ+1 and towards the center when β > σ+1. The self-similar function θ(ξ, φ) ≥ 0defines the space-time structure of the self-similar solution (4) and satisfies the following nonlinear elliptic equation(T0 = (β− 1)−1 is set here for convenience):L(θ) ≡ − 1

ξ∂∂ξ

(ξθσ ∂θ∂ξ

)− 1ξ2 ∂

∂φ

(θσ ∂θ∂φ

)+ β − σ − 12 ξ ∂θ∂ξ − C0 ∂θ∂φ + θ − θβ = 0, (6)0 < ξ <∞, 0 ≤ φ < 2π. Equation (6) has two homogeneous solutions: θ0

H ≡ 0 and θ1H ≡ 1.The case of vanishing at infinity solutions of equation (6) for C0 = 0 is widely analyzed [6, 15–17]. Two classes ofradially nonsymmetric solutions of complex symmetry were found and investigated for β > σ + 1. For β ≤ σ + 1 onlysimple finite support radially symmetric solutions were known to exist [22].The idea to seek for solutions of equation (6) tending at infinity to the nontrivial constant solution θ1

H ≡ 1 was crucialfor finding complex symmetry (C0 = 0) and spiral symmetry solutions (C0 6= 0) for β < σ + 1. This idea was sustained1376

M. Dimova, S. Dimova, D. Vasileva

by the radially symmetric case, β < σ + 1. It was shown in [22], that a continuum set of solutions, tending to a nonzeroconstant solution exists. The solutions oscillate around θ1H and the oscillations are damped. Approximate solutionsof equation (6), having similar behaviour, were found and analyzed in [9]. Numerical construction of the solutions ofequation (6) and their evolution in time were described in [7]. Detailed analysis of the spiral symmetry solutions (C0 6= 0)for β < σ + 1 is made in [5].In this work we report about a new class of solutions of equation (6) in the case β = σ + 1 and C0 = 0. The idea issustained again by the radially symmetric case, where a continuous set of solutions to equation (6), oscillating around θ1

Hand tending to it at infinity, is proven to exist [22]. These new solutions of (6) define a new special class of waves inthe nonlinear medium, described by equation (3). The outline of the paper is as follows. In Section 2 we consider theradially symmetric case. The third section deals with the 2D radially nonsymmetric case. At the end some conclusionsare made.2. 2D radially symmetric case

In the two-dimensional radially symmetric case with β = σ + 1 the equation (3) readsut = 1

r (ruσur)r + uσ+1, r ∈ R1+, t > 0, σ > 0. (7)Equation (7) admits self-similar blow-up solutions [22] of the kind

u(t, r) = ψ(t)θ(ξ) = (1− tT0)−1/σ

θ(ξ), ξ = r for β = σ + 1. (8)The function ψ(t) determines the solution amplitude, the self-similar function θ(ξ) ≥ 0 determines the space-timestructure of the self-similar solution and satisfies the following ordinary differential equation (self-similar equation):

L(θ) ≡ − 1ξ (ξθσθ′)′ + θ − θσ+1 = 0, 0 < ξ <∞. (9)

It is clear, that the solution (8) corresponds to the initial datau(0, r) = u0(r) = θ(r) (10)

and satisfies the symmetry conditionur(t, 0) = 0. (11)We seek for bounded at infinity solutions, which for a finite R 1, R <∞, satisfy the self-similarity law

u(t, R) = u(t, r∗)θ(r∗) θ(R), t > 0, (12)

for an appropriate choice of the point r∗ R .It is well known [22], that it is not possible to construct numerically a special complex structure of self-similar solutions,starting with arbitrary initial data. Arbitrary initial data could initiate only the simplest structurally stable (see below)self-similar solutions.To construct a new class of solutions of equation (7), first we must find the solutions of the self-similar equation (9)under the conditionsθ′(0) = 0, (13)θ(∞) = 1. (14)

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The main challenge in solving the nonlinear self-similar problem (9) & (13) & (14) is the multiplicity of its solutions forone and the same parameters. The following problems arise: to find a “good” approximation to each of the solutions; toconstruct an iteration process, converging fast to the desired solution (corresponding to the initial approximation) andensuring sufficient accuracy; to determine in advance where to translate the boundary condition (14) from infinity.To overcome the difficulty with the initial approximations to the different self-similar functions, we use the approach,proposed in the work [10]. Based on the hypothesis that the self-similar functions have small oscillations around thenontrivial homogenous solution θ1H , the self-similar equation (9) is linearized around θ1

H . Further, the solutions of thelinearized equation are taken as initial approximations to the self similar-functions and the asymptotic analysis of theselinear approximations gives the possibility to refine the boundary condition (14). The combination of the ContinuousAnalogue of the Newton method (CANM) and the Galerkin finite element method occurred to be very successful forsolving the self-similar problem (9) & (13) & (14).Following this strategy for the self-similar problem (9) & (13) & (14), we set θ0(ξ) = 1 + αy(ξ), α = const, |αy(ξ)| 1in (9) and get the Bessel equation y′′ + y′/ξ + σy = 0. The bounded at ξ = 0 solution is y(ξ) = J0(√σ ξ). Using theknown asymptotics [1]J0(z) ∼ √ 2

πz cos(z− π4), z →∞,

we findy(ξ) ∼ √ 2

π√σ ξ

cos(√σ ξ− π4), ξ →∞,

and finallyθ0(ξ) ∼ 1 + γ

√ 2π√σ ξ

cos(√σ ξ− π4), γ = const, ξ →∞. (15)

From (15) we derive a boundary condition of the third kind for θ(ξ),θ′(l) + θ(l)− 12l = −γ√2√σ

πl sin(√σ l− π4), l 1, (16)

and take furtherθ0(ξ) = 1 + αJ0(√σ ξ), α = const, |α| 1, (17)

as the initial approximation to the solutions of equation (9).2.1. Numerical method for the self-similar problem

To solve the nonlinear problem (9) & (13) & (16) we use the CANM [13, 20]. The idea is to reduce the stationary problemL(θ) = 0 to the evolution one,

L′(θ) ∂θ∂t = −L(θ), θ(ξ, 0) = θ0(ξ), (18)by introducing a continuous parameter t, 0 < t < ∞, on which the unknown solution depends: θ = θ(ξ, t). By settingv = ∂θ/∂t and applying the Euler method to the Cauchy problem (18), one comes to the iteration scheme

L′(θn)vn = −L(θn), (19)θn+1 = θn + τnvn, n = 0, 1, . . . , θn = θn(ξ) = θ(ξ, tn), vn = vn(ξ) = v(ξ, tn), 0 < τn ≤ 1, (20)

θ0(ξ) is an initial approximation. For the nonlinear operator L(θ) defined by (9) the equation (19) takes the form− (ξθσnv ′n)′ − (ξσθσ−1

n θ′nvn)′ + ξ

(1− (σ +1)θσn)vn = −[−(ξθσnθ′n)′ + ξ(1−θσn )θn]. (21)1378


The iteration corrections vn(ξ) must satisfyv ′n(0) = 0, v ′n(l) + vn(l)2l = −(θ′n(l) + θn(l)− 12l + γ

√2√σπl sin(√σ l− π4

)), (22)

where l is an appropriately chosen “computational infinity”.The operator L(θ) is nonsymmetric, so we use the Galerkin Finite Element method (GFEM) to discretize the problem(21)–(22). In weak form this problem reads: Find a function vn(ξ) ∈ H1 satisfying the identity(L′(θn)vn, w) = −(L(θn), w), w ∈ H1, (23)

and the boundary conditions (22). Here ( · , · ) is the standard L2 inner product, θn(ξ) is a given function satisfying theboundary condition (13), θn ∈ D = θn(ξ) : θσ+1

n , dθσ+1n /dξ ∈ L2(0, l), and H1 =

w(ξ) : w, ξ1/2w ′ ∈ L2(0, l). Theidentity (23) is∫ l

0ξθσnv ′nw ′ + ξσθσ−1

n θ′nvnw ′ + ξ(1− (σ+1)θσn)vnwdξ − lθσn (l)v ′n(l)w(l)− lσθσ−1

n (l)θ′n(l)vn(l)w(l)= −∫ l

0ξθσnθ′nw ′ + ξ (1−θσn )θnwdξ + lθσn (l)θ′n(l)w(l). (24)

For discretization of (24) & (22) linear finite elements on quasiuniform partitions ωh of the interval [0, l] are used. A systemof linear equations is thus obtained for the vector V n of nodal values of the iteration corrections vn = v(ξ, tn),A(θn)V n = −B(θn)Θn. (25)

Matrices A and B are nonsymmetric band matrices, Θn is the vector of nodal values of the function θn = θ(ξ, tn). Tosolve the system (25), LU-decomposition of the matrix A(θn) is used at every iteration step. The iteration parameter τnin (20) is determined by the following extrapolation formula [21]:τn =

min(1, τk−1 δk−1

δn

), δn < δk−1,

max(τ0, τk−1 δk−1δn

), δn ≥ δk−1.

(26)Here δn is some norm of the residual L(θn). In the computations the uniform norm of the discrete residual is used,δn = maxη∈ωh |B(θn)Θn|. The value of τ0 is taken to be between 0.01 and 0.1. When δn decreases, the algorithm (26)ensures the convergence of τn to 1 (τn → 1−), and the rate of convergence of the iteration process (19)–(20) becomesquadratic. The stop criterion is δn < δ for some small δ. When it is fulfilled we take θn = θ(ξ, tn) as the soughtafter solution of problem (9) & (13) & (16). As initial approximations for the iteration process (19)–(20) we use the linearapproximations (17).2.2. Numerical method for the parabolic problem

Here we use the GFEM [24], based on the Kirchhoff transformation of the nonlinear heat-conductivity coefficient, i.e.G(u) = ∫ u

0 sσ ds = uσ+1σ + 1 . (27)

This is essential for the further interpolation of the nonlinear coefficients and for the optimization of the computationalprocess.1379


The discretization is made on the problem (7) & (10) & (12) in a weak form: Find a function u(t, r) ∈ D, D = u :

r1/2u, r1/2∂uσ+1/∂r ∈ L2 satisfying the integral identity(rut , v) = A(t;u, v), v ∈ H1(0, R), 0 < t < T0, (28)

the initial condition (10) and the boundary conditions (11)–(12), where R is an appropriately chosen “computationalinfinity”. Here(u, v) = ∫ R

0 u(r)v(r)dr, A(t;u, v) = ∫ R

0[−r ∂G(u)

∂rdvdr + ruσ+1v

]dr + R ∂G(u(R))

∂r v(R),H1(0, R) = v : r1/2v, r1/2v ′ ∈ L2(0, R).

The lumped mass finite element method [24] with interpolation of the nonlinear coefficients is used for discretizationof (28). Let 0 = r1 < r2 < . . . < rn = R : ri+1 − ri ≤ h be a partition of the interval [0, R ] into elementsei = [ri, ri+1], i = 1, 2, . . . , n − 1. Let Sh be the space of the continuous functions on [0, R ], which are linear on ei,Sh = w(r) ∈ C ([0, R ]) : wei ∈ P1. Let φini=1 be the standard Lagrangian nodal basis of Sh and uh(t, r) be theapproximate solution in Sh for every fixed value of t. The semidiscrete problem is: Find uh : [0, T0]→ Sh such that

(ruh,t , w) = A(t;uh, w), w ∈ Sh, uh(0) = u0,h. (29)The approximate value T0 of the blow-up time T0 is found in the computations.We use the finite element interpolants of the solution u(t, r), of the nonlinear functions G(u) and q(u) = uσ+1, and ofthe initial data u0(r):

uh(t, r) = n∑i=1 ui(t)φi(r), u0,h(r) = n∑

i=1 u0(ri)φi(r) = n∑i=1 θ(ri)φi(r), (30)

G(u) ∼ GI = n∑i=1 G(ui)φi(r), q(u) ∼ qI = n∑

i=1 q(ui)φi(r). (31)Substituting (30)–(31) in (29), we find a system of ordinary differential equations (ODE)

U = M−1(−KG(U)) + q(U), U(0) = U0, (32)with respect to the vector U(t) = (u1(t), u2(t), . . . , un(t))T of the nodal values of the solution u(t, r) at time t. HereG(U) = (G(u1), . . . , G(un))T , q(U) = (q(u1), . . . , q(un))T , M is the lumped mass matrix, K is the stiffness matrix. Let usmention, that because of the Kirchhoff transformation and the interpolation of the nonlinear coefficients only the vectorsG(U) and q(U) contain the nonlinearity of the problem, while the matrix K does not depend on the unknown solution.To solve the system (32), a modification of the explicit Runge–Kutta method [19], which has second order of accuracyand an extended region of stability is used. The time step τ is chosen automatically so as to guarantee stability andrelative accuracy. The stop criterion is τ < 10−16 and then T0 is the time, reached in the computations. In the boundarycondition (12) we set r∗ = R/2. Explicit methods were preferred over the implicit ones for solving the system (32), becausethe condition for solvability of the nonlinear implicit scheme on the upper time level imposes the same restriction onthe relation “time step – step in space”, as does the condition for validity of the weak maximum principle for the explicitscheme, see [22, Chapter VII, § 5].

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2.3. Metastability of the self-similar solution

The blow-up solutions are not stable with respect to the initial data – small changes of the initial data may producesmall changes in the blow-up time, but very big differences in the solutions values near the blow-up time. For blow-upsolutions and in general, for invariant solutions, a more important property is the preservation during the evolutionof some characteristics, such as geometric form, rate of growth, localization in space. Depending on the time of suchpreservation (for all of the time of existence or for the bigger part of that time) the notions of structural stability andmetastability are introduced. For the blow-up solutions (8) the notion of structural stability is introduced in [10]. It givesa possibility to investigate the asymptotic behavior of the blow-up solutions in a special “self-similar norm”, consistent forevery t with the geometric form of the solution. In order to introduce the notions of structural stability and metastability,we define the self-similar representation [10] Θ(t, ξ) of the solution u(t, r) of equation (7), corresponding to arbitraryinitial data u0(r): Θ(t, ξ) = u(t, ξ)Γ(t) , Γ(t) = maxr u(t, r)maxr u0(r) . (33)The s.-s.s. u(t, r) (8), corresponding to the solution θ(ξ) of the problem (9) & (13) & (16) is called structurally stable [10], ifthere exists a class of initial data u0(r) 6= θ(r), so that for the self-similar representations (33), Θ(t, ξ) of the correspondingsolutions u(t, r), it holds

∥∥Θ(t, ξ)− θ(ξ)∥∥C [0,∞) → 0, t → T−0 , θ(ξ) = maxr u0(r)maxξ θ(ξ) θ(ξ).It is clear, that if u0(r) ≡ θ(r), then θ(ξ) ≡ θ(ξ) and Θ(t, ξ) ≡ θ(ξ) for all t.The s.-s.s. u(t, r) (8) is called metastable, if for every ε > 0 there exists a class of initial data u0(r) ≈ θ(r), and a timeT , T0 − T T0, so that for the self-similar representations (33), Θ(t, ξ) of the corresponding solutions u(t, r), it holds

∥∥Θ(t, ξ)− θ(ξ)∥∥C [0,∞) ≤ ε, 0 ≤ t ≤ T .2.4. Numerical investigations

We constructed numerical solutions to the self-similar problem (9) & (13) & (16) for different values of σ > 0 and param-eters α, γ. The numerous experiments performed with the GFEM and linear finite elements show the fast convergenceof the proposed iteration procedure (19) & (20) (usually 14–20 iterations are enough for δ = 10−7) and optimal, secondorder of accuracy at the nodal points. It was found that when varying the parameter α in the interval [0.01, 0.4] theiteration process converges to one and the same self-similar function. On the other hand small variations of the constantγ in the boundary condition (16) lead to different self-similar functions which differ by their amplitudes.Taking the computed self-similar function θ(ξ) as initial data for the parabolic problem (7) & (10) & (12), we have comparedboth the exact blow-up time T0 = 1/σ with T0, found in the computations, and the self-similar representations Θ(t, ξ)with θ(ξ). A good internal criterion for the accuracy of both numerical methods, for solving the self-similar problem andthe parabolic one, is the restoration of the blow-up time in the process of evolution of self-similar initial data. Becausewe have substituted T0 = 1/σ = 0.5 in the self-similar equation, the computed blow-up time T0 should be close to T0.All experiments show: T0 is close to T0, (see the examples bellow), and Θ(t, ξ) are close to θ(ξ) (their graphs coincidewithin the plotting resolution) up to times T ≥ T0 · 98 %. As all of the complex structures and waves, these waves aremetastable. Near the blow-up time they degenerate into the unique structurally stable monotone finite support solutionof equation (7) for the same value of σ .Example 2.1.Figure 1 (a) shows the function θ(ξ), computed for σ = 2, α = 0.1, γ = 0.2, l = R = 95, h = 0.1 and Figure 2 (a)its evolution in time. The exact blow-up time is T0 = 0.5 and the computed blow-up time is T0 = 0.4999550. Theself-similar representations (33), shown in Figure 2 (b), coincide within the plotting resolution with computed θ(ξ) up totime T = 0.4972434 = T0 · 99.45 %. In Figure 3 the regions near the origin for the same example are presented.

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(a) (b)

Figure 1. Self-similar function θ(ξ) computed for σ = 2: (a) α = 0.1; (b) α = −0.5.

Figure 2. Evolution in time of the self-similar solution u(t, r) from Example 2.1 and its self-similar representations.

Figure 3. Zoom in the region near the origin of the graphs in Figure 2.

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Example 2.2.In the case β > σ + 1 a special kind of solutions of the self-similar problem, having a zero region around the center ofsymmetry, were found [6, 8]. We managed to find such kind of solutions for β = σ + 1, taking negative values of α in theinitial approximations (17). Figure 1 (b) shows the function θ(ξ), computed for σ = 2, α = −0.5, γ = 0.2, l = R = 50,h = 0.1. Its evolution in time (near the origin) is presented in Figure 4 (a). The exact blow-up time is again T0 = 0.5,T0 = 0.4998097. The self-similar representations (33), shown in Figure 4 (b), coincide within the plotting resolution withcomputed θ(ξ) up to time T = 0.4931033 = T0 · 98.6 %. In the final stage the zero region disappears, the solution of theparabolic problem tends to the structurally stable finite support blow-up solution for the same σ .

Figure 4. Evolution in time of the self-similar solution u(t, r) from Example 2.2 and its self-similar representations. (Zoom in the region near theorigin.)

3. 2D radially nonsymmetric case

In the two-dimensional radially nonsymmetric case we seek for solutions of equation (3) in the formu(t, r, φ) = g(t)θ(ξ, φ), g(t) = (1− t/T0)−1/σ, ξ = r, φ = φ,

where θ(ξ, φ) satisfies the following nonlinear elliptic equation (self-similarity equation):L(θ) ≡ − 1

ξ∂∂ξ

(ξθσ ∂θ∂ξ

)− 1ξ2 ∂

∂φ

(θσ ∂θ∂φ

)+ θ − θσ+1 = 0. (34)The methodology for handling this 2D problem is analogous to the one used in the radially symmetric case. The firststep in achieving the goal of this work is to find a new class of radially nonsymmetric solutions of (34). As in the radiallysymmetric case we are looking for solutions of (34) tending to θ1

H at infinity. Note that only radially symmetric solutionshave been obtained when we set one of the following standard boundary conditions: θ(l, φ) = 1 and (∂θ/∂ξ)(l, φ) = 0 atappropriate chosen “computational infinity” l. That is why a suitable asymptotic and a corresponding boundary conditionat infinity are required.We linearize equation (34) around the homogeneous solution θ1H ≡ 1 and construct linear approximations

θ(ξ, φ) = 1 + αy(ξ, φ), α = const, |αy| 1, (35)1383


where y(ξ, φ) is a bounded at ξ = 0 solution of the following linear equation:− 1ξ∂∂ξ

(ξ ∂y∂ξ

)− 1ξ2 ∂

2y∂φ2 + σy = 0. (36)

Particular solutions y(ξ, φ) = Jk (√σ ξ) cos (kφ), k ∈ N, are found in [9]. The function Jk (√σ ξ) is the Bessel function ofthe first kind of order k . The functions y(ξ, φ) are 2π/k periodic and have k axes of symmetry. In Figure 5 the graphsof y(ξ, φ) for k = 1, 2, 3 are presented.

k = 1

k = 2

k = 3Figure 5. Solutions y(ξ, φ) = Jk (√σ ξ) cos (kφ) to the linearized equation (36), k = 1, 2, 3.

Our experience in dealing with such kind of problems [3, 5–8] shows that the linear approximations (35) are very closeto the solutions of (34). Therefore these linear approximations can be successfully used as initial approximations to thesolutions of equation (34) as well as to prescribe the asymptotic behaviour of θ(ξ, φ).1384


Using the asymptotic expansion of Jk (z) [1] for z →∞,Jk (z) ∼ √ 2

πz cos(z− kπ2 − π4), z →∞,

we gety(ξ, φ) ∼ √ 2

π√σ ξ

cos(√σ ξ− kπ2 − π4) cos (kφ), ξ →∞.

The above asymptotic expression allows us to predict the following asymptotic for θ(ξ, φ):θ(ξ, φ) ∼ 1 + γ

√ 2π√σ ξ

cos(√σ ξ− kπ2 − π4) cos (kφ), γ = const, ξ →∞. (37)

Taking into account the symmetry of functions θ(ξ, φ) we solve the self-similar equation (34) in the domain Ω =(0, l)× (0, ω), ω = π/k , under the following boundary conditions:limξ→0 ξθσ ∂θ∂ξ = 0, φ ∈ [0, ω],∂θ∂ξ + θ − 12ξ = −γ√ 2

π√σ ξ

sin(√σ ξ − kπ2 − π4) cos (kφ), ξ = l 1, φ ∈ [0, ω],

∂θ∂φ (ξ, 0) = ∂θ

∂φ (ξ, ω) = 0, 0 ≤ ξ ≤ l.(38)

The boundary condition at ξ = l 1 is derived from (37).3.1. Numerical method for the self-similar problem

To solve the nonlinear boundary value problem (34) & (38) we apply the numerical approach described in Section 2.Here we will stress only on the special features of its 2D implementation. In the considered two-dimensional case theunknown iteration corrections vn from equation (19) have to satisfy the following boundary conditions:limξ→0 ξθσ ∂vn∂ξ = 0, φ ∈ [0, ω],∂vn∂ξ + vn2ξ = −(∂θn∂ξ + θn − 12ξ + γ

√ 2π√σ ξ

sin(√σ ξ − kπ2 − π4) cos (kφ)), ξ = l 1, φ ∈ [0, ω],

∂vn∂φ (ξ, 0) = ∂vn

∂φ (ξ, ω) = 0, 0 ≤ ξ ≤ l.The integral identity (23) must be fulfilled for vn ∈ H1, θn ∈ D and every wn ∈ H1, whereD = θ(ξ, φ) : ξ1/2θ, ξ1/2 ∂θσ+1

∂ξ , ξ−1/2 ∂θσ+1∂φ ∈ L2(Ω), H1 = w(ξ, φ) : ξ1/2w, ξ1/2 ∂w

∂ξ , ξ−1/2 ∂w

∂φ ∈ L2(Ω).

In the two-dimensional case the identity (23) reads∫Ωξθσn

∂w∂ξ

∂vn∂ξ + σξθσ−1

n∂θn∂ξ

∂w∂ξ vn + σ

ξ θσ−1 ∂θn

∂φ∂w∂φ vn + θσn

ξ∂w∂φ

∂vn∂φ + ξ

(1− (σ + 1)θσ)wvndξ dφ−∫ ω

0 ξσθσ−1n

∂θn∂ξ wvndφ−

∫ ω

0θσn2 wvndφ = −∫Ω

ξθσn

∂w∂ξ

∂θn∂ξ + θσ

ξ∂w∂φ

∂θn∂φ + ξ (1− θσ )wθndξ dφ

−∫ ω

0 θσn( (θn − 1)2 + γ

√ 2π√σ√ξ sin(√σ ξ − kπ2 − π4

) cos (kφ))w dφ.1385


Bilinear finite elements are used for discretization of (23). At each step of the iteration process we get a system of linearalgebraic equations (25) with respect to the vector of the iteration corrections V n. The matrix A is stored and used insky-line form. The linear algebraic problems are solved by LU-decomposition.We set the linear approximations (35) as initial approximations to the iteration process (19), i.e. θ0(ξ, φ) = 1 +αJk (√σ ξ) cos (kφ). Let us mention that for different values of k we get different solutions θ(ξ, φ) = θk (ξ, φ) inherent tothe medium with given value of the parameter σ .The accuracy of the described approach is numerically tested using embedded grids. Table 1 shows the values of thesolution computed for parameters σ = 2, k = 3, ξ ∈ [0, 50], φ ∈ [0, π/3], α = 0.1, γ = 0.2 at some common pointsof embedded grids with steps h, h/2, h/4, h = (hξ , hφ). The order of accuracy s is computed by the Runge methods = ln∣∣(θh−θh/2)/(θh/2−θh/4)∣∣ ln−1 2.Table 1. Self-similar function θ(ξ, φ) computed for σ = 2 and k = 3 (see Figure 7, t = 0).

hξ hφ θ(0, 0) θ(3, 0) θ(10, 0.698131) θ(21, 0.837758)0.2 π/45 0.9910681854 1.1031301831 1.0125526817 0.98221247970.1 π/90 0.9924258651 1.0840508002 1.0098418003 0.98393964780.05 π/180 0.9926433751 1.0800695782 1.0091812556 0.9843510854s 2.64 2.26 2.04 2.07

3.2. Numerical method for the parabolic problem

GFEM, based on the Kirchhoff transformation (27) of the nonlinear heat-conductivity coefficient is used again to solvethe following parabolic problem:ut = 1

r (ruσur)r + 1r2 (uσuφ)φ + uσ+1, 0 < t < T0, (r, φ) ∈ Ω,

Ω = (0, R)× (0, ω), ω = πk ,

ruσur(t, 0, φ) = 0, t ∈ [0, T0), φ ∈ [0, ω],u(t, R, φ) = u(t, r∗, φ)

θ(r∗, φ) θ(R, φ), t ∈ [0, T0), φ ∈ [0, ω],u(0, r, φ) = u0(r, φ) = θk (r, φ) ≥ 0, (r, φ) ∈ Ω,uσ (t, r, 0)uφ(t, r, 0) = uσ (t, r, ω)uφ(t, r, ω) = 0, t ∈ [0, T0), r ∈ [0, R ],u0(0, φ) = const, u0(r, φ) = u0(r, 2ω−φ).

In a weak form the problem reads: Find a function u(t, r, φ) ∈ C (0, T0)×D, D = w : w, ∂wσ+1/∂r, ∂wσ+1/∂φ ∈ L2(Ω)satisfying the following integral identity:(rut , v) + A(t;u, v) = (rq, v), v ∈ H, 0 < t < T0,

(u, v) = ∫Ω uv dΩ, A(t;u, v) = ∫

Ωr∂G(u)∂r

∂v∂r + 1

r∂G(u)∂φ

∂v∂φ dΩ + ∫ ω

0 R ∂G(u(R, φ))∂r v(R, φ)dφ,

H = v : r1/2v, r1/2 ∂v∂r , r−1/2 ∂v∂φ ∈ L

2and the initial and boundary conditions above.Let Ωh be a partition of Ω into rectangles eij = [ri, ri+1]× [φj , φj+1], i = 1, 2, . . . , l, j = 1, 2, . . . , m, 0 = r1 < r2 < . . . <rl+1 = R , 0 = φ1 < φ2 < . . . < φm+1 = ω. Let Sh be the space of continuous on Ω bilinear functions on the rectangles.

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Let ψij : i = 1, . . . , l+ 1, j = 1, . . . , m+ 1 be the standard basis of Sh: ψij (ri, φj ) = 1, ψij (rp, φq) = 0 if p 6= i or q 6= j .But, ψ1j /∈ H, j = 1, 2, . . . , m+ 1, because r−1/2ψ1j,φ /∈ L2. Thus as a trial function we chooseψ1 = m+1∑

j=1 ψ1j = r2 − rr2 ∈ H

instead of ψ1j , j = 1, 2, . . . , m+ 1. Because u(t, 0, φ) does not depend on φ, it is natural to postulate u11(t) = u12(t) =. . . = u1,m+1(t). Then

uI = u11m+1∑j=1 ψ1j + l+1∑

i=2m+1∑j=1 uijψij = u11ψ1 + l+1∑

i=2m+1∑j=1 uijψij .

The rest of the basis functions are the same: ψ(i−2)(m+1)+j+1 = ψi,j , i = 2, . . . , l+ 1, j = 1, . . . , m+ 1. Finally the set ofbasis functions is ψini=1, n = l(m+1) + 1. We seek an approximate solution uh(t, φ) of the formuh(t, r, φ) = n∑

i=1 ui(t)ψi(r, φ). (39)The nonlinear functions G(u) and q(u) are interpolated on the new basis

GI = IhG = n∑i=1 G(ui)ψi(r, φ), qI = Ihq = n∑

i=1 q(ui)ψi(r, φ). (40)By using (39)–(40) and the lumped mass matrix, we come to the semidiscrete problem

U = −M−1KG(U) + q(U), U(0) = U0. (41)The system of ODE (41) is solved again by the modification of the explicit Runge–Kutta method [19].3.3. Numerical investigations

To test the reliability of the worked out numerical technique for the self-similar problem we compare the results, foundby the algorithm for the radially-symmetric case, with the results, obtained by the essentially 2D algorithm for k = 0.The aim of the numerical investigations is to analyze the evolution in time and the stability of the new class of 2Dself-similar solutions. Below we present the evolution of the self-similar solutions, corresponding to heat-conductivitycoefficient σ = 2 and parameters k = 1, 2, 3. All numerical experiments carried out show the excellent restoration of theblow-up time T0 = 1/σ = 0.5. We also compare the self-similar representationΘ(t, r, φ) = u(t, r, φ)Γ(t) , Γ(t) = maxr u(t, r, φ)maxr u0(r, φ)

with the self-similar function. For this reason we introduce the rescaled error e(t),e(t) = maxni=1 |Θ(t, ri, φi)− θ(ri, φi)|maxni=1 θ(ri, φi)− minni=1 θ(ri, φi) .

We observe that in all of the examples the rescaled errors are smaller than 0.01 for t < T0 · 90 %. This justifies theconclusion that the new self-similar solutions are metastable. They preserve their structure up to times very close to theexact blow-up time. In the final stage the solutions of the parabolic problem tend to the finite support radially symmetricstructurally stable blow-up solution for the same σ .In the examples bellow T0 = 0.5, l = R = 50, hr = 0.1, hφ = π/30, α = 0.1, γ = 0.2.1387


Example 3.1.Figure 6 shows the evolution of the self-similar solution u(t, r, φ), corresponding to the self-similar function θ(ξ, φ),computed for k = 1. The blow-up time found in the computations is T0 = 0.4999738. Table 2 contains the rescalederrors for different times t. In this case e(t) < 0.01 for t < T0 · 95 %.

t = 0 t = 0.497634

t = 0.499014 t = 0.499722

t = 499945 t =0.499973Figure 6. Evolution in time of the self-similar solution u(t, r, φ), σ = 2, k = 1.

Example 3.2.The evolution of the self-similar solution u(t, r, φ), corresponding to the self-similar function θ(ξ, φ), computed for k = 2,is investigated. The blow-up time, found in the computations, is T0 = 0.4999218.Example 3.3.Figure 7 shows the evolution of the self-similar solution u(t, r, φ), corresponding to the self-similar function θ(ξ, φ),computed for k = 3. The computed blow-up time is T0 = 0.4998930.

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Table 2. Metastability of the wave from Example 3.1.

t umax e(t) t/T0 %0.0000 1.094 0.0000 0.000.1751 1.357 0.0015 35.140.2930 1.701 0.0030 58.600.4154 2.662 0.0060 83.080.4738 4.788 0.0099 94.760.4860 6.554 0.0145 97.200.4976 16.044 0.0457 99.520.4990 25.097 0.0945 99.80

t = 0 t = 0.486123

t = 0.496300 t = 0.499292

t = 0.499822 t = 0.499885Figure 7. Evolution in time of the self-similar solution u(t, r, φ), σ = 2, k = 3.

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4. Conclusions

The idea to seek for solutions of the self-similar problem, tending at infinity to the nontrivial constant solution θ1H ≡ 1was crucial for finding complex symmetry solutions in the case β = σ+1. Thus the set of possible waves in the nonlinearmedium, described by the model of heat structures, is enriched. It is shown, that the new waves are metastable – as allof the known complex objects they preserve their structure up to times, very close to the time of their existence. Nearthat time they degenerate into the simplest radially symmetric localized on the fundamental length structures.

Acknowledgements

The work of the second author is partially supported by the Sofia University research grant No 181/2012. The first andthe third authors are partially supported by the Bulgarian National Science Foundation under Grant DDVU02/71.

References

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numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

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