Blow-Up of Finite Difference Solutions to Parabolic Equationswith Semilinear Dynamical Boundary Conditions ?

Miglena N. Koleva, Lubin G. Vulkov

University of Rousse, Department of Mathematics, 7017 Rousse, Bulgaria, mkoleva{lvalkov}@ru.acad.bg

Abstract. In the present work we analyze semidiscrete and fulldiscrete (finite difference) solutions ofthe semilinear parabolic problem with dynamical boundary conditions that can be posed on the wholedomain boundary as well as only on a part of the boundary. Necessary and sufficient conditions for blow-up of the numerical solutions are found. It is proved that the numerical solutions and the discrete blow-uptimes converge to the corresponding real ones, when mesh size goes to zero. Numerical experiments arealso discussed.

1 Introduction

Let be a bounded domain in RN (N 1) with piecewise smooth boundary = S1 S2, S1 S2 = . Weconsider the following problem:

u

t4u = f(u) in QT = (0, T ), (1)

u

t+ k

u

n= g1(u) on S1 (0, T ), (2)

au

n+ bu = g2(u), a 0, b 0, a+ b = 1, on S2 (0, T ), (3)

u(x, 0) = u0(x) on S1. (4)

Here k > 0, a, b are constants, 4 is the Laplace operator with respect to the space variables and /n is theoutward derivative to S1 (or S2).

Problems of type (1)-(4) model processes of filtration in the hydrology, in heat transfer in a solid in contactwith a fluid, chemistry, semiconductors etc., see [1], [3], [6], [7] and [9] for more information.

We shall consider nonlinearities f , g1 and g2 for which blow-up occurs in finite time. A point x isa blow-up point if there exists a sequence [(xn, tn)] such that tn Tmax < +, xn x and u(xn, tn) as n.

The important questions are:1. When the solutions blow-up?2. Where (BUS) the solutions blow-up?Blow-up results for differential problems with dynamical boundary conditions start with the paper [9],

where the special case is considered:

S1 , i.e. S2 , f(u) = g(u) = u1+, > 0 (5)

Theorem 1. (M. Kirane, 1992) Let u0 W 2p (), 1 p 0, to problem (1)-(4), (5) blows up in afinite time.

Later these results were extended in many papers, see [4], [6] and [8].

? This research was supported by the Bulgarian National Fund of Science under Project VU-MI-106/2005.

240 M. Koleva and L. Vulkov

In [12] was studied the blow-up of finite difference solution to parabolic problem (1)-(4) in 1D case. In [13]was considered blow-up of continuous and semidiscrete solutions of the problem:

4u = 0 in (0, T ),u

t+ k

u

n= g(u) on S1 (0, T ), (6)

au

n+ bu = 0, a 0, b 0, a+ b = 1 on S2 (0, T ),

u(x, 0) = u0(x) on S1.

In the present work we analyze: Semidiscrete and fulldiscrete (finite difference) solutions of the problem(1)-(4). Necessary and sufficient conditions for blow-up of the numerical solutions are found. It is proved thatthe semidiscrete solutions and the numerical blow-up times converge to the corresponding real ones, whenmesh size goes to zero. Numerical experiments are also discussed.

2 The Semidiscrete Problem

We introduce the following uniform mesh on the domain = [0, 1] [0, 1]

= {(x1,i, x2,j)| x1,i = (i 1)h1, i = 1, . . . , N, (N 1)h1 = 1;x2,j = (j 1)h2, j = 1, . . . ,M, (M 1)h2 = 1}.

and denote y = y(t) = yij(t) = y(x1,i, x2,j , t), i = 1, . . . , N , j = 1, . . . ,M the values of the numericalapproximation at the nodes x = (x1,i, x2,j) in the time t. Define the following finite differences for any meshfunction vi = v(xi) given on by vx,i = (vi+1 vi)/h, vx,i = vx,i1.

We consider the case S2 0, which is more complicated in view of deriving a second order differencescheme. In fact, equation (3) can be obtain from (2), removing the time derivative and associating the sourceterm g1(u) with g2(u) bu.

For semidiscretization of the problem (1)-(4) we use the finite difference method and obtain the followingsystem of ordinary differential equations :

dy

dt=

2k(h2yx1 + h1yx2) + 2(h1 + h2)g1(y) + kh1h2f(y)kh1h2 + 2(h1 + h2)

, i = 1, j = 1, (7)

dy

dt=

1kh2 + 2

(kh2yx1x1 + 2kyx2 + 2g1(y) + kh2f(y)),i = 2, . . . , N 1,j = 1, (8)

dy

dt=

2k(h1yx2 h2yx1) + 2(h1 + h2)g1(y) + kh1h2f(y)kh1h2 + 2(h1 + h2)

, i = N, j = 1, (9)

dy

dt=

1kh1 + 2

(kh1yx2x2 + 2kyx1 + 2g1(y) + kh1f(y)),i = 1,j = 2, . . . ,M 1, (10)

dy

dt= yx1x1 + yx2x2 + f(y), i = 2, . . . , N 1, j = 2, . . . ,M 1, (11)

dy

dt=

1kh1 + 2

(kh1yx2x2 2kyx1 + 2g1(y) + kh1f(y)), i = N,j = 2, ...,M 1, (12)dy

dt=

2k(h2yx1 h1yx2) + 2(h1 + h2)g1(y) + kh1h2f(y)kh1h2 + 2(h1 + h2)

,i = 1,j =M, (13)

dy

dt=

1kh2 + 2

(kh2yx1x1 2kyx2 + 2g1(y) + kh2f(y)), i = 2, . . . , N 1,j =M, (14)dy

dt=2k(h2yx1 + h1yx2) + 2(h1 + h2)g1(y) + kh1h2f(y)

kh1h2 + 2(h1 + h2),i = N,j =M, (15)

y(0) = u0(x), i = 1, . . . , N, j = 1, . . . ,M. (16)

Blow-Up of Finite Difference Solutions to Parabolic Eqns 241

Let S2 = {x = (x1, x2)| x1 = 1, 0 x2 1}. Then, instead of (7), (10) and (13) we have the followingequations, respectively:

dy

dt=

kh2kh2 + 2

(2h1yx1 +

2h2yx2 +

2ah2

g1(y) +2ah1

(g2(y) by) + f(y)), (17)

dy

dt= yx2x2 +

2h1yx1 +

2bah1

y 2ah1

g2(y) + f(y), (18)

dy

dt=

kh2kh2 + 2

(2h1yx1

2h2yx2

2bah1

y +2ah1

g2(y) +2kh2

g1(y) + f(y)). (19)

The local truncation error is (h) = O(h21 + h22) if u C4,1x,t , x (x1, x2), [16].

2.1 Blow-up of semidiscrete solution

We can rewrite the equations (7)-(16) in the following vector-matrix form:

MY (t) = AY (t) + G(Y (t)) (20)

where

Y (t) = [y11, y21, . . . , yN1 y1(t)

, . . . , y1j , y2j , . . . , yNj yj(t)

, . . . , y1M , y2M , . . . , yNM yM (t)

]T , (21)

Y (0) = [y1(0), y2(0), . . . , yM (0)]T . (22)

In the Theorem 2, we will need the following maximum principle:

Lemma 1. Let the assumptions of Theorem 1 hold. Then Y (t) C1[0, Tmax) and Y (t) 0 for t [0, Tmax).

Theorem 2. Let the assumptions of Theorem 1 hold. Then there exists a time Thb

242 M. Koleva and L. Vulkov

3 Time Discretization and Methods for Computation

We introduce a nonuniform mesh grid in time

t0 = 0, tn = tn1 +4tn1 =n1k=0

4tn (4tk > 0, k = 0, 1, ...).

The time increment 4tn is chosen to be decreasing according to the growth of the numerical solution, [2, 14].For example, in the case f(u) = g1(u) = g2(u) = u1+, > 0, the time step [14], 1 < p

Blow-Up of Finite Difference Solutions to Parabolic Eqns 243

0

0.5

1

0

0.5

1

1

1.5

2

2.5

3

x2

u0=sin(pi x1)+sin(pi x2)+1

x1

u 0

0

0.5

1

0

0.5

1

0

10

20

30

40

50

60

x2

T=1.20450e1

x1

y

0

0.5

1

0

0.5

1

0

200

400

600

800

1000

1200

x2

T=1.20661e1

x1

y

Fig. 1. = 2, = 0.01, N =M = 21

0

0.5

1

0

0.5

1

5

10

15

x2

u0=5[sin(pi x1)+sin(pi x2)+1]

x1

u 0

0

0.5

1

0

0.5

1

0

10

20

30

40

50

60

x2

T=2.10021e3

x1

y

0

0.5

1

0

0.5

1

0

200

400

600

800

1000

1200

x2

T=2.29968e3

x1

y

Fig. 2. = 2, = 0.01, N =M = 41

0

0.5

1

0

0.5

1

5

10

15

x2

u0=5[sin(pi x1)+sin(pi x2)+1]

x1

u 0

0

0.5

1

0

0.5

1

0

10

20

30

40

50

60

x2

T=2.68083e7

x1

y

0

0.5

1

0

0.5

1

0

200

400

600

800

1000

1200

x2

T=2.68697e7

x1

y

Fig. 3. = 5, = 0.01, N =M = 41

244 M. Koleva and L. Vulkov

Numerical experiments for our test examples, show that the BUS consist of points (point blow-up), except thecase of constant initial function for some value of , see Figure 4. BUS depends on the maximum value ofthe initial function and (i.e nonlinear source terms in general case). The solution blows up as fast as themaximum value of the initial function and the value of are bigger, see Figure 1,2,3. In the case of differentsource terms or in more general case, when S2 6 0, BUS depends also on the interaction between nonlinearities.Thus, the blow-up point may not coincides with the point, where the initial function has maximum value.

0

0.5

1

0

0.5

1

0

0.5

1

1.5

2

u0 1

x2x1

u 0

0

0.5

1

0

0.5

1

0

200

400

600

800

1000

1200

x2

=2

x1y

0

0.5

1

0

0.5

1

0

200

400

600

800

1000

1200

x2

=5

x1

y

T=5.02512e1

T=0

T=2.04004e1

T= 0

Fig. 4. Different values of at different time levels T , = 0.01, N =M = 21

4.2 Convergence Results

Now, we check the convergence rate of the numerical solution of the problem (1)-(4), f(u) = u2+F (x1, x2, t),g1(u) = u2 + G1(x1, x2, t), g2(u) = u2 + G2(x1, x2, t) and u0(x1, x2) = sin(pix1) sin(pix2). The functions F ,G1 and G2 are chosen, such that the exact solution of the problem is u(t, x1, x2) = et sin(pix1) sin(pix2). Themesh step sizes in space are h1 = h2 = h (N =M) and the ratio h2 = 1 is fixed.

In Table 1 we give the max errors (EN ) of the numerical solutions computed, using grid with N2 grid nodesin space and convergence rates (CR), calculated with the following formula: CR = log2[EN/E2N ]. Decreasing

Table 1. Max error and convergence rate at time T

T = 0.05 T = 0.1

N Max error CR Max error CR

11 4.36960871e-3 6.12952948e-3

21 1.27949323e-3 1.77193 1.59991442e-3 1.93778

41 3.31186965e-4 1.94985 4.03557181e-4 1.98715

81 8.36224738e-5 1.98568 1.01108231e-4 1.99687

161 2.09256494e-5 1.99862 2.52919547e-5 1.99915

321 5.23390773e-6 1.99931 6.32377763e-6 1.99982

the space mesh step size twice yields to decreasing the error approximately four times, which means that theconvergence rate is O( + h2).

Blow-Up of Finite Difference Solutions to Parabolic Eqns 245

References

1. Andreuci, D., Gianni, R.: Global existence and blow up in a parabolic problem with nonlocal dynamical boundaryconditions. Adv. Diff. Equations 1 5 (1996) 729-752

2. Bandle, C., Brunner H.: Blow-up in diffusion equations: A survey. J. of Comp. and Appl. Math. 97 (1998) 3-223. Crank, J.,The Mathematics of Diffusion. Clarendon Press, Oxford (1973)4. Escher, J.: Quasilinear parabolic system withdynamical boundary conditions. Comp. PDE 1b (1993) 1309-13645. Ewing, R., Lazarov, R.,Vassilevski, P.: Local refinement techniques for elliptic problems on cell-centered grids I.

Error analysis. Math. of Comput. 194, 56 (1991) 437-4616. Fila, M., Quittner, P.: Global solutions of the Laplace equation with a nonlinear dynamical boundary condition.

Math. Appl. Sci. 20 (1997) 1325-13337. Groger, K.: Initial boundary value problems from semiconductor device theory. ZAMM 67 (1987) 345-3558. Guedda, M., Kirane, M., Nabana, E., Pohozaev, S.: Nonexistence of global solutions to an elliptic equation with

nonlinear dynamical boundary condition. Bol. Soc. Paran. Mat. 22 2 (2004) 9-169. Kirane, M.: Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic

type. Hokkaido Math J. 21 (1992) 221-22910. Koleva, M.: On the Computation of Blow-Up solutions of Elliptic Equations with Semilinear Dynamical Boundary

Conditions. LNSC 2907 (2004) 473-48011. Koleva, M.: Comparison of a Rothe-Two Grid Method and Other Numerical Schemes for Solving Semilinear

Parabolic Equations, LNCS 2401 (2005) 352-36012. Koleva, M., Vulkov, L.:On the Blow-Up of Finite Difference Solutions to the Heat-Diffusion Equation with Semi-

linear Dynamical Boundary Conditions. Appl. Math. and Comp. 161 (2005) 69-9113. Koleva, M., Vulkov L.: Blow-Up of Continuous and Semidiscrete Solutions to Elliptic Equations with Semilinear

Dynamical Boundary Conditions of Parabolic Type. J. Comp. and Appl. Math, in press14. Nakagava, T.: Blowing up of a finite difference solution to ut = uxx + u

2. Appl. Math. & Optimization 2 (1976)337-350

15. Pao, C.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)16. Samarskii, A.: The Theory of Difference Schemes. Marcel Dekker Inc (2001)17. Xu, J: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 1 (1994) 231-237