A Numerical Method for Solution ofSecond Order Nonlinear Parabolic Equations

on a SphereYuri N. Skiba, Denis M. Filatov

AbstractAn efficient numerical method for solution ofsecond order nonlinear parabolic equations on a sphere ispresented. The method involves the ideas of operator splittingand swap of coordinate maps for computing in differentdirections. As a result, 1D finite difference problems withperiodic boundary conditions and matrices of a simple structureappear, so that for their solution a fast numerical algorithmis applicable. The method is tested via several numericalexperiments, including simulation of the phenomena of blow-upand temperature waves, that have many important applicationsin industry.

Index Termssecond order nonlinear parabolic equations,operator splitting, coordinate map swap, blow-up and burningprocesses.

I. INTRODUCTION

SECOND order nonlinear parabolic equations are ade-quate mathematical models of many physical phenomenamet in mechanics, biophysics, ecology and other areas ofscience [1], [2], [3], [4], [5], [6], [7], [8], [9]. Some ofthem are convenient to study in the spherical geometryseparating the differential operator into the spherical andradial components. Because the radial component is trivial,in our research we shall focus on the spherical part. Themodel has the form

Tt = (TT ) + f, (1)

where is the spherical Hamilton operator, T =T (, , t) 0, = (, ) > 0, 0, f = f(T, , , t),while and are the longitude and latitude of the sphereS, respectively.

Since equation (1) is being considered on a sphere, weare dealing with a Cauchy problem formulated in a domainthat has no boundaries. Besides, sphere is a periodic domainonly in , while it is not such in due to the presence of thepoles. Therefore, if one tries to design a numerical procedureto solve the original 2D problem using periodicity in thelongitude and joining somehow the solution at the poles, itwill definitely be computationally expensive. So, our aim isto develop an efficient numerical method for solving equation(1) that would allow computing physically correct numericalsolutions in a fast manner.

Manuscript received March 4, 2013. This work was supported by MexicanSystem of Researchers (SNI) grants 14539 and 26073. It is part of the projectPAPIIT DGAPA UNAM IN104811-3.

Yu.N. Skiba is with the Centro de Ciencias de la Atmosfera (CCA),Universidad Nacional Autonoma de Mexico (UNAM), Av. Universi-dad 3000, Cd. Universitaria, C.P. 04510, Mexico D.F., Mexico (e-mail:skiba@unam.mx).

D.M. Filatov is with Centro de Investigacion en Computacion (CIC),Instituto Politecnico Nacional (IPN), Av. Juan de Dios Batiz s/n, C.P. 07738,Mexico D.F., Mexico (phone: +52 (55) 5729-6000, ext. 56-544, fax: +52(55) 5586-2936, e-mail: denisfilatov@gmail.com).

II. SPLITTING AND MAP SWAP

Prior to performing finite difference approximation welinearise and then split the original equation by coordinatesin the double time interval (tn1, tn+1) [10]. So, hereafterwe consider two operators, L1in , and L2in , i.e.

L1T =1

R cos

(

D

R cos

T

)

, (2)

L2T =1

R cos

(

D cos

R

T

)

. (3)

Here D stands for the diffusion coefficient (T n) computedat the time moment tn (tn1, tn+1), whereas R is theradius of the sphere. The corresponding split 1D problems aresolved in time successively: the solution to the first problemis used as the initial condition for the second one, and viceversa.

The splitting allows one to treat the 1D problems asperiodic in and in , provided each of the problems isbeing considered on a separate grid: the first problem isapproximated on the grid

S(1), =

{

(k, l) : k [

2 , 2 +

2

)

,

l [

2 +2 ,

2

2

]}

, (4)

while the second one is approximated on the swapped grid

S(2), =

{

(k, l) : k [

2 ,

2

]

,

l [

2 +2 ,

32 +

2

)}

. (5)

Obviously, both grids are defined on the same set of nodes.The only change to make in (3) if using (5) is to replacecos with | cos|, as well.

Now we are ready to construct finite difference approx-imations of the derived 1D problems. To obtain the sec-ond approximation order in time, the bicyclic splitting [10](a sort of Strang splittings [11]) is used in each doubletime interval (tn1, tn+1) coupled with the Crank-Nicolsonapproximation

Tn1+i/3kl T

n(4i)/3kl =

Li

(

Tn1+i/3kl + T

n(4i)/3kl

2

)

, i = 1, 2, (6)

Tn+1/3kl T

n1/3kl = 2f

nkl, (7)

Tn+(4i)/3kl T

n+(3i)/3kl =

Li

(

Tn+(4i)/3kl + T

n+(3i)/3kl

2

)

, i = 2, 1, (8)

Proceedings of the World Congress on Engineering 2013 Vol I, WCE 2013, July 3 - 5, 2013, London, U.K.

ISBN: 978-988-19251-0-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2013

where the external forcing f is computed as fnkl =f (k, l, tn). Further, at each point (k, l) the spatialderivatives of the operators Li are approximated as

(

DT

)

1

()2(

D++k Tkl Dk Tkl

)

, (9)

(

DT

)

1

()2

(

D++l Tkl D

l Tkl

)

, (10)

where D+ = (Dk+1,l +Dkl)/2, D = (Dkl +Dk1,l)/2,+k Tkl = Tk+1,l Tkl,

k Tkl = Tkl Tk1,l, D =D |cos|, D

+= (Dk,l+1 + Dkl)/2, D

= (Dkl +Dk,l1)/2,

+l Tkl = Tk,l+1 Tkl,

l Tkl = Tkl Tk,l1.It can explicitly be shown that each of the resulting split

one-dimensional second order finite difference schemes isdissipative. More generally, it holds

||Tn+1||L2

(

S(1),l

) ||Tn1||L2

(

S(1),l

) +

2 ||fn||L2

(

S(1),l

), (11)

where T = {Tkl} and f = {fkl} are grid functions takenat the corresponding time moments. Besides, it is easy todemonstrate that the schemes result to be balanced.

Let us emphasise that a substantial profit we have achieveddue to the splitting is that we use periodic boundary con-ditions while computing in both directions, and , thushaving finite difference schemes with tridiagonal matrices.Therefore, the solution can be obtained with a fast linearsolver, so it is cheap from the computational standpoint.

Another gain of the splitting is that one can take higherthan second order finite difference stencils and hence derivehigher order finite difference schemes.

III. NUMERICAL TESTS

A. Test for the Balance Property

Let the function

T (, , t) = c1 sin cos cos2 t+ c2 (12)

with

= + cos sin t (13)

be the solution to (1). Then, having found the correspondingsource function f via a substitution of (12) into (1), we cansolve the diffusion problem numerically and compare thesolution with the known analytics.

In doing so, we take c1 = 2.5, c2 = 50, = 9, = 5, = 3, = 1, = 105. In Fig. 1 we plot graphs ofthe temporal behaviour of the total mass of the solution,computed on the grid 6 6, and the source function. Asone can see, the total mass is decaying when the (negative)sources are growing; when the sources are about zero, thetotal mass is nearly a constant, as it ought to be. Thisresult demonstrates that the constructed schemes possess theproperty of balance and hence provide physically adequatenumerical solutions. Maximum relative error of the numericalsolution is (tn) = 0.54% at = 103, as well.

0 2 4 6 8 10 12 1415.922767

15.922768

15.922769

Time

Tn

0 2 4 6 8 10 12 144

3.5

3

2.5

2

1.5

1

0.5

0

0.5x 10

7

Time

fn

Fig. 1. Balance Property Test: graphs of the total mass (top) and sources(bottom) in time

B. Test of Combustion and Temperature Waves

Now we are going to verify the schemes on modellingtwo strongly nonlinear real physical phenomena. The firstphenomenon is the propagation of a temperature wave at aconstant amplitude; the second phenomenon is combustionin a limited area, one of whose applications is metal surfaceflaming. Both phenomena were numerically simulated in [1],[2] as Cauchy problems on R1. Below we shall apply theconstructed schemes for studying them on a sphere.

Temperature wave. Take = 0, = 104 and f =c(T T 3) with c = 10. This problem is linear with respectto the diffusion coefficient, but it is nonlinear with respectto the source function. We shall consider two cases, taking ahat-like spot as the initial condition: in the first case the spotsepicentre is located in middle latitudes, while in the secondcase it is placed exactly on the North pole. The numericalsolutions, obtained on the grid 66, are shown in Fig. 2-3.There are two features to notice. First, as the time is growingthe wave fronts are covering the entire sphere, while thewaves amplitudes are kept constant in time (cf. the colour-bars values at different time moments), that is temperaturewaves at constant amplitudes are observed. Second, thephenomenon is accurately simulated independently of thelocation of the initial condition, without any perturbances ofthe solution which would have taken place if we had addedany nonphysical modes into the model at the stage of splittingand/or map swap (4)-(5).

Combustion within a limited area. Let = 1 andf = cT , where c = 4.5. The model is now nonlinear bothwith respect to the diffusion coefficient and sources. The

Proceedings of the World Congress on Engineering 2013 Vol I, WCE 2013, July 3 - 5, 2013, London, U.K.

ISBN: 978-988-19251-0-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2013

= =

=

2 =

2

t = 0 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.2 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.4 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.6 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.8 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 1 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 1.5 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 2 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. Temperature Wave Test: mid-lat numerical solution at several timemoments

parameter determines distinct regimes of combustion: thecase 0 < < + 1 is the expansion regime (or HS-regime[1], [2])the area of combustion is getting larger in time; thecase > +1 is the reduction regime (or LS-regime)thearea of combustion is getting smaller; the case = +1 isthe stationary regime (S-regime), when combustion is limitedwithin an area of a constant size. In all three cases the sourcefunction leads to an infinite increase of the temperature T ,that is a blow-up occurs.

In Fig. 4 we show the initial condition, while in Fig. 5-7 there are numerical solutions corresponding to = 1

= =

=

2 =

2

t = 0 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.2 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.4 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.6 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 0.8 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 1 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 1.5 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= =

=

2 =

2

t = 2 days

N

= 0 = 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Temperature Wave Test: North pole numerical solution at severaltime moments

(HS-regime), = 3 (LS-regime) and = 2 (S-regime),respectively. In Fig. 8 we plot graphs of the solutions L2-norms. It is seen that a blow-up is tended to be achieved inall the casesthe solutions amplitudes unboundedly growin time, while the behaviour of the combustion area dependson the parameter and agrees with the theory. Hence,the constructed schemes allow properly simulating all theregimes of combusion, including the border sensitive S-regime.

Proceedings of the World Congress on Engineering 2013 Vol I, WCE 2013, July 3 - 5, 2013, London, U.K.

ISBN: 978-988-19251-0-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2013

= =

=

2 =

2

t = 0 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

Fig. 4. Combustion Test: initial condition

= =

=

2 =

2

t = 0.2 days

N

= 0 = 0

0

0.5

1

1.5

= =

=

2 =

2

t = 0.4 days

N

= 0 = 0

0

0.5

1

1.5

2

2.5

3

= =

=

2 =

2

t = 0.6 days

N

= 0 = 0

0

1

2

3

4

5

= =

=

2 =

2

t = 0.8 days

N

= 0 = 0

0

1

2

3

4

5

6

7

8

= =

=

2 =

2

t = 0.9 days

N

= 0 = 0

0

2

4

6

8

10

= =

=

2 =

2

t = 1 days

N

= 0 = 0

0

2

4

6

8

10

12

Fig. 5. Combustion Test: numerical solution for HS-regime at several timemoments

IV. CONCLUSION

A numerical method for solution of nonlinear parabolicequations on a sphere has been presented. The keypointof the method is the operator splitting by coordinates andsubsequent map swap that allows representing the sphereas if it were a periodic domain in both directions. Hence,we constructed second order finite difference schemes ap-proximating 1D diffusion problems with periodic bound-

= =

=

2 =

2

t = 0.03 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

= =

=

2 =

2

t = 0.06 days

N

= 0 = 0

0.2

0.4

0.6

0.8

1

1.2

= =

=

2 =

2

t = 0.1 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

= =

=

2 =

2

t = 0.12 days

N

= 0 = 0

0.5

1

1.5

2

= =

=

2 =

2

t = 0.14 days

N

= 0 = 0

0.5

1

1.5

2

2.5

3

3.5

= =

=

2 =

2

t = 0.15 days

N

= 0 = 0

1

2

3

4

5

6

7

8

9

Fig. 6. Combustion Test: numerical solution for LS-regime at several timemoments

ary conditions in the longitude and latitude. Therefore weavoided difficulties related to imposing suitable boundaryconditions at the poles. The constructed schemes keep thesignificant properties of the original differential problemthey are balanced and dissipative, and thus provide physi-cally adequate numerical solutions. Due to the tridiagonalstructure of the matrices of the linear systems the schemesare computationally inexpensive. The numerical experimentsdemonstrated accurate simulation of two important physicalphenomenatemperature waves and combustion with blow-up.

REFERENCES

[1] A. A. Samarskii et al., Blow-up in Quasilinear Parabolic Equations,Walter de Gruyter, 1995.

[2] A. A. Samarskii et al., Thermal Structures and the FundamentalLength in a Medium with Nonlinear Heat Conductivity and VolumeHeat Sources, in: Regimes with Sharpening. An Evolution of the Idea.Co-evolution Laws of Complex Structures, Nauka: Moscow, 1999, pp.39-46 (in Russian).

[3] J. Bear, Dynamics of Fluids in Porous Media, Courier Dover Publica-tions, 1988.

[4] M. E. Glicksman, Diffusion in Solids: Field Theory, Solid-State Prin-ciples and Applications, John Wiley & Sons, 2000.

Proceedings of the World Congress on Engineering 2013 Vol I, WCE 2013, July 3 - 5, 2013, London, U.K.

ISBN: 978-988-19251-0-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2013

= =

=

2 =

2

t = 0.05 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

= =

=

2 =

2

t = 0.1 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

= =

=

2 =

2

t = 0.15 days

N

= 0 = 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

= =

=

2 =

2

t = 0.2 days

N

= 0 = 0

0

0.5

1

1.5

2

= =

=

2 =

2

t = 0.25 days

N

= 0 = 0

0

0.5

1

1.5

2

2.5

3

3.5

= =

=

2 =

2

t = 0.3 days

N

= 0 = 0

0

1

2

3

4

5

6

Fig. 7. Combustion Test: numerical solution for S-regime at several timemoments

[5] J. R. King, Instantaneous Source Solutions to a Singular NonlinearDiffusion Equation, Journal of Engineering Mathematics, vol. 27,1993, pp. 31-72.

[6] A. A. Lacey, J. R. Ockerdon, A. B. Tayler, Waiting-time Solutionsof a Nonlinear Diffusion Equation, SIAM Journal of Applied Mathe-matics, vol. 42, 1982, pp. 1252-1264.

[7] G. A. Rudykh, E. I. Semenov, Non-self-similar Solutions of Multidi-mensional Nonlinear Diffusion Equations, Mathematical Notes, vol.67, 2000, pp. 200-206.

[8] A. Kh. Vorobyov, Diffusion Problems in Chemical Kinetics, MoscowUniversity Press, 2003 (in Russian).

[9] Z. Wu et al., Nonlinear Diffusion Equations, World Scientific Publish-ing: Singapore, 2001.

[10] G. I. Marchuk, Methods of Computational Mathematics, Springer,1982 (translated from Russian, Nauka: Moscow, 1977).

[11] G. Strang, On the Construction and Comparison of DifferenceSchemes, SIAM Journal of Numerical Analysis, vol. 5, 1968, pp. 506-517.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

Time||T

n|| L

2(S

(1)

,

)

0 0.05 0.1 0.15 0.20.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time

||Tn|| L

2(S

(1)

,

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

||Tn|| L

2(S

(1)

,

)

Fig. 8. Combustion Test: graphs of the L2-norms of the solutions in timefor HS- (top), LS- (middle) and S- (bottom) regimes

Proceedings of the World Congress on Engineering 2013 Vol I, WCE 2013, July 3 - 5, 2013, London, U.K.

ISBN: 978-988-19251-0-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2013