-
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:[email protected]).
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:
[email protected]).
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