skade eaction ffusion ne-dimensional System - ZIB

Tehnicl Repot TR 9-9 (July 193)
Jens Lang
skade eaction ffusion
ne-dimensional System

Jens Lang
KARDOS — KAskade Reaction Diffusion
One-dimensional System
Abstract. A software package for the adaptive solution of time-dependent
reaction-diffusion systems and inear elliptic systems in one space dimenson
is presented. The used algorithm is based on fundamental arguments in
LANG/WALTER [4]. Here, only brief outlines of the algorithm are given.
This software package is based on the KASKADE olbox [3]
Key words: Rothe method, embedded Runge-Kutta method, adaptiv
ultilevl Fnite Element method.

Contents
Introducti
efng a problem
eading ri
ommad lngua
For programmers oly 10
efining a tim problem 10
efining a time ntegrator 11
Time integration process
id manageent 17
exampl
er funtions for a population ecology odl 18
efining he new probl 22
oviding two omon arse grids 22
alog wih the sysem
efereces 25

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ntroduction
Usng the program KARDOS, sysems of semlinear parabolic initial boundary
value problems of the for
{x)ut ~ {D{X)U) F(u X G 0 C R\ t G [t t
Ci(, ^GTiCön
un + a(x)u £2(^ x x £ T2 C df
u(tx) = u0(x
are solvable. Here, u = (u u2,... ur) is a vector function The matrix
function P(x) may vanish on subset of ft. The discretizaton is done by
the Rothe method. In contrast to he widespread method of lines, time is
discretized first than space. The main advantage of this sequence is the
possibility o compute the pace discretizaton optimal during the time integration
b an adaptiv multileve finite element method. Therefore the
KASKADE toolbox is modified to handle one-dimensional problems. In
KARDOS a special embedded Runge-Kutta method of order 32) has bee
implemented. This method keeps its accuracy even in the case of ifferenial
algebraic equations (P(x) = 0 somewhere).
The above differetial equations can be reformulated into an abstract Cauchy
problem poibl of differenal-algebraic type in an appropriate Hilbrt
pace H
Pu f(u u G H ,t e[tt
u(0) = u0
Now, a three-stage embedded Rosenbroc ethod for his pure tim problem
looks as follows:
{P ~ 7" r /«(«o))A;8 = r/(u0 + Yl a
1.
) + T f{uo Y l i = 1 3
3=1 J=
Uo +
u0 + Y
3 =
1.3)

Yet this frm not suited to be implemented straight-forward, becuse
along with function evaluations there are still matrix*vector producs on the
ighhand ide. The ransformation
leads to the new ysem
/ "fi * — A
1 \
P-fu(u0) {u0 + J2 a J2— « = 1
r 7 i=1 j=1
uo ^ r (1.
In KARDOS a special set of parameters, which guarantees L-stabiity of the
chosen method, is used. Setting 711 = 722 = 733 := 7 only two function
valuations and one matrix inversion are needed on the righthand side. The
corresponding method was given by ROCHE [], who gives further sateent
about convergence and consisteny.
efficent
0 . 5 2 8 4
704526589 9 3 9 4 1
7309 0.940143465
09140443903 0.
1.99625219 0.874044410
1.99625219 4797
0. 223756213
ue to the drence of the pproximaed ues of er 3 and
timeError := ||u (1-
we have a good estimator of the main error term descibing the ocal error
of the econd order method. The proposal for the new te ize is
timeTol \
afeFa — — 1-
timeError

The nom whih KARDOS uses s a weighted L-no, giv by
whe
it
uAbsi
uRelMxi
\\ u
r-
||«
||^||o < f ^ i
uAfe, < ||u||o < uRelMxi
uRelM ||u
ReM RTOL o ,
uA6s ATOL * (O) 1 /
:i-7)
Scaling can be turned on or off. If the scaled error norm is used, the tolerances
RTOLi and ATOLi have to be seleced very carefully to reflect accurately
the scale of the problem. The tolrane ATOLi should indicate the absolute
value at which the i-th component is essentially insignifican. O the other
hand, the value RTOLi affects the relative accuracy of the i-th component
with respect to its maximal value in time. This control turns out to be quite
efficient and robust for a wid lass of problems. Howeer, it is cear that i
is not a univeral mthod.
To implement one time step, one elliptic proble has to be solved at each
tage of the Rosenbrock method. This is done by means of an adaptiv
ultilevel finite element method elaborated in recent years, see DEUFLHARD
et al. []. This method is an excellent tool to adapt the space discretization
for the current solution in uch a way, that a prescribed tolerance spaceTol
is achieved. KARDOS uses standard linear elements connected with local
error estimator of BABUSKA-HEINBOLDT [1] yp. We get at each tage,
discretizations of the form
B T v ) = n(v) e SnVvn e S i = 3 1.8)
where Sn denotes the space of al ntinuous, piecewise linear functions on
partition An of 0, satisfying the Dirichlet boundary conditions. In S% hey
vanish on the Dirichlet boundary. The equations only differ in the respective
righthand id and in the pecial boundary conditions taken in onsider
ation.

To cntrol the pace dicretiztion, we restrict urselves to the firt stage of
the Rosenbrock method which is exactly the semiimplicit Euler method applied
to the system (1.2). According to a local principle, on each current grid
an error estimation is computed. Therefore we consider the corresponding
elliptic differental equation on each element, imposng the urrent solution
of the Euler method UE as boundary conditions, and solve it approximately
by a quadratic finite element method. The difference ek between the linear
and quadratic approximation over the kth elent satisfies
(e

paceTol
The final grid is used for solving the other elliptc problems too. Only one
grid is appled for all unknowns of the ystem
Setting the parameter globTol, KARDOS ontrols the space and time is
retization due t he special tolerances
aceTol = globTol/3.0 , timeTol = globTol/0 .
This specific splitting results from the chosen parameters of the method. O
the other hand, the parameters spaceTol and timTol allow specal tuning,
desired. For more details see LANG/WALTER [4,]
In the case of inconsisent sarting values for differential-algebraic systems
it is possible to use an implicit Euler method in the firt time step coupled
with a damped Newton thod.
As a program, which handles a very general class of problems, KARDOS
cannot succed for all problems. Experiments have hown, that the compuations
have often been improved settng the paramters ery carefully. In
reme cases the iterativ olver may fail

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efining roblem
At first the user has to check that in the main file kardos.c the static param
eter time is set t one. Next the desired number of equations noEqn used in
he funcions
nitProblemO'kardos",100,noEqn,250,FULL)
nitTimeProblem("kardos",100,noEqn)
to prepare the memory has to be inserted. At the momnt the maximal
number is 10. It can be changed easly in nods.h settng the fixed parameter
MAX_NODE_GROUPS.
The actual problem is defined by the functions
Parabolic
Laplace
onvection
elmholtz
ource
acobian
nitialFunc
auchy
irichlet
P(x)
diffusion matrix D(x)
not carried out! (IGNORE)
put in the righhand id (IGNORE
force term F(u)
jacobian of the force te (u
initial values u0(x)
boundary values ^2(t}x)}o(x
boundary values £(tx)
The procedur SetTimeProblem announces the problem defined that way to
he KARDOS toolbox. According to the fact, that one example is better
han thousand explanatons, the reader is referred to sud ntensively the
ile stdtimeprob.c.
nt: etting time=0 KARDOS can also be used o solv systems of linear
liptic partial dfferential equations i one spac imenson omarabl to
LLKASK [3]

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din id
KARDOS requires two common starting grids, which first have to be read
with the help of the command readtri. During the computation one grid concerns
the starting values, on the other grid he urrent solution is computed
adaptively.
Beside a geometrical descripton of the coarse space discretization, the sor
of boundary conditions has to be declared. For that, the parameters B, I, D
and C, which stand for Boundary, Interor, Dirichlet and Cauchy, are avail
able. Furthermore, it's possible to characterize different subdomains through
lassparameter. A general input reads as follows:
grid name
Dimension:(numbe of points,number of edges
point index:x-coordinate,B or I to characterize
he location of the point, I or D or C to characterize
he boundary conditions for all components, classparamete
to characterize a specal propery onl on the boundar
END
edge index:(left point index,right point index), cassparameter
to characterize a special propery of a subdomain
END
n example oncerning two components with mixed boundary conditions on
[0,1] and two different materials in [0,0.) and (051] looks lik the following
input stream
xamplel
Dimension: 4,3)
0:0.B,DC,
0.1,11
2:0.1,11
3:1.B,CD1
END
0,
1,,0
1
ND
Inner boundary points are available.

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C o n d ng
All specal commands which K ARD OS needs for the time integration are ex
plained n the following. A list of all commands can be found in kardos.def
All other ommands not explained here are used as in KASKADE.
KARDOS reads at the start a file kardos.startup which contains a sequence
of ommands and executes these commands. If no quit is incuded, more
ommands are requesed from sandard input.
timeproblem
A tim probl is selected.
inftimeproblem
informs about the urrent tim probl
seliterate
For the time inegration the paramters ssortim and stabtime are avail
abl
selrefine
For different refineent the paramete anval or extrapol are available.
seltimeinteg
A time integrator is sected. The parameters rodas and euler are available.
The Euler method can be used only for the sarting tep.
timestepping
This command corresponds to the solve-command of KASKADE. The tim
inegraton is carried out. The following parameters can be used:

verboseP
scaling
masteps nt
tstart eal
tend eal
timestep eal
globtol
spacetol eal
timetol eal
itertol eal
itermasteps nt
showsol nt
atoli eal
rtoli eal
Important da prnted. A he momnt,
always true.
ep size control: 0 absolute,
1 for relative. If scaling is set to be 1, so
all corresponding parameters atoli, rtoli
il,2,...,number of unknowns have to be et too.
umber of maximal tim teps
starting tim
final time
initial ime ep
accuracy demanded for the integrator (internal
pacetol and timetol ar et automatically)
desred space accurac
desired time accuracy (if spacetol
and timetol are selected, hen internal globtol
is set)
desred relative accuracy of he ierativ
olver for the linear sysems
standard is 0.0001
maximal number of iterations, standard is 1000
graphic is plotted every Int seps, standard is 1
i=l,...,number of unknowns, necessary input for
every unknown to control the time step size
relatively (scaling 1). The tolerance atol should
ndicate the absolute value at which the
corresponding component is essntiall
insignificant. The tolerance rtol affects
the relative accurac with respect to the
maximal value in time.

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r o g e r only
This chapter gives some information to get a feeling for the internal structur
of the program Of course, it cannot mention all details. Furthermore
for a better understanding it is necessary o read al header and c-files of
K ARD OS very carefully.
5.1 Defining a time problem
To include all data of a initial boundary value problem two new data struc
ures are added.
struct timeProblem
char *name;
int (*Parabolic)(...);
int (*Laplace)(...);
int (*onvection)(...);
int (*elmholtz)(...);
int (*ource)(...);
int (*Jacobian)(...);
real (*InitialFunc)(...);
int (*auchy)(...);
int (*irichlet)(...);
int (*ol)(...);
struct timeProblemType
char *name;
int maxNoOfTimeProblems;
int noOfEquations;
The access to these srucures is realized by he pointers
timeProblem *actTimeProblem, *timeProblems;
timeProblemType *theTimeProblem;
Furthermore, there are some functions t announc ertain problms within
he program
int etTimeProblem(char *name, char **varames
int *Parabolic,
10

A tim probl is announced.
int *Laplace,
int *onvection,
int *elmholtz
int *ource,
int *acobian,
real *InitialFunc,
int *auchy,
int *irichlet,
int *ol)
int SetStdTimeProblemsO
ome standard time problems are announced.
void SetTimeProblemAddressesO
Special addresses are set to sect time problems within the ommand language.
int nitTimeProblem(char *name,
int noOfTimeProblems,
int noOfEquations)
The current timeProblemTyp-sructure is built up
5.2 Defining a time integrato
To define a time integrator the ructure timentegMethod is used. It oniss
of the following elements:
char *name
ame of the im ntegrator
int verboseP
ontrol of the informaton output, sandard: true
int scaling
ontrol of the nor for the time error, 0: absolute, 1: relativ
int showSol
ontrol of the picture plot, standard: 1

int nonNegativity
ontrol of the nonnegativi of the olution, unused
int noOfTimeteps
number of the urrent tim teps
int maxSteps
aximal number of time steps
int noOfStepReductions
number of the time step reducons within an integraton tep
int maxStepReductions
maximal allowed im ep redtions within an integraton tep
tandard:
int stepTypePar
differeiation from the starting step
STAR_STEP: starting step, LATER_STEP: later sep
int backDepth
grid depth, on which ever olution process is sarted again, sandard: 0
real time
local starting time
real newTime
local nal tim
real tEnd
global final time
real timeStep
length of the time step
real newTimeStep
new propoed im ep

eal tRest
emained global im
real gammaStab
tabili arameter of the mthod
real timeTol
allowed olrane of the local time error
real spaceTol
allowed olrane of the local space error
real globTol
allowed olrane of the global error
real timeError
urren ocal time error
real globError
urrent global error
real trueError
rue error if solution known, unused
real normSol
nor of the solution
real timeStepFactor
factor by which the old time step is divided
real rho
imeTolglobTol, standard:
real maxTimeStep
aximal time step, standard: 0.
real timeStepSafe
afety factor for the onrol of the tim tep, sandard:

eal spaceTolFac
paceTolglobTol, sandard: 1.
TRIANULATION *compTrian
urren omputation grid
TRIANGULATON *resultTrian
grid for the sartng values
int *InitTimenteg
inializes the tim ntegraton process
int *Timente
ealizes the im ntegration proces
int *CloseTimeInteg
oses the im ntegration proces
int *AssRhsl
omputes the righthand side of the firs tage
int *AssRhs2
omputes the righthand side of the second sage
int *AssRhs3
omputes the righthand side of the thid stage
problem *firstStage
iptic problem of he firt stage
problem *secondtage
iptic problem of he second stage
problem *thirdStage
iptic problem of he third sage
The access to the dfferent tim ntegrators is done by he pointe
timentegMethod *actTimentegtor, *timenteMethods;

To bui up im ntegr the fllowing funcions a avable
int fTimentegMethod(char *name,
int *InitTimeInteg,
int *TimeInteg,
int *CloseTimenteg,
char *nameFirstStage,
char *nameSecondStage,
char *nameThirdtage,
int *sshsl,
int *sshs2,
int *AssRhs3,
real spaceTolFactor,
real gammatability)
announces time inegrator. Beside a name, functions for the realiza
ion of the method (initTimenteg, Timenteg, loseTimelnteg), the
corresponding elliptic problems ("nameFirstStage" , "nameSecondStage" ,
"nameThirdStage") the functons for the computation of the righthand
ides ssRhsl, AssRhs2, AssRhs3), the parameter spaceTolFac and the
tabilit parameter gammaStab have to be inserted,
uch as DefTimeIntegMethod("rodas",InitIntegtep,R°dasStep,lose
ntegStep,"rodasl", "rodas2","rodas3",AssRHSodasl,ssRHS
odas2,ssRHSodasTHIRDDAS_GA) .
int SetStageProblemsO
The elliptc problems occurring in each stage of the method are announed
within ettageProblems by calling the funcon etProblem.
5.3 ime interaion process
Picture 1 shows the whole process of he time integration method. The program
comes to an end when the final time or the maximal step number are
eached or when the maximal number of sep ize reducions wihin one negraton
ep is exceeded.

espping

5.4 Grid managent
The program KASKADE supports work with several grids which are connected
to a list. A time integration requires to exchange values on differently
efined grids. This leads to cos eek algorihms. To be more efective here,
he srucure EDG is enlarged he new elent
*twin
If the considered edge exists in he other triangulation, then this elemen
points at he corresponding element, otherwise it is nil. In the structur
TRIANGULATION the new element int twinRelation stands for the e
istence of such a relation between two triangulations.
To work wih tw grids the folowing funcons can be used:
int SetTwinRelOnCoarseGrids(TRIANGULATI *t,RIUL *t)
inializes the twirelaton of two arse grids.
DG *FindBrotherEdge(real ,EDG *edFrom
nds the edge of the other grid n which a given poin ies.
real InterpolateInEdge(real E *edn,int var,int inde
gives back an nterpolated value.
17

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le
In this chapte how how to add a new problem to the list of predefined
problems:
define a s of functions to characerize the inial boundary value probem
add the problem to the list of predefined problems (call etProblem,
using nitUserTime in user. of the kardos-directory,
add tw ommon car grids in the grids-directory,
use kardos . startup for an automatic ommand language dialog wi
the sysem
6.1 ser functions for opulation ecolo del
Let us olve the folowing equatons
- 0.
+ 1-u
in the domain 0 ) for t > 0, with the boundary condons
and the initial condition
= 0 for t > x = 0 and x =
5 for
4x + 1 for
-4x + 11 for
for
10 for
4x + for
-4x + for
for
18

This system of nonlinear equations modls a cerain planktonit predator
prey suations in which crowding is a facor. The KARDOS user defines in
user. he following funtions:
static int EcoParabolic(real x, int classA, real t, real *fVals,
int equation, int variable)
if (equation!=variable) return F_IGN0RE;
switch (variable)
{
case 0: fVals[0] = 1.0;
break;
case 1: fVals[0] = 1.0;
break;
return F_C0ISTAIT;
static int EcoLaplace(real x, int classA, real t, real *fVals,
int equation, int variable)
if (equation!=variable) return F_IGN0RE;
switch (variable)
{ case 0: fVals[0] = 0.0125;
break;
case 1: fVals[0] = 1.0;
break;
return F_C0ISTAIT;
static int EcoConvection(real x, int classA, real t, real *fVal,
int equation, int variable)
return F_IGI0RE;
static int EcoHelmholtz(real x, int classA, real t, real *fVal,
int equation, int variable)
return F_IGI0RE;
static int EcoSource(real x, int classA, real t, EDG* ed,

eal valO, vail;
real *fVal, int equation,
real (*Func)(real,EDG*,int,timeIntegMethod*))
switch (equation)
{
case 0: valO = Func(x,ed,0,actTimeIntegtor);
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = ((35.0+16.0*val0-val0*val0)/9.0-vall)*val0;
break;
case 1: valO = Func(x,ed,0,actTimeIntegtor);
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = (val0-1.0-0.4*vall)*vall;
break;
return F_VARIABLE;
static int EcoJacobian(real x, int classA, real t, EDG* ed, real *fVal,
int equation, int variable,
real (*Func)(real,EDG*,int,timeIntegMethod*))
real valO, vail;
switch (equation)
{
case 0: if (variable==0)
{
valO = Func(x,ed,0,actTimeIntegtor);
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = (35.0+32.0*val0-3.0*val0*val0)/9.0-vall;
else if (variable==l)
{
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = -vail;
else ZIBStdOut("error in EcoJacobian\n");
break;
case 1: if (variable==0)
{
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = vail;

eturn F_VARIABLE;
else if (variable==l)
{
valO = Func(x,ed,0,actTimeIntegtor);
vail = Func(x,ed,l,actTimeIntegtor);
fVal[0] = val0-1.0-0.*vall;
else ZIBStdOut("error in EcoJacobian\n");
break;
static real EcoInitialFunc(real x, int classA, int variable)
{
real val = 0.0;
switch (variable)
{
case 0: if ((0.0

fVals[l] = 0.0; /* */
else ZIBStdOutCerror in EcoCauchy\n");
break;
case 1: i ((x==0.0)I I(x==2.5))
fVals[0] =0.0; /* Sigma */
fVals[l] = 0.0; /* i */
return F_COISTAIT;
eise ZIBStdOutC"error in EcoCauchy\n");
break;
6.2 Defining the new problem
ext we defne the new problem wih the help of the following procedure
int InitUserTimeQ
{
f (!SetTimeProblem("example1",varName,
EcoParabolic,
EcoLaplace,
EcoConvection,
EcoHelmholtz,
EcoSource,
EcoJacobian,
EcoInitialFunc,
EcoCauchy,
EcoDirichlet,
(int(*)(real,int,real,real*,int))nil));
return true;
A call of nitTimeUser is carried out automatically from he main program
6.3 roviding tw como coar rid
Now we inser two arse grids n the grids-decory using the files ecologyl
and cology2
ecologyl ecology2
Dimension:(11,10) Dimension:(11,0)
22

0 0.0,B,CC 0 0.0,B,CC
1:0.25,1,11 1:0.25,1,11
2:0.5,1,11 2:0.5,1,11
3:0.5,1,11 3:0.5,1,11
4:1.0,1,11 4:1.0,1,11
5:1.25,1,11 5:1.25,1,11
6:1.5,1,11 6:1.5,1,11
:1.5,1,11 1.5,1,11
:2.0,1,11 :2.0,1,11
9:2.25,1,11 9:2.25,1,11
0:2.5,B,CC 10:2.5,B,CC
EID EID
0 (0,1) 0 (0,1)
1 (1,2) 1 (1,2)
2 (2,3) 2 (2,3)
3 (3,4) 3 (3,4)
4 (4,5) 4 (4,5)
5 (5,6) 5 (5,6)
6 (6, 6 (6,
( (
(9) (9)
9 (9,0) 9 (9,10)
EID EID
6.4 ialo with the ste
KARDOS reads at the tart a file kardos.startup which contains a sequenc
of commands and executes these commands. That way we get an automat
dialog wih the system. Of course, a step-by-step ialog is available to.
deposit in kardos.starup the following omands:
read ../grids/ecologyl.g
read ../grids/ecology2.g
timeproblem examplel
seliterate pbicgstabtime
selmatmul sparse
selestimate babuska
selrefine meanval
seltimeinteg rodas
window new automatic name solutionl
graphic solution triangulation
seldraw 0
window new automatic name solution2
graphic solution triangulation

seldraw 1
timesteppin timestep 0.0001 maxsteps 200 globTol 0.02 tEnd 10.0
ow we can start the program

ferences
Babus, W.C. Rheinboldt: Error estimas for adptive fin et
comutons. SIAM J. Numer. Anal 15 p. 7364 (1978)
] P. Deuflhard, P. Leinen, H. Yserentant: Concepts an Adaptie Herarcal
F e Elen Code. IMPACT, , p. 3-3 1989)
3] B. Erdmann, J. Lang, R. Roitzsch: KASKADE - Manual. To appear as
Technical Report TR 93-5, Konrad-Zus-Zentrum (ZIB) (1993)
[4] J. Lang, A. Walter: A Fnie Eent Method Adaptive n Space and
Tim for Nonlinear Reaction-Dffuson Systems. IMPAC of Comput
ing Scence and Engineerng, , p. 9-314 (199
] J. Lang, A Walter: An Adaptive Rothe Method for Nonlinear Reacon-
Diffuson Sstms. Appled Numerial Mathematics 13 p. 1 1 4
1993)
M. Roche: Rosenbrock Methods for ifferenal Albraic Equatons.
umer Math. 52, p. 4 (1
Acknowledgements. I acknowledge R. Roitzsch for inspiring disussions
on the field of programming and for making available always he urrent
version of KASKADE. I also thank U. Nowak for supplying me wih referen
olutions obtained y his finite dfferene method.