TECHNISCHE UNIVERSIT Ä T BERLIN
Error Analysis and Mo del A daptivit y for
Flo ws in Gas Net w orks
Jero en J. Stolwijk and V olk er Mehrmann
Preprin t 2017/02
Preprin t-Reihe des Instituts für Mathematik
T ec hnisc he Univ ersität Berlin
http://www.math.tu-berlin.de/preprints
Preprin t 2017/02 Jan uar 2017

Error Analysis and Mo del A daptivit y for Flo ws
in Gas Net w orks ∗
Jero en J. Stolwijk and V olk er Mehrmann
Abstract
In the sim ulation and optimization of gas flo w in a pip eline net w ork,
a hierarc h y of mo dels is used that emplo ys different form ulations of the
Euler equations. While the optimization is p erformed on piecewise linear
mo dels, the flow sim ulation is based on the simulation of one to three di-
mensional Euler equations including the temp erature distributions. T o
decide whic h mo del class in the hierarc hy is adequate to ac hieve a de-
sired accuracy , this pap er presen ts an error and p erturbation analysis
for a t w o level model hierarch y including the isothermal Euler equations
in semilinear form and the stationary Euler equations in purely alge-
braic form. The fo cus of the w ork is on the effect of data uncertain ty ,
discretization and rounding errors in the n umerical sim ulation of these
mo dels and their in teraction. T w o simple discretization sc hemes for the
semilinear mo del are compared with resp ect to their conditioning and
temp oral stepsizes are determined for whic h a w ell-conditioned problem
is obtained. The results are based on new component wise relative con-
dition n um b ers for the solution of nonlinear systems of equations. More-
o v er, the model error b et w een the semilinear and the algebraic mo del
is computed, the maxim um pip eline length is determined for which the
algebraic mo del can b e used safely , and a condition is deriv ed for which
the isothermal mo del is adequate.
Key W ords: gas net w ork, isothermal Euler equations, algebraic approxima-
tion of Euler equations, error analysis, condition n umber, data uncertain t y ,
comp onen t wise error analysis, sto chastic error analysis.
2010 Mathematics Sub ject Classification: 35Q31, 65G50, 65M15.
∗ The authors thank the Deutsc he F orsch ungsgemeinsc haft for their supp ort within
pro ject B03 in CR C TRR 154.
0

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 1
1 In tro duction
Gas pla ys a crucial role in the energy supply of the w orld. It is sufficien tly
and readily a v ailable, it is traded, and it is storable. In German y e.g., after
oil, natural gas is the second most used energy supplier, with a total share of
more than 20 % of the energy consumption in 2015 [8]. The high demand for
gas and the deregulation of the energy mark ets call for a reliable mathematical
mo deling, simulation, and optimization (MSO) of the gas transp ort through
existing pip eline net w orks.
In view of this demand, in the last decades considerable research on the
sim ulation and optimization of gas net w orks has b een p erformed, see e.g. [1,
3, 4, 10, 11, 15, 16, 17, 19, 22, 28, 33, 37], where different sim ulation mo dels
for the flo w through a pip e or a net w ork of pip es and compressor stations hav e
b een prop osed. Since the simulation models are a key factor in optimization
to ols, adequate accuracy and high efficiency is very important. So, using
error estimation, t ypically the grid is adapted in space and time and as a new
comp onen t of the sim ulation pro ce ss w e will discuss the adaptation of the
mo del within a mo del hierarc h y . W e will fo cus on the pure pip e flo w, where
the mo del hierarc h y is easily constructed and where it can b e used to find
an appropriate trade-off b et w een accuracy and computational complexity , see
[5, 7, 12, 14].
It is w ell kno wn, see e.g. [20, p. 5], that the n umerical solution of a com-
putational problem con tains errors from all or some of the follo wing sources:
mo deling, discretization, iteration, data uncertain t y , and rounding errors, see
Figure 1 for a sc hematic o v erview. These errors should b e balanced to ac hiev e
an adequate sim ulation result. W e derive error estimates and a sensitivit y
analysis within the t ypical mo del hierarc hy , with resp ect to the discretiza-
tion sc heme and the iteration error for the solution of the resulting nonlinear
systems of equations. T o demonstrate the new tec hniques and to keep the
presen tation simple w e presen t a deterministic as well as statistical error and
sensitivit y analysis only for t w o sp ecific comp onen ts of the mo del hierarc h y , a
purely algebraic mo del and an isothermal semilinear mo del, but the analysis
can b e carried out also for more complex comp onen ts in the mo del hierarch y .
F or these tw o mo dels, mo del and discretization error estimators for an arbi-
trary cost functional ha v e b een derived in [12, 13]. Ho w ever, the effect of data
uncertain t y and rounding errors in the solution of these tw o mo dels has not
b een considered and is the main topic of this pap er.
T o estimate the errors, w e p erform a bac kward error analysis, see e.g. [20],
and deriv e first order upp er b ounds as w ell as statistical estimates for the error
in the solution due to data uncertain t y , mo deling, discretization, rounding and
iteration errors. A p erturbation analysis in which also higher order error terms

2 J. J. Stol wijk and V. Mehrmann
Real w orld problem
Mathematical mo del
(ODE/D AE/PDE)
Mo deling error
Discretized mo del
(linear/nonlinear)
Solution with ite-
rativ e metho d
Iteration, rounding,
and data uncer-
tain t y errors
Algebraic mo del
Solution with ite-
rativ e metho d
Iteration, rounding,
and data uncer-
tain t y errors
Discretization error Mo deling error
Figure 1: Ov erview of different error sources con tained in the n umerical sim-
ulation of a real w orld problem.
are included usually leads to v ery p essimistic upp er error b ounds, see [26], and
is therefore not considered. W e deriv e comp onen twise condition n um b ers and,
based on these, deterministic and statistical error estimates. The adv an tage
of the comp onen t wise relative condition n um b er ov er the traditional norm wise
relativ e condition n um b ers for nonlinear systems is demonstrated.
This pap er is organized as follo ws. Section 2 introduces different models
that describ e the gas flo w through a pip eline. Section 3 giv es a concise in-
tro duction in to error analysis and conditioning. Moreo v er, several kinds of
condition n um b ers are deriv ed. Section 4 presen ts a sensitivity analysis for
t w o differen t discretization schemes applied to the semilinear model and ap-
plies the deriv ed condition n um b ers to these discretizations. Moreov er, the
effect of rounding errors and the iteration error is in v estigated and the relativ e
mo del error b et ween the semilinear and the algebraic model is determined. In
Section 5 a theoretical as w ell as statistical error analysis for the stationary
Euler equations in purely algebraic form is presen ted. Some conclusions are
giv en in Section 6.

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 3
2 The Mo del Hierarc h y
As a mo del problem for the balanced error analysis in a mo del hierarc hy , the
gas flo w through a pip eline is mo deled via the one dimensional Euler equations
that represen t a system of nonlinear h yp erb olic partial differential equations
for the b eha vior of compressible, non-viscous fluids. The mo del consist of,
see e.g. [27], the con tin uity equation, the impulse equation, and the energy
equation, resp ectiv ely ,
∂ ρ
∂ t + ∂
∂ x ( ρv ) = 0 , (1a)
∂
∂ t ( ρv ) + ∂
∂ x ( p + ρv 2 ) = − λ
2 D ρv | v | − g ρh 0 , (1b)
∂
∂ t  ρ ( 1
2 v 2 + e )  + ∂
∂ x  ρv ( 1
2 v 2 + e ) + pv  = − k w
D ( T − T w ) . (1c)
Moreo v er, the state equation for real gases is added, whic h is given b y
p = RρT z ( p, T ) . (2)
In this system of equations the v ariables ha v e the follo wing ph ysical meaning:
ρ is the densit y of the gas, t is the time, v the velocity of the gas, x the space
co ordinate along the pip eline, p the pressure of the gas, λ the pip e friction
co efficien t, D the diameter of the pip eline, g the gra vitational constan t, h
the heigh t of the pip eline, h 0 ( x ) the slop e of the pip eline, c v the v olumetric
heat capacit y , e = c v T + g h the in ternal (thermal plus p oten tial) energy , T
the temp erature of the gas, k w the heat conductivit y co efficient, T w the w all
temp erature of the pip eline, and R the gas constan t. Finally , z ( p, T ) denotes
the compressibilit y factor for whic h w e use the mo del of the American Gas
Asso ciation (A GA)
z ( p, T ) = 1 + 0 . 257 p
p c − 0 . 533 pT c
p c T , (3)
where p c and T c denote the pseudo-critical pressure and temp erature, which
pro vides a go o d appro ximation of z for pressures up to 70 bar [2, 36]. The
full Euler equations (ev en in the one-dimensional case (1)) are mathematically
quite in v olv ed and their numerical solution requires large computational effort.
F or this reason, in particular when the solution is part of an optimization
pro cedure, usually sev eral simplifications are made. Suc h simplifications are
e.g. to use an appro ximate semilinear mo del as deriv ed in Subsection 2.1 or a
purely algebraic mo del as considered in Subsection 2.2.

4 J. J. Stol wijk and V. Mehrmann
2.1 Deriv ation of the Semilinear Isothermal Mo del
Starting from the full one dimensional Euler equations (1), to derive the
isothermal mo del, the temp erature T = T 0 is assumed to b e constan t within
the pip eline, suc h that the energy equation (1c) can b e dropp ed and the
isothermal Euler equations (1a) and (1b) are obtained. A ccording to the In-
ternational Standard Metric Conditions for natural gas [21], the v alue T 0 =
15 . 00 ◦ C (whic h is equal to 288 . 15 K) is tak en for this constant temperature.
Then, in the isothermal case, the compressibilit y factor z in the A GA mo del (3)
only dep ends on p and w e get
z ( p ) = 1 + αp, with α = 0 . 257
p c − 0 . 533 T c
p c T 0
. (4)
If one also assumes that this compressibilit y factor z ( p ) = z 0 is constan t in p ,
then one can use the a v erage
z 0 = z (0) + z (70 bar ) | T 0 =15 ◦ C
2 = 0 . 928
as its v alue. F or constan t temp erature T 0 and compressibilit y factor z 0 , the
state equation for real gases (2) then reduces to
p ( ρ ) = RT 0 z 0 ρ. (5)
If also the en trop y of the gas is assumed to b e constant, whic h is a reasonable
assumption when the temp erature of the gas is constan t [12, p. 7], then the
sp eed of sound is giv en b y c = p ∂ p/∂ ρ , see also [27, (14.32)]. F rom (5) it
follo ws that
c = p RT 0 z 0 = p p/ρ. (6)
Hence, w e ha v e ρ = p/c 2 , and inserting this in (1b), the momen tum equation
can b e rewritten as
∂
∂ t ( ρv ) + ∂
∂ x ( p (1 + v 2 /c 2 )) = − λ
2 D ρv | v | − g ρh 0 .
As further simplifications often the term v 2 /c 2 is neglected in the case of small
gas flo w v elo cities v , see [31, 32], and it is assumed that h 0 ( x ) ≡ 0 , i.e., the
pip eline is assumed to b e (essen tially) horizon tal. These simplifications result
in the isothermal semiline ar mo del
∂ ρ
∂ t + ∂
∂ x ( ρv ) = 0 , (7a)
∂
∂ t ( ρv ) + ∂ p
∂ x = − λ
2 D ρv | v | . (7b)

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 5
In tro ducing the mass flow r ate q = Aρv , with a constan t cross-sectional
area A , and using (6), system (7) ma y b e rewritten in the form
∂ p
∂ t + c 2
A
∂ q
∂ x = 0 , (8a)
∂ q
∂ t + A ∂ p
∂ x = − λc 2
2 D A
q | q |
p , (8b)
q ( x R , t ) = q s ( t ) , (8c)
p ( x L , t ) = p s ( t ) , (8d)
where as b oundary conditions the mass flo w rate is prescrib ed by q s ( t ) at the
right-hand side of the pip eline x R and the pressure is prescrib ed b y p s ( t ) at
the left-hand side of the pip eline x L .
When considering all these drastic mo del simplifications it has to b e an-
alyzed whether these p erturbations in the mo del ha ve a large effect on the
sim ulation results.
2.2 Deriv ation of the Algebraic Mo del
Another simplified mo del (presen ted in an ev en more reduced form in [7]) is
obtained b y neglecting the terms ∂
∂ x ( ρv 2 ) , ∂
∂ x ( ρv 3 ) , and ∂
∂ t ( ρv ) in (1). This
results in the mo del
∂ ρ
∂ t + ∂
∂ x ( ρv )=0 , (9a)
∂ p
∂ x = − λ
2 D ρv | v | − g ρh 0 , (9b)
∂
∂ t ( ρe ) + ∂
∂ x ( ρv e + pv ) = − k w
D ( T − T w ) . (9c)
If, as further simplification, a stationary mo del is assumed, i.e., the time-
deriv ativ es ∂
∂ t are set to zero, the pip eline is again assumed to b e horizon tal,
i.e., h 0 = 0 , and the compressibilit y factor z is set to b e constant, then a set
of ordinary differen tial equations is obtained, which can b e solv ed analytically
via
ˆ q = ρ in v in , (10a)
p ( x ) = r p 2
in − λc 2
2 r ρv | ρv | ( x − x 0 ) , (10b)
T ( x ) = ( T in − T w ) e − k w
D c v ρv ( x − x 0 ) + T w . (10c)
Here the constan t ˆ q = ρv is the mass flux, ρ in is the inlet densit y , v in is the
inlet v elo cit y , p in is the inlet pressure, c is the constan t sp eed of sound, r is

6 J. J. Stol wijk and V. Mehrmann
the radius of the pip eline, x 0 is the starting p oint of the pipeline, and T in
is the inlet temp erature. Equations (10) are referred to as the temp er atur e
dep endent algebr aic mo del of the one dimensional Euler equations. Again an
isothermal simplification is obtained b y taking the temp erature T constan t.
This lea v es us with (10a) and (10b), whic h are referred to as the isothermal
algebr aic mo del . A detailed deriv ation of this mo del is given in [14]. In the
optimization of gas net w orks, this nonlinear algebraic mo del is often further
appro ximated b y piecewise linear functions, see e.g. [18] and [24, p. 115]. Al-
though usually within the optimization metho ds the appro ximation accuracy
b y the piecewise linear appro ximations is con trolled, the mo deling error of the
nonlinear algebraic mo del is usually not considered. If this mo deling error is
large, then the linear relaxation tec hniques used in the optimization metho ds
inevitably lead to inaccurate results. This motiv ates our conside ration of the
mo del error in Subsection 4.5.
The discussed mo del hierarc h y is depicted schematically in Figure 2. W e
will analyze the errors in this mo del hierarc h y , ho w ev er, it should b e clear that
the analysis can b e extended b y considering all simplifications separately and
b y also starting from a more detailed original mo del.
Euler equations (1)
T emp erature dep enden t
algebraic mo del (10)
[ ∂
∂ x ( ρv 2 ) , ∂
∂ x ( ρv 3 ) , ∂
∂ t , h 0 ]=0 ,
z = const.
Semilinear isother-
mal mo del (7)
Isothermal algebraic
mo del (10a) and (10b)
∂
∂ t = 0
T = const.
[ ∂
∂ x ( ρv 2 ) , h 0 ]=0 ,
[ z , T ] = const.
Figure 2: T w o lev el mo del hierarc hy for the sim ulation of flo ws in gas net works.
3 Error Analysis and Conditioning
The aim of err or analysis is to construct an estimate or upp er b ound of the
effect that mo deling, rounding, data uncertaint y , and discretization errors ha v e
on the solution of a giv en problem, see e.g. [6, 20, 35]. The term c ondition
numb er is used to describ e the sensitivit y of problems to uncertain ties in the
input parameters [35]. Supp ose that the solution of a problem is obtained by

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 7
ev aluating the function of a single v ariable f ( d ) . Denoting the deriv ativ e of f
with resp ect to d b y f 0 ( d ) , then the quan tity [20]
κ rel ( f ; d ) =    
d f 0 ( d )
f ( d )     ,
with | . | denoting the absolute v alue, is the r elative c ondition numb er of f and
it measures, for small p erturbations ∆ d , the relativ e c hange in the output for
a giv en relativ e c hange in the input. If, on the other hand, the solution of
a problem is obtained b y ev aluating a function of sev eral v ariables f ( d ) with
d ∈ R n , the quan tities [26]
κ rel ( f ; d i ) =    
∂ f ( d )
∂ d i
d i
f ( d )     , i = 1 , . . . , n, (11)
are the individual r elative c ondition numb ers of f with resp ect to d i . Using
the classical concepts of bac kw ard and forw ard error, w e hav e the follo wing
rule of th um b [20],
forw ard error . condition n um b er × bac kward error ,
with . meaning "less than or equal to except for higher order terms". It
insigh tfully sho ws that despite of a small bac kw ard error (which is often giv en
b y the residual), a problem can ha ve a large forw ard error due to a high
condition n um b er.
F or systems of nonlinear equations, norm wise relativ e condition n umbers
w ere first studied in [40], and the results are extended and summarized in [20].
W e develop new component wise condition n umbers for nonlinear systems of
equations
F ( x ; d )=0 (12)
where F : D x × D d → C m with D x × D d an op en subset of C m × C n .
Giv en a solution x ∗ ∈ C m w e are in terested in the sensitivit y of x ∗ with
resp ect to p erturbations in the data v ector d ∈ C n , i.e., w e are interested in
the condition n um b er κ rel ( x ∗ ; d ) of x ∗ with resp ect to p erturbations ˜
d in the
data d . So instead of (12) one solv es the problem
F ( ˜
x ; ˜
d )=0 , (13)
and w e determine a relation b et ween the norms || ˜
x ∗ − x ∗ || for the solutions
˜
x ∗ , x ∗ and the deviation in the data || ˜
d − d || , where the norm || · || should b e
c hosen suc h that it fits the problem [40, p. 374]. T aylor series expansion giv es
F ( ˜
x ∗ ; ˜
d ) = F ( x ∗ ; d ) + F 0
x ( x ∗ ; d )( ˜
x ∗ − x ∗ ) + F 0
d ( x ∗ ; d )( ˜
d − d ) + O (∆ 2 ) , (14)

8 J. J. Stol wijk and V. Mehrmann
where F 0
x and F 0
d denote the Jacobians of F with resp ect to x ∗ and d and ∆
is either ˜
x ∗ − x ∗ or ˜
d − d . Since b oth F ( x ∗ ; d ) = 0 and F ( ˜
x ∗ ; ˜
d ) = 0 , (14)
can b e rewritten as
F 0
x ( x ∗ ; d )( ˜
x ∗ − x ∗ ) = − F 0
d ( x ∗ ; d )( ˜
d − d ) + O (∆ 2 ) , (15)
If F 0
x is in v ertible and b ounded in ( x ∗ ; d ) , then we obtain that
|| ˜
x ∗ − x ∗ || ≤ || F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) || || ˜
d − d || + O ( || ∆ || 2 ) , (16)
so that
|| ˜
x ∗ − x ∗ ||
|| x ∗ || . || d || || F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) ||
|| x ∗ || || ˜
d − d ||
|| d || , (17)
where the matrix norm || F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) || is the one induced b y the
v ector norm. F rom (16) and (17) it follo ws that the norm wise absolute and
relativ e condition n um b ers of the solution x ∗ with resp ect to the data d are
giv en b y
κ abs,n ( x ∗ ; d ) = || F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) || (18)
and, see [40, p. 377] and [20, Eq. (25.11)],
κ rel,n ( x ∗ ; d ) = || d || || F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) ||
|| x ∗ || , (19)
resp ectiv ely .
Considering individual comp onen ts, one can determine the sensitivit y of
the i -th comp onen t x ∗
i of the solution v ector x ∗ of the problem (12) with
resp ect to small p erturbations in the data v ector d .
Rewriting (15) as
˜
x ∗ − x ∗ = − F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )( ˜
d − d ) + O (∆ 2 ) , (20)
for the i -th comp onen t ( ˜
x ∗ − x ∗ ) i of the v ector ˜
x ∗ − x ∗ w e obtain
( ˜
x ∗ − x ∗ ) i = −  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )( ˜
d − d )  i + O (∆ 2 )
= −  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  i, : ( ˜
d − d ) + O (∆ 2 ) , (21)
where M i, : denotes the i -th ro w of the matrix M . The triangle inequality giv es
| ( ˜
x ∗ − x ∗ ) i | .     F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  i, : ( ˜
d − d )    ,
and the Cauc h y-Sc hw arz inequalit y [39, p. 107] results in first order b ounds
for the absolute error
| ˜ x ∗
i − x ∗
i | . 

  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  T
i, : 

 2 

 ˜
d − d 

 2 (22)

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 9
and the relativ e error
| ˜ x ∗
i − x ∗
i |
| x ∗
i | . k d k 2 

  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  T
i, : 

 2
| x ∗
i | 

 ˜
d − d 

 2
k d k 2
. (23)
F rom (22) and (23) it follows that the absolute condition n um b er κ abs ( x ∗
i ; d )
and the relativ e condition n um b er κ rel ( x ∗
i ; d ) of comp onen t x ∗
i with resp ect to
d are giv en b y
κ abs ( x ∗
i ; d ) = 

  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  T
i, : 

 2 (24)
and
κ rel ( x ∗
i ; d ) = k d k 2 

  F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d )  T
i, : 

 2
| x ∗
i | , (25)
resp ectiv ely .
Let us no w consider, analogous to [9, Example 3.7], comp onent wise condi-
tion n um b ers in b oth the input parameters and the output parameters. Since
w e are in terested in the maxim um comp onent wise error in the output param-
eters, w e tak e the infinit y norm in (20), which yields
k ∆ x ∗ k ∞ . 
 F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) 
 ∞ k ∆ d k ∞ ,
where ∆ x ∗ = ˜
x ∗ − x ∗ and ∆ d = ˜
d − d . Th us, the component wise absolute
condition n um b er is given b y
κ abs,c ( x ∗ ; d ) = 
 F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) 
 ∞ . (26)
In order to deriv e the comp onen twise relativ e condition n umber, w e define
matrices D x ∗ = diag ( x ∗
i ) and D d = diag ( d i ) , analogous to [9, Def. 3.1]. Then,
(20) is equiv alen t to
D − 1
x ∗ ∆ x ∗ = − D − 1
x ∗ F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) D d D − 1
d ∆ d + O (∆ 2 ) ,
where w e assume that all comp onen ts of x ∗ and d are nonzero, such that the
in v erses D − 1
x ∗ and D − 1
d exist. T aking the infinity norm again yields

 D − 1
x ∗ ∆ x ∗ 
 ∞ . 
 D − 1
x ∗ F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) D d 
 ∞ 
 D − 1
d ∆ d 
 ∞ . (27)
Hence, the c omp onentwise r elative c ondition numb er is giv en b y
κ rel,c ( x ∗ ; d ) = 
 D − 1
x ∗ F 0
x ( x ∗ ; d ) − 1 F 0
d ( x ∗ ; d ) D d 
 ∞ . (28)
Note that b y c ho osing the infinity norm in (19), w e ha ve the relation
κ rel,c ( x ∗ ; d ) ≤ κ rel,n ( x ∗ ; d ) (29)
due to the sub-m ultiplicativit y of the infinit y norm.

10 J. J. Stol wijk and V. Mehrmann
Remark. If the nonline ar system (12) is solve d using (a variant of ) Newton ’s
metho d, then in e ach Newton iter ation the line ar system
F 0 ( x j )∆ x j = − F ( x j ) , (30)
has to b e solve d, wher e ∆ x j = x j +1 − x j . It is shown in [40] that the data
unc ertainty err or in x ∗ dep ends on the sensitivity of the nonline ar system (12)
and not on the sensitivity of the line ar system (30).
While the condition n um b er leads to a first order worst-case perturbation
b ound, a statistical analysis results in an a v erage p erturbation estimate. The
statistical analysis in this pap er is p erformed using the Univ ariate Reduced
Quadrature (UR Q) metho d [34]. This metho d presen ts a con venien t trade-
off b et w een computational complexity and accuracy . In con trast to the large
sample size that is required for a Mon te Carlo Sim ulation (MCS), the URQ
metho d only utilizes a sample size of N = 2 n + 1 , where n is the n umber of
input parameters. This mak es the URQ method computationally muc h less
exp ensiv e than a MCS. The mean µ f and the v ariance σ 2
f of an output param-
eter f are appro ximated in the UR Q metho d using the quadrature form ulas
in [34, (20) and (21)].
Ha ving established norm wise and comp onen twise condition n um b ers for
general nonlinear systems of equations, in the next sections w e apply these
results to study the sensitivit y of the t w o classes of Euler equations with
resp ect to p erturbations in the data.
4 Error Analysis for the Semilinear Isothermal Mo del
In this section, an error analysis is p erformed for the isothermal Euler equa-
tions in semilinear form, called the semiline ar mo del. Subsections 4.1 and 4.2
discuss t w o simple discretization sc hemes applied to the semilinear mo del (7).
These simple discretization sc hemes, here called the 1S-scheme and the MP-
scheme , are t ypically used in the optimization of large gas netw orks, see [17,
30]. A theoretical and statistical sensitivity analysis for both systems is pre-
sen ted in Subsection 4.3. A rounding and iteration error analysis for the t w o
resulting nonlinear systems is con tained in Subsection 4.4 and a first order up-
p er b ound for the relativ e mo del error b et w een the semilinear and the algebraic
mo del is deriv ed in Subsection 4.5.
4.1 Discretization using a One-Sided Ev aluation
F or notational conv enience, we consider one space interv al [ x L , x R ] as a piece of
length H of a pip eline and discretize system (8) first in space. There are many

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 11
differen t p ossibilities to obtain suc h a discretization, here w e approximate the
space deriv ativ e ∂ q /∂ x b y
∂ q
∂ x ≈ q ( x R , t ) − q ( x L , t )
H . (31)
Moreo v er, the pressure p ( x, t ) is ev aluated at x R and the mass flo w rate q ( x, t )
is ev aluated at x L . Inserting the b oundary conditions (8c) and (8d) in to (8a)
and (8b), resp ectiv ely , results in a system of ordinary differen tial equations
(ODEs), whic h is giv en b y
˙ p ( x R , t ) + c 2
AH ( q s ( t ) − q ( x L , t )) = 0 ,
˙ q ( x L , t ) + A
H ( p ( x R , t ) − p s ( t )) = − λc 2
2 D A
q ( x L , t ) | q ( x L , t ) |
p ( x R , t ) .
Using the implicit Euler discretization in time, and in tro ducing the v ectors
x i − 1 = [ p ( x R , t i − 1 ) , q ( x L , t i − 1 )] T and x i = [ p ( x R , t i ) , q ( x L , t i )] T yields the
nonlinear system of equations
F 1 ( x i , d ) = 1
τ ( x i
1 − x i − 1
1 ) + c 2
AH ( q i
s − x i
2 )=0 , (32a)
F 2 ( x i , d ) = 1
τ ( x i
2 − x i − 1
2 ) + A
H ( x i
1 − p i
s ) + λc 2
2 D A
x i
2 | x i
2 |
x i
1
= 0 , (32b)
where the (uncertain) data are collected in the v ector
d = [ A, λ, D , c, p i
s , q i
s , x i − 1
1 , x i − 1
2 ] T . (33)
These are the cross-sectional area A , the Darcy friction factor λ , the diam-
eter D , the sp eed of sound c , the b oundary conditions, and the solution of
the previous time step x i − 1 . The first three parameters are uncertain because
their v alues cannot b e determined accurately for pip elines that lie deep in the
ground for a long p erio d of time. The sp eed of sound c within the gas is
uncertain b ecause the temp erature T and the compressibilit y factor z are set
to a constan t in (6) and th us a mo deling error is made. The b oundary v alues
are sub ject to measuremen t errors (or simulation errors when the pipeline is
split in to smaller pieces), and x i − 1 is uncertain due to the accum ulation of
discretization errors, as well as the rounding and data uncertain ty err ors in
the previous time steps. W e call this discretization sc heme the 1S-scheme in
the follo wing. It is similar to the discretization in [17, 30]; the only difference
is that p and q are there b oth ev aluated in x R , giv en that the gas flo ws from
x L to x R .

12 J. J. Stol wijk and V. Mehrmann
Equations (32) define a t w o-dimensional nonlinear system with solution x i .
The Jacobian F 0
x i ( x i , d ) of F = [ F 1 , F 2 ] T with resp ect to x i is giv en b y
F 0
x i ( x i , d ) = 

1
τ − c 2
AH
A
H − λc 2
2 D A
x i
2 | x i
2 |
( x i
1 ) 2
1
τ + λc 2
D A
| x i
2 |
x i
1

 . (34)
F or the solution x i of the nonlinear system (32) we use Newton’s metho d, see
e.g. [23], with stopping criterion

 x i
j − x i
j − 1 
 ∞ ≤ tol . (35)
In our sim ulations that w e presen t b elow w e use tol = 10 − 3 and the concrete
parameters v alues
τ = 15 s , p i − 1
R = 5 · 10 6 P a , q i − 1
L = 300 kg/s , (36a)
c = p RT 0 z 0 = √ 518 . 3 · 288 . 15 · 0 . 928 = 372 . 28 m/s, see (6) , (36b)
H = 500 m , p i
s = 5 . 07 · 10 6 P a , q i
s = 302 kg/s , (36c)
A = π
4 m 2 , λ = 0 . 06 , D = 1 m , (36d)
and starting v alues x i
0 = [5 · 10 6 P a , 300 kg/s ] T . These v alues result in an
appro ximate solution x i whic h is giv en by
x i =  p i
R
q i
L  =  5 . 01 · 10 6 P a
3 . 03 · 10 2 kg/s  . (37)
An imp ortan t question is, ho w sensitive this solution is with respect to small
p erturbations in the uncertain data d in (33). T o determine this sensitivity ,
the Jacobian of F with resp ect to d is computed, whic h is given b y
F 0
d ( x i , d ) =





















c 2
A 2 H ( x i
2 − q i
s ) 1
H ( x i
1 − p i
s ) − λc 2
2 D A 2
x i
2 | x i
2 |
x i
1
0 c 2
2 D A
x i
2 | x i
2 |
x i
1
0 − λc 2
2 D 2 A
x i
2 | x i
2 |
x i
1
2 c
AH ( q i
s − x i
2 ) λc
D A
x i
2 | x i
2 |
x i
1
0 − A
H
c 2
AH 0
− 1
τ 0
0 − 1
τ





















T
. (38)
The results of the sensitivit y analysis are presen ted in Section 4.3.

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 13
4.2 Discretization using the Midp oin t Rule
As an alternativ e space discretization of the system (8) w e use the midp oin t
rule for the pressure p ( x, t ) and the mass flo w rate q ( x, t ) , e.g., for p ( x, t ) w e
obtain
p ( x, t ) ≈ p ( x R , t ) + p ( x L , t )
2 .
Moreo v er, the b oundary conditions (8c), (8d) are inserted in to (8a), (8b). This
results in the system of ODEs
1
2 ˙ p ( x R , t ) + c 2
AH ( q s ( t ) − q ( x L , t )) + 1
2 ˙ p s ( t )=0 ,
1
2 ˙ q ( x L , t ) + A
H ( p ( x R , t ) − p s ( t )) + 1
2 ˙ q s ( t ) +
λc 2
4 D A
( q s ( t ) + q ( x L , t )) | q s ( t ) + q ( x L , t ) |
p ( x R , t ) + p s ( t ) = 0 .
Using again the implicit Euler sc heme for the time discretization yields the
nonlinear system
F 1 ( x i , d ) = 1
τ ( x i
1 − x i − 1
1 ) + 2 c 2
AH ( q i
s − x i
2 ) + ˙ p i
s = 0 , (40a)
F 2 ( x i , d ) = 1
τ ( x i
2 − x i − 1
2 ) + 2 A
H ( x i
1 − p i
s ) + ˙ q i
s
+ λc 2
2 D A
( q i
s + x i
2 ) | q i
s + x i
2 |
x i
1 + p i
s
= 0 , (40b)
with data v ector
d = [ A, λ, D , c, p i
s , q i
s , ˙ p i
s , ˙ q i
s , x i − 1
1 , x i − 1
2 ] T . (41)
W e call this discretization scheme the MP-scheme . It is equiv alen t to the
implicit b o x sc heme in [25]. The Jacobian F 0
x i of F = [ F 1 , F 2 ] T with resp ect
to x i in this case is giv en b y
F 0
x i ( x i , d ) = 

1
τ − 2 c 2
AH
2 A
H − λc 2
2 D A
( q i
s + x i
2 ) | q i
s + x i
2 |
( x i
1 + p i
s ) 2
1
τ + λc 2
D A
| q i
s + x i
2 |
x i
1 + p i
s

 . (42)
W e again use Newton’s metho d for the solution x i of (40) with stopping crite-
rion (35). F or the n umerical sim ulations presented below w e use the parameter
v alues (36),
˙ p i
s = 100 , ˙ q i
s = 0 . 05 , (43)

14 J. J. Stol wijk and V. Mehrmann
and the starting v alues x i
0 = [5 · 10 6 P a , 300 kg/s ] T . This results in an appro x-
imate solution x i giv en b y
x i =  p i
R
q i
L  =  5 . 01 · 10 6 P a
3 . 03 · 10 2 kg/s  . (44)
In order to determine the sensitivit y of x i with resp ect to p erturbations in
the data, the Jacobian of the function F with resp ect to d in (41) is calculated
as
F 0
d ( x i , d ) =




























2 c 2
A 2 H ( x i
2 − q i
s ) 2
H ( x i
1 − p i
s ) − λc 2
2 D A 2
( q i
s + x i
2 ) | q i
s + x i
2 |
x i
1 + p i
s
0 c 2
2 D A
( q i
s + x i
2 ) | q i
s + x i
2 |
x i
1 + p i
s
0 − λc 2
2 D 2 A
( q i
s + x i
2 ) | q i
s + x i
2 |
x i
1 + p i
s
4 c
AH ( q i
s − x i
2 ) λc
D A
( q i
s + x i
2 ) | q i
s + x i
2 |
x i
1 + p i
s
0 − 2 A
H − λc 2
2 D A
( q i
s + x i
2 ) | q i
s + x i
2 |
( x i
1 + p i
s ) 2
2 c 2
AH
λc 2
D A
| q i
s + x i
2 |
x i
1 + p i
s
1 0
0 1
− 1
τ 0
0 − 1
τ




























T
. (45)
The sensitivit y results are presen ted in Section 4.3.
4.3 Sensitivit y Analysis for the t w o Discretizations
This subsection con tains b oth a w orst case first order and a statistical sensi-
tivit y analysis for the 1S- and the MP- discretization sc heme.
W e use the Jacobians in (34), (38), (42), and (45) to calculate the individ-
ual condition n um b ers (25) of the comp onen ts p i
R and q i
L of the solutions x i
in (37) and (44) with resp ect to p erturbations in the uncertain data. Using the
parameter v alues in (36) and (43) w e obtain the results presen ted in T able 1.
The largest individual condition n um b er is κ rel ( q i
L ; p i
s ) for b oth sc hemes.
Moreo v er, one observ es that the mass flow rate q i
L is more sensitiv e to small
p erturbations in the parameters than the pressure p i
R . W e note that a scaling
of the parameter v alues or using differen t units, e.g. b y choosing the unit
metric ton rather than kg, do es not c hange the results.

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 15
T able 1: Individual relative condition n um b ers in (25) for the 1S- and the MP-
sc heme. The condition n um b ers are computed for the solution comp onen ts p i
R
and q i
L with resp ect to the uncertain input parameters d in (33) and (41). The
v alues in (36) and (43) are used.
κ 1S
rel κ MP
rel
p i
R q i
L p i
R q i
L
A 2 . 30 · 10 − 2 7 . 65 · 10 − 2 2 . 40 · 10 − 2 4 . 07 · 10 − 2
λ 1 . 15 · 10 − 2 3 . 60 · 10 − 2 1 . 20 · 10 − 2 1 . 88 · 10 − 2
D 1 . 15 · 10 − 2 3 . 60 · 10 − 2 1 . 20 · 10 − 2 1 . 88 · 10 − 2
c 2 . 28 · 10 − 2 8 . 09 · 10 − 2 2 . 40 · 10 − 2 4 . 38 · 10 − 2
p i
s 9 . 44 · 10 − 1 2 . 94 1 . 00 1 . 57
q i
s 2 . 53 · 10 − 2 9 . 16 · 10 − 1 2 . 53 · 10 − 2 9 . 57 · 10 − 1
˙ p i
s − − 6 . 23 · 10 − 6 4 . 58 · 10 − 4
˙ q i
s − − 3 . 13 · 10 − 6 4 . 89 · 10 − 6
p i − 1
R 7 . 93 · 10 − 2 2 . 87 2 . 08 · 10 − 2 1 . 53
q i − 1
L 2 . 37 · 10 − 3 7 . 39 · 10 − 3 1 . 25 · 10 − 3 1 . 96 · 10 − 3
Calculating the norm wise relativ e condition n umbers (19) of x i with re-
sp ect to d in (33) and (41) for the 1S- and the MP-sc heme with parameter v al-
ues (36) and (43) yields κ 1S
rel,n ( x i ; d )=1 . 39 · 10 6 and κ MP
rel,n ( x i ; d )=1 . 45 · 10 6 .
These norm wise condition n um b ers are at least three orders of magnitude
larger than the individual condition n um b ers in T able 1. Hence, the norm wise
condition n um b er considerably ov erestimates the sensitivit y of the corresp ond-
ing nonlinear ro ot finding problem, i.e., it constitutes a v ery p essimistic upp er
b ound. Considering also (29), this leads to the conclusion that it is more ad-
equate to use the comp onen t wise relative condition n um b er (28) in order to
determine the sensitivit y of x i with resp ect to d .
The comp onen t wise condition num b ers for the tw o differen t discretization
sc hemes are calculated for spatial stepsizes H b et ween 1 and 1 000 m and for
temp oral stepsizes τ b et ween 10 − 2 and 30 s. The results are depicted as t wo
con tour graphs in Figure 3. One sees that giv en H and τ , the MP-sc heme has a

16 J. J. Stol wijk and V. Mehrmann
smaller condition n um b er than the 1S-scheme. Also, using Figure 3 for a giv en
discretization sc heme, the spatial stepsize H and the temp oral stepsize τ can
b e c hosen suc h that the corresp onding nonlinear system has a lo w condition
n um b er. Note that while reducing H and τ decreases the discretization error,
it increases the sensitivit y of the problem and th us b oth the error due to data
uncertain t y and the effect of rounding are amplified. Hence, a balance b et ween
the discretization and the data uncertain t y error should b e determined to find
appropriate v alues for H and τ .
200 400 600 800
0
10
20
3
6
9
12
15

Spatial stepsize H [m]
T emp oral stepsize τ [s]
κ 1S
rel,c  x i ; d 
200 400 600 800
0
10
20
3
6
9
12
15

H [m]
τ [s]
κ MP
rel,c  x i ; d 
Figure 3: Con tour graphs for the comp onen twise relativ e condition n um-
b ers (28) of the 1S-sc heme (left) and the MP-sc heme (right) as a function of
the spatial and temp oral stepsize. The condition n um b er is computed for the
solution x i = [ p i
R , q i
L ] T with resp ect to the uncertain data d in (33) and (41).
The lines are con tour lines with the n um b ers denoting the v alues of κ rel,c .
W e also p erform a statistical error analysis for the 1S- and the MP-sc heme
using the UR Q metho d, see Section 3. The relativ e standard deviations
σ d i /µ d i for the input parameters is set to 0.5 % . Subsequen tly , the relative
standard deviations σ x i
k /µ x i
k , k = 1 , 2 , of the solution are computed. The
uncertain t y amplification factor
φ = max
k ∈{ 1 , 2 } ( σ x i
k /µ x i
k
σ d i /µ d i ) (46)
is calculated for H ∈ [1 , 1 000 m ] and τ ∈ [10 − 2 , 30 s ] . The results are depicted
as con tour graphs in Figure 4. The results are similar to the theoretical results

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 17
in Figure 3. Since φ is a measure for the mean amplification factor and the
condition n um b er is a first order w orst case b ound, w e observ e that the v alues
in Figure 4 are smaller than the v alues in Figure 3.
200 400 600 800
0
10
20
3
6
9
12
15

Spatial stepsize H [m]
T emp oral stepsize τ [s]
φ 1S
200 400 600 800
0
10
20
3
6
9
12
15

H [m]
τ [s]
φ MP
Figure 4: Con tour graphs for the amplification factor φ in (46), calculated
with the UR Q metho d, for the 1S-scheme (left) and MP-sc heme (right) as a
function of H and τ . The v alues in (36) are tak en for the mean v alues of
the uncertain input parameters. The lines are con tour lines with the n umbers
denoting the v alues of φ .
4.4 Rounding and Iteration Error Analysis
In this subsection a first order upp er b ound for the rounding errors and the
iteration error that are committed in the Newton metho d is deriv ed. The
result is applied to b oth the 1S- and the MP-sc heme.
A rounding error analysis for the solution of the linear system (30) arising
in Newton’s metho d is presen ted in [41], together with a condition for whic h
the in termediate solution x j cannot b e impro v ed due to rounding errors, see
[41, p. 117]. If e j is an upp er b ound for the rounding error || F ( x j ) − ˜
F ( x j ) || ,
F is the exact function ev aluation, and ˜
F is the computed function ev aluation,
then this condition is giv en b y
|| ˜
F ( x j ) || ≤ e j or || ˜
F ( x j ) || ≥ || ˜
F ( x j − 1 ) || . (47)
In the follo wing w e assume that the first condition of (47) is satisfied b efore

18 J. J. Stol wijk and V. Mehrmann
the second condition of (47). Define the set
S := { x ∈ R n | || ˜
F ( x ) || ≤ e j } (48)
and assume that x ∗ ∈ S . Then, provided that the Jacobian of ˜
F is in v ertible
for all x ∈ S , it follo ws from the implicit function theorem that
k x j − x ∗ k . || [ ˜
F 0 ( x j )] − 1 || || ˜
F ( x j ) − ˜
F ( x ∗ ) ||
≤ || [ ˜
F 0 ( x j )] − 1 ||  || ˜
F ( x j ) || + || ˜
F ( x ∗ ) || 
≤ || [ ˜
F 0 ( x j )] − 1 ||  || ˜
F ( x j ) || + e j  (49)
for all x j ∈ S . It is w ell-kno wn, see e.g. [23, Theorem 5.1.1], that outside of S
and under certain assumptions on the function F Newton’s metho d con v erges
quadratically , i.e., k x j +1 − x ∗ k = O ( k x j − x ∗ k 2 ) . This implies that
k x j − x ∗ k≤k x j − x j +1 k + O ( k x j − x ∗ k 2 ) , for all x j / ∈ S . (50)
Th us, w e ha ve the follo wing theorem.
Theorem 1. Given a solution x j of the nonline ar system arising in the gas
flow simulation that is c ompute d with the Newton metho d. L et η ri denote the
err or in x j b oth due to r ounding err ors in the solution of (30) and due to the
pr eliminary stopping of the Newton iter ation. L et e j b e an upp er b ound for
the r ounding err or || F ( x j ) − ˜
F ( x j ) || , then
η ri . ( || [ ˜
F 0 ( x j )] − 1 ||  || ˜
F ( x j ) || + e j  if x j ∈ S ,
k x j − x j +1 k if x j / ∈ S . (51)
Pr o of. If x j / ∈ S , then Newton’s metho d conv erges quadratically despite round-
ing errors in the solution of (30), see [41, p. 117]. Th us, η ri is giv en by (50).
On the other hand, if x j ∈ S , then condition (47) is satisfied and w e do not
ha v e quadratic con vergence. An upp er b ound for η ri is then giv en b y (49).
T o apply this result, we compute η ri for the 1S- and the MP-sc heme. F or
the 1S-sc heme w e ma y write
˜
F 1 ( x i , d ) = x i
1 − x i − 1
1
τ (1 + 3 ε ) + c 2 ( q i
s − x i
2 )
AH (1 + 6 ε ) ,
˜
F 2 ( x i , d ) = x i
2 − x i − 1
2
τ (1 + 4 ε ) + A ( x i
1 − p i
s )
H (1 + 5 ε ) + λc 2 x i
2 | x i
2 |
2 D Ax i
1
(1 + 8 ε ) ,

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 19
where | ε | ≤ u and u denotes the unit roundoff. Hence, w e ha v e
   F 1 − ˜
F 1    ≤    
x i
1 − x i − 1
1
τ     3 u +    
c 2 ( q i
s − x i
2 )
AH     6 u =: α 1S ,
   F 2 − ˜
F 2    ≤    
x i
2 − x i − 1
2
τ     4 u +    
A ( x i
1 − p i
s )
H     5 u +    
λc 2 x i
2 | x i
2 |
2 D Ax i
1     8 u =: β 1S
and, th us, e 1S
j = 
 [ α 1S , β 1S ] T 
 .
F or the MP-scheme w e obtain
˜
F 1 = x i
1 − x i − 1
1
τ (1 + 4 ε ) + 2 c 2 ( q i
s − x i
2 )
AH (1 + 7 ε ) + ˙ p i
s (1 + ε ) ,
˜
F 2 = x i
2 − x i − 1
2
τ (1 + 5 ε ) + 2 A ( x i
1 − p i
s )
H (1 + 6 ε ) + ˙ q i
s (1 + 2 ε )
+ λc 2 ( q i
s + x i
2 ) | q i
s + x i
2 |
2 D A ( x i
1 + p i
s ) (1 + 11 ε ) ,
and, hence
   F 1 − ˜
F 1    ≤    
x i
1 − x i − 1
1
τ     4 u +    
2 c 2 ( q i
s − x i
2 )
AH     7 u +   ˙ p i
s   u =: α MP ,
   F 2 − ˜
F 2    ≤    
x i
2 − x i − 1
2
τ     5 u +    
2 A ( x i
1 − p i
s )
H     6 u +   ˙ q i
s   2 u
+    
λc 2 ( q i
s + x i
2 ) | q i
s + x i
2 |
2 D A ( x i
1 + p i
s )     11 u =: β MP ,
and e MP
j = 
 [ α MP , β MP ] T 
 . Assuming the use of IEEE standard double pre-
cision arithmetic, suc h that u = 2 . 22 · 10 − 16 , see [20, p. 39], and c ho osing the
infinit y norm w e obtain the follo wing error estimates. F or the 1S-scheme
with v alues (36) and solution x i in (37) w e ha v e e 1S
j = 1 . 16 · 10 − 12 and
|| ˜
F ( x i ) || = 1 . 74 · 10 − 11 . Thus, x i is not an elemen t of the set S and w e
ha v e η ri . 1 . 56 · 10 − 10 . F or the MP-sc heme with v alues (43) and solution x i
in (44) w e ha v e e MP
j = 1 . 73 · 10 − 12 and || ˜
F ( x i ) || = 2 . 22 · 10 − 11 . Hence, x i
again is not con tained in S and w e ha ve η ri . 7 . 82 · 10 − 11 .
It can b e concluded that for b oth sc hemes the rounding and iteration errors
can b e neglected in comparison with the data uncertain t y error.
4.5 Mo deling Error b etw een the Semilinear and Algebraic Mo del
In this subsection w e analyze the mo deling error that is committed when the
isothermal semilinear mo del (7) is simplified to the isothermal algebraic mo del
(10a)-(10b) that is obtained b y assuming a stationary gas flo w, see Figure 2.

20 J. J. Stol wijk and V. Mehrmann
W e consider the semilinear and the algebraic mo del on the spatial inter-
v al [0 , L ] , with pip eline length L , and the temp oral in terv al [0 , T ] . W e define
gridp oin ts ( x i , t k ) , i = 0 , . . . , N and k = 0 , . . . , M , with stepsizes H = L/ N
and τ = T / M . Let the solution of the semilinear mo del at the gridp oin ts b e
denoted b y y sem ( x i , t k ) , the solution of the discretized semilinear mo de l with
stepsizes H and τ at the gridp oin ts ( x i , t k ) b e denoted by y k
i ( H , τ ) , and the
solution of the algebraic mo del at the gridp oin ts b e denoted b y y alg ( x i ) , with
y ( x, t )=[ p ( x, t ) , q ( x, t )] T .
W e define the relative model error η m b etw een the semilinear and the
algebraic mo del b y
η m := max
i,k 

 D − 1
y k
i ( H/ 2 ,τ / 2)  y sem ( x i , t k ) − y alg ( x i )  

 ∞ ,
with D x := diag ( x ) . Using the triangle inequalit y , w e ha v e
η m ≤ max
i,k  

 D − 1
y k
i ( H/ 2 ,τ / 2)  y sem ( x i , t k ) − y k
i ( H / 2 , τ / 2)  

 ∞
+ 

 D − 1
y k
i ( H/ 2 ,τ / 2)  y k
i ( H / 2 , τ / 2) − y alg ( x i )  

 ∞  . (54)
The term y sem ( x i , t k ) − y k
i ( H / 2 , τ / 2) in (54) denotes the discretization error of
the semilinear mo del in the gridp oin t ( x i , t k ) . Supp ose that the discretization
sc heme for the semilinear mo del con v erges with order γ in space and order δ
in time. Then, the discretization error has an asymptotic expansion of the
form
y sem ( x i , t k ) − y k
i ( H / 2 , τ / 2) = e ( x i , t k )  ( H / 2) γ + ( τ / 2) δ  + O  H γ +1 + τ δ +1  ,
with co efficien t function e ( x, t ) that is indep endent of H and τ , see [38, p. 114].
Hence, w e ha v e the first order approximations
y sem ( x i , t k ) − y k
i ( H / 2 , τ / 2) .
= e ( x i , t k )  ( H / 2) γ + ( τ / 2) δ  and (55)
y sem ( x i , t k ) − y k
i ( H , τ ) .
= e ( x i , t k )  H γ + τ δ  , (56)
where .
= denotes "is equal to except for higher order terms". Subtracting (55)
from (56) and rewriting yields
e ( x i , t k ) .
= y k
i ( H , τ ) − y k
i ( H / 2 , τ / 2)
H γ + τ δ − ( H / 2) γ − ( τ / 2) δ .
Inserting this in to (55) and (54) results in the first order upp er b ound for the
relativ e mo del error
η m . max
i,k  ( H / 2) γ + ( τ / 2) δ
H γ + τ δ − ( H / 2) γ − ( τ / 2) δ

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 21
· 

 D − 1
y k
i ( H/ 2 ,τ / 2)  y k
i ( H , τ ) − y k
i ( H / 2 , τ / 2)  

 ∞
+ 

 D − 1
y k
i ( H/ 2 ,τ / 2)  y k
i ( H / 2 , τ / 2) − y alg ( x i )  

 ∞  . (57)
In order to apply this result, we discretize the semilinear model with the
1S-sc heme from subsection 4.1. This discretization sc heme is consistent of
order 1 b oth in space and time. The stability of the 1S-sc heme is secured b y
the use of the implicit Euler metho d in time. Hence, w e ha ve con v ergence
of order 1 in space and time, i.e., γ = δ = 1 . Using e.g. concrete v alues
p in = 5 . 06 · 10 6 P a and q = 300 kg/s for the algebraic mo del and v alues (36)
for the semilinear mo del, w e compute the solutions y k
i ( H , τ ) , y k
i ( H / 2 , τ / 2) ,
and y alg ( x i ) . F rom (57), this results for these concrete data in η m . 1 . 16 % .
Ha ving analyzed differen t error sources for the semilinear mo del, in the
next section w e step do wn one lev el in the mo del hierarch y in Figure 2 and
p erform a similar analysis for the algebraic mo del.
5 Error Analysis for the Algebraic Mo del
In this section an error analysis is p erformed for the temp erature dep endent al-
gebraic mo del (10). This analysis is p erformed b oth in terms of bac kw ard and
forw ard errors in Subsection 5.1 and statistically in Subsection 5.2. F urther-
more, it is analyzed in Subsection 5.3 under whic h condition the temp erature
dep enden t mo del can safely b e simplified to the isothermal algebraic mo del.
F urther details and examples can b e found in [29].
5.1 Bac kw ard Error Analysis
In this subsection a bac kw ard error analysis is p erformed for the algebraic
mo del (10). The rounding errors due to finite precision arithmetic and the
uncertain ties in the data are in terpreted as p erturbations in the input param-
eters. Then, the relativ e errors in the output parameters are calculated and
analyzed.
In the equation for the mass flux
ˆ q ( d ) = ρ in v in , (58)
only one m ultiplication is p erformed with relativ e error ε 1 , which yields
˜
ˆ q ( ρ in , v in ) = ρ in (1 + ε ρ in ) v in (1 + ε v in )(1 + ε 1 )
= ρ in v in (1 + ε ρ in + ε v in + ε 1 + O ( ε 2 ))
= ˆ q ( ρ in , v in (1 + ε 2 )) , (59)

22 J. J. Stol wijk and V. Mehrmann
with ε 2 = ε ρ in + ε v in + ε 1 + O ( ε 2 ) . Here, ε ρ in is the relativ e measuremen t error
in ρ in , ε v in the relativ e data error in v in , and | ε 1 | < u the relative error of the
m ultiplication, with u the rounding unit in finite precision arithmetic. F or the
absolute relativ e error in ˆ q , using (59), w e obtain
| ˆ q ( d ) − ˆ q ( d + ∆ d ) |
| ˆ q ( d ) | ≤ | ∂ ˆ q
∂ ρ in
1
ˆ q ( d ) ∆ ρ in
| {z }
0 | + | ∂ ˆ q
∂ v in
1
ˆ q ( d ) ∆ v in | + O  (∆ d ) 2 
= |  ∂ ˆ q
∂ v in
v in
ˆ q ( d ) 
| {z }
ρ in v in
ρ in v in =1
∆ v in
v in
| {z }
ε 2
| + O  (∆ d ) 2 
= | ε 2 | + h.o.t. ≤ | ε ρ in | + | ε v in | + | ε 1 | + h.o.t. ,
where h.o.t. stands for higher order terms in the ε j . Assuming that the round-
off error ε 1 is so small that it can b e neglected in comparison with errors ε ρ in
and ε v in , then w e ha v e the constraint
| ε ρ in | + | ε v in | . e lim ,
where e lim is a limit for the relativ e error in ˆ q .
F or the computation of the pressure
p ( d ) = r p 2
in − λc 2
2 r ρv | ρv | ( x − x 0 ) (60)
w e use Algorithm 1. Using the T a ylor series expansion 1
1 − ε = 1 + ε + O ( ε 2 ) , this
leads to a bac kw ard error due to roundoff errors in finite precision arithmetic
with unit roundoff u , giv en b y
˜ p ( d ) = r ( p in (1 + ε 13 )) 2 − λ (1 + ε 14 ) c 2
2 r ρv | ρv | ( x − x 0 ) , (61)
where
2 | ε 13 | = | ε 1 + ε 11 + 2 ε 12 + O ( ε 2 ) | ≤ 4 u + O ( u 2 ) ,
so that | ε 13 | ≤ 2 u + O ( u 2 ) and | ε 14 | ≤ 13 u + O ( u 2 ) .
In tro ducing relativ e data errors and denoting the relative measuremen t
error for the parameter α b y ε α , con tin uing with (61), gives
˜ p ( d ) 2 = ( p in (1 + ε 15 )) 2 − λ (1 + ε 16 ) c 2
2 r ρv | ρv | ( x (1 + ε x ) − x 0 (1 + ε x 0 )) ,
with
| ε 15 | = | ε p in + ε 13 + O ( ε 2 ) |≤| ε p in | + 2 u + h.o.t.

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 23
Algorithm 1 : Computing the pressure p in (60)
Input: p in , λ , c , D , ρ , v , x , x 0
1: z 1 ← p in · p in
2: z 2 ← c · c
3: z 3 ← λ · z 2
4: z 4 ← 2 · r
5: z 5 ← z 3 /z 4
6: z 6 ← ρ · v
7: z 7 ← x − x 0
8: z 8 ← z 5 · z 6
9: z 9 ← z 8 · | z 6 |
10: z 10 ← z 9 · z 7
11: z 11 ← z 1 − z 10
12: z 12 ← √ z 11
13: p ( d ) ← z 12
Output: p
and
| ε 16 |≤| ε λ | + 2 | ε c | + | ε r | + 2 | ε ρ | + 2 | ε v | + 13 u + h.o.t.
F or the backw ard error of p ( d ) , considered as a function of p in , λ , x , and x 0 ,
w e ha v e the expression
˜ p ( p in , λ, x, x 0 ) = p ( p in (1 + ε 15 ) , λ (1 + ε 16 ) , x (1 + ε x ) , x 0 (1 + ε x 0 )) .
Assuming that v > 0 , the relativ e error in p ( d ) due to p erturbations in the
data d = [ p in , λ, x, x 0 ] T is giv en b y
p ( d ) − p ( d + ∆ d )
p ( d ) =
4
X
i =1
∂ p ( d )
∂ d i
d i
p ( d )
∆ d i
d i
+ O  (∆ d ) 2 
=  p in
p ( d )  2
| {z }
κ rel ( p ; p in )
ε 15 − λc 2 ρ 2 v 2 ( x − x 0 )
4 r p ( d ) 2
| {z }
κ rel ( p ; λ )
ε 16 − λc 2 ρ 2 v 2 x
4 r p ( d ) 2
| {z }
κ rel ( p ; x )
ε x
+ λc 2 ρ 2 v 2 x 0
4 r p ( d ) 2
| {z }
κ rel ( p ; x 0 )
ε x 0 + h.o.t. , (62)
where κ rel ( p ; p in ) , κ rel ( p ; λ ) , κ rel ( p ; x ) , and κ rel ( p ; x 0 ) are the individual relative
condition n um b ers, see (11).

24 J. J. Stol wijk and V. Mehrmann
If w e require that k rel ( p ; p in ) ≤ tol, where tol should dep end on ε p in , then
the algebraic mo del can b e used safely for a maxim um pip eline length
L = x − x 0 ≤ 2 r p 2
in (1 − 1 / tol ) / ( λc 2 ρ 2 v 2 ) .
Cho osing e.g. the concrete nominal v alues d nom giv en b y
p in nom = 2 · 10 5 , λ nom = 0 . 03 , c nom = 343 , r nom = 0 . 5 , (63a)
ρ nom = 1 , v nom = 10 , x nom = 10 5 , and x 0 nom = 0 , (63b)
then κ rel ( p ; x ) = κ rel ( p ; λ ) and κ rel ( p ; x 0 ) = 0 . The relativ e condition n um b ers
κ rel ( p ; p in ) and κ rel ( p ; λ ) grow quic kly with the pip eline length L , see Figure 5.
The t w o graphs ha ve a v ertical asymptote at L = 113 km. Given that w e
require that 
 [ κ rel ( p ; p in ) , κ rel ( p ; λ )] T 
 ∞ ≤ 2 , it can b e concluded for these
concrete data that the algebraic mo del can only b e used safely for pip elines
up to 60 km length.
0 50 100
0
5
10
15
Pip eline length L [km]
κ rel ( p ; p in )
0 50 100
0
2
4
6
8
L [km]
κ rel ( p ; λ )
Figure 5: Individual relativ e condition n umbers κ rel ( p ; p in ) and κ rel ( p ; λ )
in (62) considered as a function of the pip eline length L = x − x 0 , with
nominal v alues (63).
F or the computation of the temp erature
T ( d )=( T in − T w ) e − k w
D c v ρv ( x − x 0 ) + T w , (64)
w e apply Algorithm 2. In tro ducing in ev ery step of Algorithm 2 a relative
error ε , and using T aylor expansion, w e obtain
˜
T ( d ) =  T in (1 + ε 11 − T w
T in
( ε 11 − ε 10 )) − T w (1 + ε 10 )  e − k w (1+ ε 12 )
D c v ρv ( x − x 0 )

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 25
Algorithm 2 Computing the temp erature T in (64)
Input: T in , T w , k w , D , c v , ρ , v , x , x 0
1: z 1 ← T in − T w
2: z 2 ← D · c v
3: z 3 ← z 2 · ρ
4: z 4 ← z 3 · v
5: z 5 ← k w /z 4
6: z 6 ← x − x 0
7: z 7 ← z 5 · z 6
8: z 8 ← e − z 7
9: z 9 ← z 1 · z 8
10: z 10 ← z 9 + T w
11: T ( d ) ← z 10
Output: T
+ T w (1 + ε 10 ) ,
where
| ε 10 | ≤ u ,
| ε 11 | = | ε 1 + ε 8 + ε 9 + ε 10 + O ( ε 2 ) | ≤ 4 u + O ( u 2 ) , and
| ε 12 | = | ε 2 + ε 3 + ε 4 + ε 5 + ε 6 + ε 7 + O ( ε 2 ) | ≤ 6 u + O ( u 2 ) .
Including data errors for the input parameters giv es
˜
T ( d )=( T in (1 + ε 13 ) − T w (1 + ε 14 )) e − k w (1+ ε 15 )
D c v ρv ( x (1+ ε x ) − x 0 (1+ ε x 0 ))
+ T w (1 + ε 14 ) ,
where
1 + ε 13 = 1 + ε T in + ε 11 − T w
T in
( ε 11 − ε 10 ) + O ( ε 2 ) ,
1 + ε 14 = (1 + ε T w )(1 + ε 10 ) = 1 + ε T w + ε 10 + O ( ε 2 ) , and
1 + ε 15 = 1 + ε k w + ε 12 + ε D + ε c v + ε ρ + ε v + O ( ε 2 ) .
This results in the bac kw ard error
˜
T ( T in , T w , k w , x, x 0 ) =
T ( T in (1 + ε 13 ) , T w (1 + ε 14 ) , k w (1 + ε 15 ) , x (1 + ε x ) , x 0 (1 + ε x 0 )) ,

26 J. J. Stol wijk and V. Mehrmann
so that the relativ e error in the temp erature T ( d ) due to finite precision arith-
metic and data errors is giv en b y
∆ T
T ( d ) = T in
T in +  e k w ( x − x 0 )
D c v ρv − 1  T w
| {z }
κ rel ( T ; T in )
ε 13 + T w − T w e − k w ( x − x 0 )
D c v ρv
( T in − T w ) e − k w ( x − x 0 )
D c v ρv + T w
| {z }
κ rel ( T ; T w )
ε 14 −
( T in − T w )( x − x 0 ) k w
D c v ρv  T in + ( e k w ( x − x 0 )
D c v ρv − 1) T w 
| {z }
κ rel ( T ; k w )
ε 15 − ( T in − T w ) k w x
D c v ρv T ( d ) e − k w ( x − x 0 )
D c v ρv
| {z }
κ rel ( T ; x )
ε x
+ ( T in − T w ) k w x 0
D c v ρv T ( d ) e − k w ( x − x 0 )
D c v ρv
| {z }
κ rel ( T ; x 0 )
ε x 0 + O  (∆ d ) 2  , (65)
with ∆ T = T ( d ) − T ( d + ∆ d ) .
T o see whether the relative errors ε 13 , ε 14 , ε 15 , ε x , and ε x 0 in the input pa-
rameters are amplified in the relativ e error for the temp erature T , we consider
e.g. the concrete nominal v alues
D nom = 1 , ρ nom = 1 , v nom = 10 , T in nom = 293 , (66a)
T w nom = 283 , k w nom = 0 . 0341 , c v nom = 1700 , and x 0 nom = 0 , (66b)
in the individual relativ e condition n um b ers in (65). With x 0 nom = 0 , then
κ rel ( T ; x 0 ) = 0 and the four remaining relativ e condition n umbers are depicted
in Figure 6 as a function of the pip eline length L = x − x 0 . The figure shows
that all relativ e condition n um b ers remain b elo w one, which means that the
relativ e errors in the input parameters are not amplified. The relativ e condi-
tion n um b ers κ rel ( T ; k w ) and κ rel ( T ; x ) are so small compared to κ rel ( T ; T in )
and κ rel ( T ; T w ) that they can b e neglected.
Our bac kw ard analysis and the computation of the asso ciated condition
n um b ers show that the v alues for the pressure are most affected b y data and
rounding errors and presen t restrictions to the pip eline length that can b e
safely considered. This theoretical analysis presen ts a first order w orst case
analysis. F rom a practical p oin t of view the w orst case analysis is imp ortant
to obtain w arnings, but in view of the large uncertain ty that the data will ha v e
a statistical analysis seems more adequate. Suc h an analysis is p erformed in
the next subsection.

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 27
0 50 100
0 . 8
0 . 85
0 . 9
0 . 95
1
Pip eline length L [km]
κ rel ( T ; T in )
0 50 100
0
0 . 05
0 . 1
0 . 15
0 . 2
L [km]
κ rel ( T ; T w )
0 50 100
0
2
4
6 · 10 − 3
L [km]
κ rel ( T ; x )
0 50 100
0
2
4
6 · 10 − 3
L [km]
κ rel ( T ; k w )
Figure 6: The individual relativ e condition num b ers κ rel ( T ; T in ) , κ rel ( T ; T w ) ,
κ rel ( T ; k w ) , and κ rel ( T ; x ) in (65) considered as a function of the pip eline length
L = x − x 0 for concrete v alues (66).
5.2 Statistical Error Analysis
In this subsection w e p erform a statistical analysis for the algebraic mo del (10)
whic h complemen ts the theoretical analysis carried out in the previous sub-
section.
The efficien t UR Q metho d, see Section 3, enables us to calculate the rel-
ativ e standard deviation of the pressure p and the temp erature T for man y
differen t pip eline lengths L . The mean of the remaining input parameters is
set to the nominal v alues in (63) and (66). The relativ e standard deviation
σ d i /µ d i is set to 0 . 5% for ev ery input parameter d i of d . The mass flux ˆ q is
not considered here, b ecause it is constan t with resp ect to L .
The results of the UR Q sim ulation for p and T (with the concrete data
in (63) and (66)) are depicted in Figure 7 and sho w a similar b ehavior as the
w orst case analysis in subsection 5.1; the uncertaint y in the pressure p gro ws

28 J. J. Stol wijk and V. Mehrmann
quic kly for increasing pip eline length L and the uncertaint y in the temp era-
ture T decreases sligh tly for increasing L .
0 50 100
0
2
4
6
8
Pip eline length L [km]
σ p /µ p in %
0 50 100
0 . 4
0 . 42
0 . 44
0 . 46
0 . 48
0 . 5
L [km]
σ T /µ T in %
Figure 7: Relativ e standard deviation of the pressure p (left) and the tem-
p erature T (righ t) as a function of the pip eline length L , computed with the
UR Q metho d. The relative standard deviation of the input parameters is set
to 0.5 % with mean v alues in (63) and (66).
Ha ving p erformed the analysis for the algebraic mo del including temp er-
ature and ha ving observ ed that the temp erature dep endence is rather insen-
sitiv e, we can also extend the simplification of the algebraic model to the
isothermal v ersion b y assuming the temp erature T to b e constan t. The error
inflicted b y this simplification is analyzed in the follo wing subsection.
5.3 Error b et w een T emp erature Dep enden t and Isothermal Alge-
braic Mo del
Supp ose that the v alue T ( d ) , for certain parameter v alues d , is tak en for the
constan t temp erature, whereas the actual parameter v alues are giv en b y ˜
d .
Then, using T a ylor series expansion, a first order upp er b ound for the relativ e
error in T is giv en b y
| T ( d ) − T ( ˜
d ) |
| T ( d ) | .
n
X
i =1    
∂ T ( d )
∂ d i
d i
T ( d )     | d i − ˜
d i |
| d i | .
Inserting the nominal v alues d nom in (66) together with x nom = 7 · 10 4 ( 70 km),
the individual relativ e condition n um b ers for T with resp ect to the parameter
v ector d are giv en b y
   
∂ T ( d )
∂ ρ
ρ
T ( d )     =    
∂ T ( d )
∂ v
v
T ( d )     =    
∂ T ( d )
∂ D
D
T ( d )     =    
∂ T ( d )
∂ x
x
T ( d )    

ERR OR ANAL YSIS FOR FLO WS IN GAS NETW ORKS 29
=    
∂ T ( d )
∂ k w
k w
T ( d )     =    
∂ T ( d )
∂ c v
c v
T ( d )     = 4 . 18 · 10 − 3 ,
   
∂ T ( d )
∂ x 0
x 0
T ( d )     = 0 ,    
∂ T ( d )
∂ T in
T in
T ( d )     = 8 . 73 · 10 − 1 , and
   
∂ T ( d )
∂ T w
T w
T ( d )     = 1 . 27 · 10 − 1 .
It follo ws that only p erturbations in the parameter T in create an equiv alen t
relativ e p erturbation in the temp erature T . Perturbations in the other input
parameters only cause a small relativ e error in T . This means that if the input
temp erature T in is not sub ject to change, then the temp erature can safely
b e set constan t. If, ho wev er, the input temp erature c hanges, for example
for differen t pip elines, then the temp erature cannot b e set constan t and the
temp erature dep enden t algebraic mo del should b e c hosen.
6 Conclusions and Outlo ok
This pap er presen ts an error and p erturbation analysis for the Euler equations
in semilinear and algebraic form.
The partial differen tial equations of the semilinear mo del are discretized
using t w o simple sc hemes. It is sho wn that the norm wise relativ e condition
n um b ers of the resulting nonlinear systems of equations lead to a considerable
o v erestimation of the sensitivit y of the problems. The no vel component wise
relativ e condition n um b ers constitute more accurate measures for the sensi-
tivit y . F urthermore, it is sho wn that the mass flow rate has higher condi-
tion n um b ers with resp e ct to the uncertain parameters than the pressure and
w e can determine stepsizes for whic h w ell-conditioned problems are obtained.
Moreo v er, it is shown that the rounding and iteration errors can be neglected
compared to the data uncertain t y error and w e find that the mo deling er-
ror b et w een the semilinear and the algebraic mo del is approximately 1 % for
certain concrete parameter v alues.
The error analysis for the pressure in the algebraic mo del results in an error
that gro ws quic kly with increasing pip eline length, such that the algebraic
mo del can only b e used safely for short pip elines (for certain parameter v alues
up to 60 km length). The error in the temp erature decreases sligh tly with
increasing pip eline length. These results are obtained b oth via a first order
w orst case and via a statistical analysis. Moreo ver, it is sho wn that only if
the pip eline input temp erature is not sub ject to c hange, then the temp erature
can safely b e set constan t and the isothermal algebraic mo del can b e used.
F uture work will implemen t the deriv ed error estimators into a robust
error con troller, whic h allows to adaptiv ely switc h b etw een differen t mo dels

30 J. J. Stol wijk and V. Mehrmann
in the gas pip eline net w ork in order to achiev e a prescrib ed accuracy while
minimizing the computational cost.
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Jero en J. STOL WIJK,
Institut für Mathematik, MA 4-5,
T ec hnische Univ ersität Berlin,
Str. des 17. Juni 136, D-10623 Berlin, German y
Email: [email protected] erlin.de
V olk er MEHRMANN,
Institut für Mathematik, MA 4-5,
T ec hnische Univ ersität Berlin,
Str. des 17. Juni 136, D-10623 Berlin, German y
Email: [email protected] erlin.de

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