JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
ARTICLE NO.
217, 395]404 Ž1998.
AY975715
On the Front-Tracking Algorithm
Paolo Baiti
S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy
and
Helge Kristian Jenssen
Department of Mathematical Sciences, NTNU, N-7034 Trondheim, Norway,
and S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy
Submitted by Barbara Lee Keyfitz
Received October 23, 1996
In this paper we present an improved version of the front-tracking algorithm for
systems of conservation laws. The formulation and the theoretical analysis are here
somewhat simpler than in previous algorithms. At the same time, our version leads
to a more efficient numerical scheme. Q 1998 Academic Press
1. INTRODUCTION
We are concerned with the construction of a global weak solution to the
Cauchy problem for a strictly hyperbolic n = n system of conservation
laws
u t q Ž f Ž u . . x s 0,
u Ž 0, ? . s u.
Ž 1.
The basic idea of front tracking for systems of conservation laws is to
construct approximate solutions within a class of piecewise constant functions. One approximates the initial data by a piecewise constant function
and solves the resulting Riemann problems. Rarefactions are replaced by
many small discontinuities. One tracks the outgoing fronts until the first
time two waves interact. This defines a new Riemann problem, etc. One of
395
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Copyright Q 1998 by Academic Press
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BAITI AND JENSSEN
the main problems in this construction is to keep the number of wavefronts finite for all times t ) 0. For this purpose there are presently three
types of front-tracking algorithms available w1, 2, 5, 9x.
In w1, 2x one defines the notion of generation order which tells how
many interactions were needed to produce a wavefront. In order to keep
the number of waves finite, one solves in an accurate way the Riemann
problems arising from interactions between waves of low order, and in a
less accurate way those arising from interactions between high-order
waves. This simplified solution is constructed by letting the incoming waves
pass through each other, slightly changing their speeds, and by collecting
all the remaining waves into a so-called ‘‘non-physical’’ front. All nonphysical waves propagate with a constant speed greater than all characteristic speeds.
In w9x one does not use the concept of generation order. Instead, for
each time where two waves interact one considers a functional depending
on the future interactions. If this functional is small enough, some of the
small waves are removed and the algorithm is restarted. This guarantees
that the approximate solution can be prolonged for a positive time. One
then shows that it is only necessary to apply this restarting procedure a
finite number of times. In the special case of 2 = 2 systems w5x, the number
of fronts remains automatically finite, and no restarting procedure is
needed.
By the algorithms defined in these papers piecewise constant approximate solutions un Ž t, x . are defined for all t G 0. By a compactness argument one then shows that a subsequence converges to a global weak
entropic solution of Ž1.. Indeed, by the results in w3, 4x the entire sequence
will converge to the unique semigroup solution.
From a theoretical point of view these methods yield the same existence
result. However, from a numerical point of view they are not efficient. The
method presented in w9x demands knowledge of the future history of the
approximate solution, whereas in w1, 2x one has to keep track of the past
history by counting the generation order of each wave.
In this paper we present a new algorithm which avoids these problems.
More precisely, we introduce a threshold parameter « which dictates how
to solve the Riemann problems. If the product of the strengths of the
colliding waves is greater than « then we use an accurate approximation to
the Riemann problem, whereas if the product is smaller than « we use a
crude approximation. We prove that if « tends to zero sufficiently fast,
compared with other approximation parameters Žthe number of initial
jumps and the maximal size of rarefaction fronts., then our sequence of
approximations un converges to an entropy weak solution of the Cauchy
problem Ž1..
FRONT TRACKING ALGORITHM
397
2. THE ALGORITHM
We consider a strictly hyperbolic n = n system of conservation laws Ž1.
in which each characteristic family is either genuinely nonlinear or linearly
degenerate in the sense of Lax w8x, and where the flux f is C 2 on a set
V ; R n. The function u is assumed to be of sufficiently small total
variation. We recall that, given two states uy, uq sufficiently close, the
corresponding Riemann problem admits a self-similar solution given by at
most n q 1 constant states separated by shocks or rarefaction waves w8x.
More precisely, there exist C 2 curves s ¬ c i Ž s .Ž uy. , i s 1, . . . , n,
parametrized by arclength, such that
uqs cn Ž sn . ( ??? ( c 1 Ž s 1 . Ž uy . ,
Ž 2.
for some s 1 , . . . , sn . We define u 0 s
˙ uy and u i s
˙ c i Ž si .( ??? ( c 1Ž s 1 .Ž u 0 ..
When si is positive Žnegative. and the ith characteristic family is genuinely nonlinear, the states u iy1 and u i are separated by an i-rarefaction
Ž i-shock. wave. If the ith characteristic family is linearly degenerate the
states u iy1 and u i are separated by a contact discontinuity. The strength of
the i-wave is defined as < si <.
For given initial data u let un be a sequence of piecewise constant
functions approximating u in the L1-norm. Let Nn be the number of
discontinuities in the function un , and choose a parameter dn ) 0 controlling the maximum strength of rarefaction fronts.
In order to construct a piecewise constant approximation for all positive
times, we introduce various ways of solving the Riemann problems generated by wavefront interactions. More precisely, we will use an accurate
solver when the product of the strengths of the incoming fronts is ‘‘large,’’
a simplified solver yielding non-physical waves when the product is ‘‘small,’’
or when one of the incoming waves is non-physical.
2.1. The Riemann Sol¨ ers
In the definition of the Riemann solvers we will introduce non-physical
wa¨ es. These are waves connecting two states uy and uq, say, and
ˆ ) 0 strictly greater than all characteristic
traveling with a fixed speed l
speeds in V. Such a wave is assigned strength < s < s
˙ < uyy uq < and is said
to belong to the Ž n q 1.th family. We notice that since all the non-physical
ˆ, they cannot interact with each other.
fronts travel with the same speed l
Assume for a positive time t we have an interaction at x between two
waves of families ia , ib and strengths saX , sbX , respectively, 1 F ia , ib F n q
1. Here saX denotes the left incoming wave. Let uy, uq be the Riemann
problem generated by the interaction and let s 1 , . . . , sn and u 0 , . . . , u n be
defined as in Ž2.. We define the following approximate Riemann solvers.
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BAITI AND JENSSEN
ŽA. Accurate sol¨ er: if the ith wave belongs to a genuinely nonlinear
family and si ) 0 then we let
pi s
˙ u sirdn v ,
Ž 3.
where u s v denotes the smallest integer number greater than s. For l s
1, . . . , pi we define
ui , l s
˙ ci Ž l sirpi . Ž u iy1 . ,
xi, l Ž t . s
˙ x q Ž t y t . li Ž u i , l . ,
Ž 4.
where l i Ž?. denotes the ith characteristic speed. Otherwise, if the ith
characteristic family is linearly degenerate, or it is genuinely nonlinear and
si F 0, we define pi s
˙ 1 and
xi, l Ž t . s
˙ x q Ž t y t . l i Ž u iy1 , u i . .
ui , l s
˙ ui ,
Ž 5.
Here l i Ž u iy1 , u i . is the speed given by the Rankine]Hugoniot conditions
of the i-shock connecting the states u iy1 , u i . In this case the approximate
solution of the Riemann problem is defined in the following way:
¡u
¨aŽ t , x . s
˙
y
if x - x 1, 1 Ž t . ,
q
if x ) x n , p nŽ t . ,
~u
ui
¢u
i, l
Ž 6.
if x i , p iŽ t . - x - x iq1, 1 Ž t . ,
if x i , l Ž t . - x - x i , lq1 Ž t .
Ž l s 1, . . . , pi y 1 . .
ŽB. Simplified sol¨ er: for every i s 1, . . . , n let siY be the sum of the
strengths of all incoming ith waves. Define
u9 s
˙ cn Ž snY . ( ??? ( c 1 Ž s 1Y . Ž uy . .
Ž 7.
Let ¨ aŽ t, x . be the approximate solution of the Riemann problem Ž uy, u9.
given by Ž6.. Observe that in general u9 / uq hence we introduce a
non-physical front between these states. We thus define the simplified
solution in the following way:
¨sŽ t, x . s
˙
½
¨aŽ t , x .
ˆŽ t y t . ,
if x y x - l
uq
ˆŽ t y t . .
if x y x ) l
Ž 8.
Notice that by construction this function contains at most two physical
wavefronts and a non-physical one.
FRONT TRACKING ALGORITHM
399
2.2. Construction of the Approximate Solutions
Given n we construct the approximate solution un Ž t, x . as follows. At
time t s 0 all Riemann problems in un are solved accurately as in ŽA.. By
slightly perturbating the speed of a wave, we can assume that at any
positive time we have at most one collision involving only two incoming
fronts. Suppose that at some time t ) 0 there is a collision between two
waves belonging to the ia th and ib th families. Let sa and sb be the
strengths of the two waves. The Riemann problem generated by this
interaction is solved as follows. Let «n be a fixed small parameter that will
be specified later.
if < sa sb < ) «n and the two waves are physical, then we use the
accurate solver ŽA.;
v
if < sa sb < - «n and the two waves are physical, or one wave is
non-physical, then we use the simplified solver ŽB..
v
We claim that this algorithm yields a converging sequence of approximate solutions defined for all times t ) 0, for any «n .
LEMMA 2.1. The number of wa¨ efronts in un Ž t, x . is finite. Hence the
approximate solutions un are defined for all t ) 0.
Proof. We introduce the standard functional Q measuring the interaction potential. For a fixed n and time t ) 0 at which no interaction occurs
in un Ž t, ? ., let x 1Ž t . - ??? - x m Ž t . be the discontinuities in un Ž t, ? ., and
denote by s 1 , . . . , sm and i1 , . . . , i m their strengths and families, respectively. Two waves sa , sb with xa - xb are said to be approaching either if
ia s ib - n q 1 and one of them is a shock, or if ia ) ib .
The interaction potential Q is defined as
QŽ t . s
˙
Ý
Ž a , b .g A
< sa sb < ,
Ž 9.
where A denotes the set of all approaching waves at time t. As in w1, 2x,
one can prove the following. Let u be any piecewise constant approximate
solution u of Ž1. constructed as above in the strip w0, T w=R , and with
sufficiently small total variation. If at time t two waves of strengths sa and
sb interact, then
DQ Ž t . F y
< sa sb <
2
.
Ž 10 .
For each n consider the set of collisions for which the interaction potential
between the incoming waves is greater than «n . By the bound in Ž10., Q
decreases by at least «nr2 in these interactions. Since new physical waves
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BAITI AND JENSSEN
can only be generated by this kind of interactions, it follows that their
number is finite. Furthermore, since non-physical waves are introduced
only when physical waves interact, the number of non-physical waves is
also finite. Finally, since two waves can interact only once, the number of
interactions is finite, too. This implies that the approximate solutions are
defined for all positive times, i.e., T s ` for each n .
We can now state the main result of this paper.
THEOREM 2.2. Let u be of small total ¨ ariation, and let un con¨ erge to u
in the L1-norm. Let Nn be the number of jumps in un , dn the parameter
controlling the maximum strengths of rarefaction fronts, and «n the threshold
parameter. If
lim dn s 0,
lim «n Nn q
nª`
n ª`
ž
1
dn
k
/
s 0,
Ž 11 .
for e¨ ery positi¨ e integer k, then the sequence of piecewise constant approximations un constructed by the abo¨ e algorithm con¨ erges to an entropy weak
solution of the Cauchy problem Ž1..
3. PROOF OF THE THEOREM
We recall here that a weak solution u of Ž1. is a distributional solution,
i.e., for any fixed smooth function f with compact support in R = R it
satisfies
q`
Hy` u Ž x . f Ž 0, x . dx
q`
q`
H0 Hy` Ž u Ž t , x . f Ž t , x . q f Ž u Ž t , x . . f Ž t , x . . dx dt s 0.
q
t
x
Ž 12 .
As in w1, 2x, if the approximate initial data un have sufficiently small
total variation, by Helly’s theorem it follows that there exists a subsequence of un Ž t, x . which converges in L1loc to some function uŽ t, x .. To
prove that u is a weak solution of Ž1. it suffices to show that
q`
Hy` Ž u Ž x . y u Ž x . . f Ž 0, x . dx
n
q
T
H0 Ý Ž ˙x
a
a
un Ž t , xa . y f Ž un Ž t , xa . .
. f Ž t , xa . dt Ž 13.
401
FRONT TRACKING ALGORITHM
tends to zero as n ª `, for any fixed smooth function f with compact
support. Here the xa s xa Ž t . denote the lines of discontinuity of un in the
strip w0, T x = R, and w?x denotes the jump across these discontinuities. By
assumption the first term tends to zero as n ª `.
To estimate the second term, let RŽ t . and N Ž t . be the sets of indices a
corresponding to rarefactions and non-physical fronts at time t, respectively, and let sa be the strength of the wave at xa . Proceeding as in w2x,
since the total variation of un Ž t, ? . is uniformly bounded in t and n , we
obtain
T
H0 Ý Ž ˙x
a
a
un Ž t , xa . y f Ž un Ž t , xa . .
F C Ž max < f < .
T
H0
F CT Ž max < f < .
ž
ž
Ý
a g RŽ t .
. f Ž t , xa . dt
< sa Ž t . < 2 q
Ý
ag N Ž t .
sup < sa Ž t . < q sup
tg w0, T x
a g RŽ t .
< sa Ž t . < dt
/
Ý
tg w0, T x a g N Ž t .
Ž 14 .
< sa Ž t . < ,
/
where C denotes constants independent of n .
So, to have convergence to a weak solution of Ž1., we need to prove that
both the maximal size of refraction waves and the total amount of
non-physical waves present in un tend to zero as n ª `.
By Ž3. it is clear that the strength of any rarefaction wave in un is
bounded by dn . To estimate the second term in the right-hand side of Ž14.
we prove the following lemma.
LEMMA 3.1. The total strength of non-physical wa¨ es in un at time t tends
to zero uniformly in t as n ª `.
Proof. We introduce the notion of generation order of wavefronts. Fix
a n . First we assign order 1 to all the waves at time t s 0q. For waves
generated at times t ) 0 we assign orders inductively as in w1, 2x; i.e., the
order of an outgoing wave is the minimum order of the incoming waves of
the same family, if any, and is one more than the maximum order of all
incoming waves otherwise. Observe that, since the numbers of waves is
finite, there is a maximal generation order for the waves in un , call it sn .
Let Vk Ž t . be the sum at time t of the strengths of all waves in un Ž t, ? . of
order G k. As in w1, 2x, it follows that for all k G 2,
Vk Ž t . F 4C1 2yk ,
; t G 0,
Ž 15 .
uniformly in n , for some constant C1. This bound will be used to estimate
the total strength of higher-order non-physical fronts.
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BAITI AND JENSSEN
For lower-order non-physical waves we need another estimate. Let
Mi , Si denote the total number of fronts and the number of non-physical
fronts of order i, respectively, in the approximate solution un . Since a
kth-order wave can be generated only from an interaction between one of
order k y 1 and one of order j F k y 1 and since only one non-physical
wave can be generated from each interaction, we have the two relations
Žsee also w1, 2x.
Mk F n dny1 Ž M1 q ??? qMky1 . Mky1 ,
Ž 16 .
S k F Ž M1 q ??? qMky1 . Mky1 .
Ž 17 .
Now define P1 s
˙ M1
established that Mk F
Pk satisfies the bounds
and Pk s n dny1 Ž P1 q ??? qPky1 . Pky1.
Pk F Pkq1 F n dny1 Ž k q 1. Pk2 for every
˙
Pk F Ž kn dny1 P1 .
2 ky 1
F Ž kn2dny2 Nn .
2 ky 1
It is easily
k and that
.
Ž 18 .
In turn, using Ž17., this implies that
Sk F
2
kPky1
F kŽ
2 ky 1
kn2dny2 Nn
.
F C Ž k . Nn q
ž
1
dn
pŽ k .
/
,
Ž 19 .
k
where C Ž k . s Ž kn. 2 and pŽ k . s 2 kq 1. We note that the estimate Ž17. is
useful only for each fixed k; since the constants C Ž sn . and pŽ sn . grow too
fast as n ª `, we cannot use this to estimate the total amount of
non-physical waves of all orders.
By standard interaction estimates w1, 2x, the strength of a non-physical
wave generated by an interaction between two physical waves is bounded
by C1 «n . As a non-physical front interacts with physical ones, its strength
can increase. However as in w2, Lemma 2, pp. 115]116x, there exists a
constant C2 such that for all times the strength of the wave remains
bounded by C1C2 «n .
Now we can estimate the total strength of the non-physical waves at
time t in the following way. By Ž15. and Ž17. it follows that
Ý
ag N Ž t .
< sa < F
Ý CŽ k.
kFk 0
ž
Nn q
pŽ k .
1
dn
/
? C1C2 «n q
Ý
4C1 2yk , Ž 20 .
kGk 0
for some integer number k 0 .
Given r ) 0, choose k 0 such that Ý k G k 04C1 2yk F rr2. Next, by the
second condition in Ž11., take n so large that
Ý
kFk 0
C Ž k . Nn q
ž
1
dn
pŽ k .
/
? C1C2 «n -
r
2
.
Ž 21 .
FRONT TRACKING ALGORITHM
403
By Ž20. and Ž21. it follows
Ý
ag N Ž t .
< sa < F r ,
Ž 22 .
for large n , uniformly in t. This completes the proof of the lemma.
Since both the maximal size of rarefaction waves and the total amount
of non-physical waves present in un tend to zero, then Ž13. and Ž14. show
that u is a weak solution of Ž1..
In a similar way one can prove that if Žh , q . is a flux, entropy-flux pair,
then we have
h Ž u . t q q Ž u . x F 0,
Ž 23 .
in the distributional sense Žsee also w1, 2x.. This completes the proof of the
theorem.
Remark 1. The approximation error in the above scheme consists of
two different parts, due to the approximation of rarefaction waves and the
introduction of non-physical waves. The error from approximation of
rarefaction waves is O Ž dn ., whereas the error from the non-physical waves
is split into two parts. By Ž19., the non-physical waves of generation G k
contribute by an amount O Ž2yk .. From Ž17. the error due to non-physical
waves of generation F k is bounded by
k
O Ž kn . 2 dny2
ž
kq 1
«n ,
/
Ž 24 .
if we assume that Nn is O Ž dny1 .. By asking the three errors to be of the
same order of magnitude, it follows that k s O Ž<log dn <. and that an
appropriate choice for «n is given by
«n s
dn1q 2r dn
Ž <log dn < n .
1r dn
.
Ž 25 .
Remark 2. The second condition in Ž11. is also necessary. Assume that
both dn and the strength of any wave at time t s 0 are O Ž Nny1 ., which is
the case for smooth initial data approximated by their values at equally
spaced intervals. By the interaction estimates one gets that the strength of
a wave of generation k is O ŽŽ C1 Nny1 . k .. If Ž11. fails, i.e., «n has only
polynomial growth w.r.t Nny1 , then for n large enough the approximate
solution un can contain waves of only a finite number of orders, independent of n . Hence, in general this cannot yield a weak solution as n ª `.
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BAITI AND JENSSEN
Remark 3. The algorithm presented in this paper is numerically more
efficient than the previous theoretical algorithms since one does not
consider the generation order and keeps track only of the ‘‘big waves.’’
This last feature}of keeping only the waves which have a large potential
for influencing the solution}is the main advantage with respect to computational effort. This reflects what is actually done in practice when one
implements front tracking for systems of conservation laws Žsee w7x and
references therein ..
ACKNOWLEDGMENTS
We thank A. Bressan for suggesting this problem and for many helpful discussions. The
second author thanks A. Bressan and N. H. Risebro for the opportunity of visiting S.I.S.S.A.
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