SIAM J. NUMER. ANAL.
Vol. 33, No. 3, pp. 843-863, June 1996
()
1996 Society for Industrial and Applied Mathematics
001
ON THE CONVERGENCE OF OPERATOR SPLITTING APPLIED
TO CONSERVATION LAWS WITH SOURCE TERMS*
J. O. LANGSETHt, A. TVEITO*, AND R. WINTHER
Abstract. The order of convergence for operator splitting applied to conservation laws with source terms is
studied. The operator splitting procedure is based on local solutions of the associated homogeneous conservation
law and an ordinary differential equation. We prove that, for scalar problems, the error introduced by the splitting is
linear with respect to the time step. The theoretical results are illustrated by numerical examples.
Key words, hyperbolic conservation laws, operator splitting, error estimate
AMS subject classifications. 65M15, 35L65
1. Introduction. The purpose of this paper is to study an operator splitting procedure
applied to hyperbolic conservation laws with source terms. We consider Cauchy problems for
systems of the form
(1.1)
ut
+ f (U)x
g(u),
where f and g are smooth functions of u. The operator splitting procedures are defined from
local solutions of the corresponding homogeneous equation
(1.2)
Ut
+ f (U)x
0
and the ordinary differential equation
(1.3)
ut
g(u).
The motivation for such procedures is the desire to extend sophisticated numerical methods
developed for hyperbolic systems of the form (1.2) to more general systems of the form (1.1).
This paper is mainly devoted to theoretical results for scalar equations. However, in order
to illustrate that the splitting techniques considered here also have applications to systems, we
do include numerical examples involving the shallow water equations and the Euler equations
of gas dynamics.
It is well known that if a formally first-order finite difference scheme, like the Godunov
method, is applied to a scalar equation of the form (1.1), the convergence rate is no better than
O(h 1/2) (cf. Lucier [14]). Here h represents the mesh size. This slow convergence of finite
difference schemes reflects the fact that the solution is not sufficiently regular, and it motivates
the search for alternative numerical procedures.
For homogeneous equations of the form (1.2), Dafermos [3] has proposed a front-tracking
procedure based on a piecewise linear approximation of the tlux function f and a piecewise
constant approximation of the initial data. He demonstrates that in this case the exact solution
of the Cauchy problem can be computed. Holden, Holden, and Hegh-Krohn [5] developed
*Received by the editors August 30, 1993; accepted for publication (in revised form) July 5, 1994.
tDepartment of Informatics, University of Oslo, P.O. Box 1080 Blindern, N-0316 Oslo, Norway (janolav@
lfi.uio.no). The research of this author was supported by the Royal Norwegian Council for Scientific and Industrial
Research (NTNF) program STP.28291.
SINTEF SI, P.O. Box 124 Blindern, N-0314 Oslo, Norway. The research of this author was supported by the
Royal Norwegian Council for Scientific and Industrial Research (NTNF) program STP.28402.
Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, N-0316 Oslo, Norway. The research
of this author was supported by the Royal Norwegian Council for Scientific and Industrial Research (NTNF) program
STP.28402.
843
844
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
this idea into a numerical method in the scalar case. Furthermore, Lucier 13] used the results
of Kruzkov [9] and Kuznetsov [10] to show that Dafermos method is a first-order accurate
numerical scheme.
The computation ofthe Dafermos solution is essentially based on a series of local Riemann
problems. For a piecewise linear flux function, the Riemann solution of (1.2) consists of a
finite number of shock curves, which are a straight line in (x, t)-space. Hence, the complete
solution of the Cauchy problem, with piecewise constant initial data, consists of a sequence
of piecewise linear, possibly interacting shock curves.
The most straightforward generalization of the Dafermos method to more general equations of the form (1.1) introduces solutions with shock curves which are locally nonlinear. As
an alternative to this approach, in this paper we shall study a generalization based on operator
splitting in which we alternate between solutions of (1.2) and (1.3). Hence, if we also use
Dafermos method to solve (1.2), we obtain a true generalization of this method to equations of
the form (1.1). In our theoretical results we will simply use the forward Euler scheme to solve
the ordinary differential equations (1.3). The purpose of our work is to analyze the efficiency
of this approach.
Operator splitting, or fractional steps, is frequently used as part of a computational strategy
for many classes of partial differential equations. For a general approach to the analysis of
such methods, we refer to Chorin et al 1]. Operator splitting has been used extensively on
multidimensional conservation laws; cf. e.g. Crandall and Majda [2] and Holden and Risebro
[6]. The main purpose of these papers is to generate multidimensional solutions from a
sequence of one-dimensional solutions. A recent result by Teng [19] shows that for twodimensional scalar problems, the convergence order of the two most commonly used splitting
algorithms is (.9(At 1/2), where At is the splitting time step. For applications on inhomogeneous
problems see e.g. Holden and Risebro [7], LeVeque and Yee [11], and Westerberger and
Ballmann [20].
The main result in this paper is to establish that the error introduced by the operator
splitting, when (1.2) is solved exactly, is O(At), where At denotes the time step. Furthermore,
we show that the obtained generalization of the Dafermos method to equations of the form
1) is indeed a linearly convergent method.
The precise statements of the main results are given in 2. The proofs of these results
are basically given in 3, although the derivation of a single technical result is delayed
until the final section. The theoretical results are supported by numerical experiments presented in 4.
.
,
2. Preliminaries and statement of the main result. We begin by introducing some
notation. If/C is a domain in then LP(/C), 1 < p _< c denotes the classical L p space of
real-valued functions on/C. The norm on L
LP (]) is denoted by II. The localized
version of L ’, consisting of functions on which are in L" (/C) for any compact subset/C of
are denoted by
denote the L norm Ill,
c. It what follows, let
Furthermore, BV BV(]) denotes the subspace of c consisting of functions with
,
’
Lo
Lo
bounded variation, i.e.,
BV
L]o
{v
TV(v) <
where
(2.1)
TV(v)
sup ]
v(x + z)
v(x)
dx.
z4:0
The class of Lipschitz-continuous functions on 1 is denoted by Lip. More precisely
Lip
{v
6
L [IVllLip
<
cx},
845
CONVERGENCE OF OPERATOR SPLITTING
where
IILip denotes the seminorm
(2.2)
sup
lvlltip
xy
Iv(x)
X
v(y)]
Yl
Furthermore, for r > 0 define Br {u 6 B V Ilu Iloo < r}.
In what follows, the mollifier function we oge (x) is important. It is defined using a
0 for
nonnegative and symmetric function f2 C with integral one, satisfying f2 (x)
Ix _> 1. For e > 0, define
(2.3)
lf2(x/e).
toe(x)
,
Then oge has the following properties:
(i) o)e e C roe >_ O,
(ii) o (x) o (-x),
oge(x) dx 1,
(iii)
(iv) o(x) m 0 for Ixl e,
(v) we(x) Male, I’(x)l Male 2,
(vi)
Ix I (x) dx e,
where M is a finite constant independent of e. Since there will be no ambiguity, the e-subscript
on we will be suppressed.
The purpose of this paper is to study the convergence of operator splitting as applied to a
scalar conservation law with a source term, i.e.,
f
f
(2.4)
ut "4-
f (U)x
g(u),
u(x, O)
uo(x).
We shall study numerical approximations of the entropy solution u for 0 _<
< To, where
To
> 0 will be considered fixed in the rest of this paper. The initial function u0 is assumed
to be in B V, while the flux function f and the source term g are smooth. Define the domain
S = I-m, m], where m is a fixed positive number. Let Kg
SUPue s Ig(u)l. In order
to guarantee that the solution u and the numerical approximations introduced below always
remain in the state space S, we assume that rn > ToKg and that u0
for r rn ToKg.
We use the Kruzkov form [9] in order to define the entropy solution of the Cauchy
r
problem (2.4). The entropy solution is bounded, has bounded total variation, and is L
continuous in time. A precise definition of the Kruzkov form and the properties of the solution
will be given in the next section.
We will consider an operator splitting procedure for the initial value problem (2.4), where
we alternate between solving the homogeneous equation (1.2) and the ordinary differential
equation (1.3).
Our purpose is to derive an error bound for the splitting procedure, when the ordinary
differential equation is approximated by the explicit Euler scheme.
In order to define the operator splitting procedure let E(t) Bm-tKg
]m denote the
Euler operator given by
(2.5)
vo + g(vo)
E(t)vo
m
Kg. Furthermore, let H ’/m Bm be the solution operator of the homogeneous
problem, i.e., v(., t) H(t)vo is the entropy solution of the initial value problem
for <_
(2.6)
l)t
+ f (V)x
O,
v(x, O)
vo.
846
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
N where
The operator splitting solution v(x, tn)}n--0
tn
(2.7)
n At and tu <
To is defined by
[H(At) E(At)I"vo.
v(., tn)
Note that this approximate solution is defined only at discrete t-values. The main result below
states that the operator splitting solution, when (2.6) is solved exactly, converges linearly in
At to the entropy solution of (2.4).
THEOREM 2.1. Let u u(x, t) be the entropy solution of(2.4) and v = v(x, tn) be the
operator splitting solution (2.7). There exists a positive constant Ko, depending on T V (uo),
T V (vo), f, g, To, such that for T N At < To,
(2.8)
Ilu(., T)
v(., T)II
K0(llu0
011 + zXt).
This theorem will be proved in the next section.
The typical application of the splitting procedure is to use a numerical method in order to
approximate the solution of the homogeneous problem (2.6). Let Hh 13m "-> 13m denote the
corresponding numerical solution operator. Then Hh is said to be an approximation of order
ot in h to H if there exists a positive constant K1 depending on T V (v0) such that
(2.9)
Iln(t)vo- nh(t)woll
< IIv0- w011 + Kht
for all u0, w0 6 Bm. Here h denotes some discretization parameter and w0 is a suitable
approximation of v0. Using Hh in order to solve the homogeneous problem (2.6), we define
N
the approximate operator splitting solution {w(., t)}=o
by
(2.10)
[Hh(At) E(At)]nwo.
w(., tn)
The next theorem regarding this approximation follows essentially from the main result above.
THEOREM 2.2. Let u u (x, t) be the entropy solution of (2.4), and let Hh be an approximation of order in h to H; that is, (2.9) is satisfied. Let wo be a suitable approximation of
uo. Then there exists a positive constant K2, depending on T V(uo), T V (wo), f, g, To, Ko,
and K1, such that
(2.11)
[lu(’, T)
w(., T)II
< K2([lu0 w0ll + h a + At)
NAt <_ To.
The proof of this theorem is also found in the next section.
Assume that the discretization parameters h and At are of the same order of magnitude.
If Dafermos’ method is used to approximate (2.6), then the estimate (2.11) holds with a 1,
cf. 13]. Hence, we obtain an algorithm for (2.4) which is first order in both h and At.
for T
3. Proof of the main result. In this section we will give the main steps of the proof of
Theorems 2.1 and 2.2.
We begin with the definition of the entropy solution of the hyperbolic conservation law
with source term. The definition deals with a slightly more general problem than (2.4), since
such problems will appear in the proof below. Consider
(3.1)
ut
+ f (u, x, t)x
g(u, x, t),
u(x, O)
uo(x),
where both f and g are assumed to be smooth functions in all their arguments. Furthermore
let uo Bm-rolCg. The constant Kg now represents the quantity Kg sup{Ig(u, x, t)l x
0_<t < To, u cS}.
,
847
CONVERGENCE OF OPERATOR SPLITTING
DEFINITION 1. A bounded measurable function u u (x, t) is called an entropy solution
of (3.1) in ]R [0, T] if, for any constant k ]R and any smooth function ok(x, t) > 0 with
compact support in IR x [0, T], the following inequality holds:
Lrflu-kldPt+sign(u-k)(f(u,x,t)-f(k,x,t))4)xdxdt
k)f(k, x, t)xdp dxdt +
sign(u
>
sign(u
lu0 kldp(x, O)
lu(x, T)
kldp(x, T) dx
k)g(u, x, t) dxdt.
This definition is due to Kruzkov (cf. [9]). We assume that the solution has the following
properties:
(3.2)
u(x,t)
S, Vx
]R,
TV(u(.,t)) <_ M,
(3.3)
(3.4)
u(., s)ll _< M(t
Ilu(., t)
O < <To,
O <_ <To,
s),
0 <_ t, s <_ To,
where M is a positive constant; cf. Oleinik 15] and Kruzkov [9].
In order to prove the theorems, we will need some auxiliary results. In our first lemma
we consider a function v which solves the homogeneous problem, i.e., v(., t) H (t)vo. Let
ap
(x, t) be a smooth function and define
q(x, t)
v(x, t) -t- O(x, t).
Consider the problem of finding an equation governing the evolution of q. This question, of
course, should be analyzed in terms of the precise definition of an entropy solution. But it
is enlightening to formally consider this question under the assumption of smooth solutions.
Then it follows that q solves the equation
qt
+ f (q(x, t)- O(x, t))x
grt(x, t)
with q(x, O)
vo(x) + gr(x, 0) initially. The following lemma states that this observation
also holds for nonsmooth solutions when the solution is defined according to Definition 1.
LEMMA 3.1. Assume that v(x, t) is the entropy solution of the problem
(3.5)
vt 4"
f (V)x
O,
v(x, O)
vo(x).
Let gt (x, t) be a smooth function and define
q(x, t)
v(x, t) + O(x, t).
Then q is the entropy solution of
(3.6)
q,
+ fi(q, x, t)x = o (x, t),
q(x, O)
vo(x) + (x, 0),
,
where the flux function f (q, x, t) f (q ap(x, t)) and the source term (x, t) lrt(x t).
The proof of this lemma is rather lengthy and is given in 5. The next auxiliary result
concerns the problem of bounding the integral of a Lipschitz-continuous function and the
derivative of the mollifier function
848
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
]m. Furthermore, let F S x ]R
LEMMA 3.2. Assume that u
N be a measurable
function satisfying
sup IIF(’, Y)IILip _< L < x.
(3.7)
Then
frfrtF(u(x),y)o)t(x-y)dxdy
(3.8)
where w
<_ L TV(u),
o9 is defined above.
Proof. Let
J=
Since
fe oY (x) dx
fF(u(x),y)o)’(x-y)dxdy
0, we have
flfeF(u(y),y)o)(x-y)dxdy
=0.
Therefore
y)
F(u(y), y))w’(x
y)dxdy
From (3.7) we obtain
Lflu(x)-u(y)llo)’(x-y)ldxdy
=L lu(y+Z)lzl- Izllo’(z)l
J<_
u(y)l
dydz.
Note that properties (ii) and (iii) of w yield
Izllo)’(Z)l dz
ZO)’(z)dz
w(z)dz
1.
Hence from the definition of the total variation, we finally obtain
J < L TV(u)
fe
IzlloY(z)l dz
Z TV(u).
Recall that in order to compute the operator splitting solution v at T N At we do N
steps. In each step we first apply the Euler operator E for a time step At. Then we use the
resulting function as an initial condition for the homogeneous conservation law which is also
solved for a time step At (cf. (2.7)).
A main step in the proof of Theorem 2.1 is to estimate the error between u and v after one
single time step At. Hence, we are interested in estimating Ilu(., zxt) H(At) E(At)v01l. In
what follows we write
r/(x, t)
H(t) E(At)vo.
With this definition, observe that O(x, At) = v(x, At).
In order to estimate the difference between u(., At) and v(., At) we introduce a comparison function q (x, t) given by
849
CONVERGENCE OF OPERATOR SPLITTING
(3.9)
q(x, t)
rl(x, t)
+ @(x, t),
--(At
t)g(vo(x)).
where
@(x, t)
Since p(x, At)
v(x, At). Furthermore,
0 we have q(x, At)
0(x, 0) + ap (x, 0)
q (x, 0)
vo(x) + Atg(vo) + (x, O)
vo(x).
The difference
u(., At)
v(., At)
u(., At)
q(., At)
will be estimated by deriving bounds for u (., t) q (., t) for all 6 [0, At]. This will be
done by observing that q is the entropy solution of a suitable equation of the form (3.1). It is
reasonable to assume that q is the entropy solution of the problem
(3.11)
qt
where
(3.12)
_
+ f(q,x,t)x
g(vo(x)),
f (q, x, t) f (q
q(x, O)
vo(x),
p(x, t)).
If v0 is smooth, then p is smooth and we can apply Lemma 3.1 directly to achieve this. Let
us therefore for a moment assume that v0 has this property.
Since u
u(x, t) is the entropy solution of (2.4), we obtain from Definition 1 that u
satisfies
foatflu-klckt+sign(u-k)(f(u)-f(k))ckxdxdt
(3.13)
+
>
L
f/tfr
lu0 k]O(x, O) -lu(x, At) klqb(x, At)dx
sign(u
k)g(u)q dxdt
for any smooth function 4
4 (x, t) _> 0 with compact support and any constant k
Correspondingly, from the observations above, q q (y, r) satisfies
+
(3.14)
+
>
k)(fi(q, y, r)
sign(q
k)fi(k, y, r)yqb(y, r)} dydr
k[qb(y, O) -Iq(Y, At)
sign(q
.
fi(k, y,
sign(q
Ivo
6
k[qb(y, At)dy
k)g(vo) dydr
.
for any smooth t#
(y, r) > 0 with compact support and any constant k
with selecting special choices of the constants k and the smooth functions
I. We proceed
850
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
In (3.13) choose k
q(y, ) and p(x, t)
o(x
y)w(t
r). Then integrate over
x [0, At] in the variables (y, r)"
fo t fo ’
qlw(x
lu
+ sign(u
y)M(t
r)
f(q))of(x
q)(f(u)
r)dxdtdydr
y)w(t
(3.5)
{luo
/
>_
y)w(z)
qlw(x
]u(x, At)
y)w(At
qlw(x
r)} dxdydr
--foAtffoAtftsigll(U--q)g(u)oo(X-y)og(t-z)dxdtdydz,
where u = u (x, t) and q
u (x, t) and b (y, r)
q (y, ). Correspondingly choose k
w(x y)w(t z) in (3.14) and then integrate over x [0, At] in the variables (x, t)"
-fo tf fo tf tlq--ulog(x--y)of(t--z)
(3.16)
+ sign(q u)(f(q, y, z) f(u, y, z))w’(x y)w(t r)
+ sign(q u)f(q, y, r)ytO(x y)w(t z)} dxdtdydr
+
f0 ’
{Iv0(y)
y)w(t) -Iq(Y, At)
ulco(x
u]oo(x
y)w(t
At)} dydxdt
>_-fo’Xtffo’Xtfsign(q-u)g(vo(y))o(x-y)o(t-)dxdtdyd.
Add (3.15) and (3.16) and observe that the first terms in both expressions cancel. Rearranging
the remaining terms and taking absolute values yields
(3.17)
L(e) < R(e)
+ I(e) + 12(e) + 13(e).
These expressions are
foAtfrf
L(e)
lu(x, At)
qlco(x
y)w(At
r)dxdydz
Iq(Y, At)
ulw(x
y)w(At
t)dxdydt
N
+
(3.18)
L (E)
R(e)
(3.19)
" L2(e).
fo tf f
+ foAtffN
+
qlw(x
y)w(z)dxdydr
vo(Y)lW(x
y)w(t) dxdydt
lu0(x)
lu
R (e) R2(e).
851
CONVERGENCE OF OPERATOR SPLITTING
(3.20)
of(x
X
(3.21)
y)oo(t
foAt f foAt fsign(u
f
/2(e)
r)dxdtdydz
q)f (u, y, Z)yW(X
y)w(t
z)dxdtdydr
At
(3.22) I3(e)=
sign(u
q)(g(u)
g(vo(y))w(x
y)oo(t
z)dxdtdydz
Note that the first term in (3.15) and the first term in (3.16) vanish. Passing to the limit
in e we get the following lemma concerning the difference u v after one time step in the
splitting procedure.
LEMMA 3.3. Let u be the solution of (2.4) and let 0(’, t)
H(t) E(At)vo for vo
O,
a
on
exists
constant
There
C
>
depending
TV(uo),
TV(vo),
f, and g such that
13m-zXtr.
(3.23) Ilu(’, At)
v(., At)ll
Ilu(’, At)
0(’, At)ll
(1
+ CAt)lluo o011 + CAt 2.
Proof Our intention is to derive the desired estimate from (3.17). This inequality was
derived under the additional assumption that v0 was smooth. However, observe that r/(., t)
H(t) E (At)v0 depends continuously in L on the initial function v0. Since any B V-function
can be approximated to any accuracy in L by smooth and B V-bounded functions, it is
therefore enough to establish (3.23) when v0 is smooth. Hence, in particular, we can assume
that (3.17) holds.
In order to prove this lemma, we treat each term in (3.17) separately starting with L(e).
0. To see this consider
Both terms L and L2 approach u (., At) q (., At)II when e
tl()
and note that since fot w(At
1
llu(’, At)
q(., At)ll
r)dr
1
q(., At)ll
=ll/,t(., zt)
1/2 for At
> e we have
foAtfNf
lu(x, At)
Iq(Y, z)
q(x, At)lw(x
y)w(At
z)dxdydz
Iq(Y, r)
q(y, At)lw(x
y)w(At
z)dxdydz
q(x, At)lw(x
y)w(At
Thus
<_
fo’ti.i.
<_
+
SoAtff
Iq(Y, At)
L 1,1 (?) q" L 1,2(?).
q(x, At)lw(x
y)w(At
z)dxdydz
r) dxdydr.
852
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
Using the L continuity of q and property (vi) of o) it is easily seen that there exists a constant
N’ such that L 1,1 _< N’?. Correspondingly, using the total variation of q and property (vi) of
q(., At)ll
09, L1,2 _< M’e. The same arguments are used to bound L2. Since Ilu(’, zXt)
Ilu(’, At) v(., At)ll it follows that
(3.24)
L(e)
e---0
Ilu(-, At)
v(., At)ll.
The term R is treated in the same way as L. Both RI(e) and R2(e) approach
e
0, hence
(3.25)
R(e)
Ilu0
e---0
llu0 v011 as
v0ll.
The rest of the terms in (3.17) depend on the source term g. We first consider ll.
Let r 6 [0, At] be fixed. Define
F(u, y)
where q
f (u, y, z)
q)(f (u)
sign(u
(f (q)
f (q, y, r))),
q (y, r). Then F is Lipschitz continuous in u. Furthermore,
sup liE(’, Y)llLip _<
sup
Y6/
Since fu(U, y, r)
(y, r)
f’(u + (At
6
If’(u)
fu(U, y, z)[.
N x [0, At]
r)g(vo(y))) it follows that
sup F (., y)Ilup _< C
where the constant C depends on f and g. From Lemma 3.2 we obtain
(3.26)
Ii(e) <_ CAt
foAtfo
At
TV(u(., t))o(t
z)dtdz < CMAt 2
C1At 2.
From the definitions of 12 we obtain that
12(e) <
fo ’ fo
w(t
[f (u(x, t), y, r)ylO)(X
r)
y)dxdy dtdr.
Here
f (k, y, "t2)y
Since
f’(k- gr(y, Z’))lry(y,
-f’(k ap(y, r))(At t)g’(vo(y))v’o(y ).
f and g are smooth functions, there is a finite constant C’ independent of At such that
If(k, y, r)yl < C’AtlVo(Y)l.
Using this we achieve that
h(e)
_< C’At
(3.27)
C’TV(vo)At
o)(t
foAtfo
r)
IV’o(y)lo)(x
y)dxdy dtdr
At
co(t
z)dtdz <_ C2At 2.
853
CONVERGENCE OF OPERATOR SPLITTING
For the last term we obtain that there exists a constant C depending on g such that
C
13(e)
<_ C
(3.28)
+
C
foo’tffo’tf
foAtffoAtf
fo’ tf, fo"’f
[u(x, t)
vo(Y)loo(x
y)w(t
3)dxdtdyd3
lu(x, t)
uo(x)lto(x
y)og(t
3) dxdtdyd3
lu0(x)
vo(y)lo(x
3)dxdtdyd3
y)og(t
C(/3,1 +/3.2).
Using the L 1-continuity in time u, we achieve that
At
I3,1---
f0Atf0 o(t-3)(flu(x,t)-uo(x)ldx)
At
<_ M
(3.29)
fo
dtd3
At
tw(t
3) dtdz
< MAt 2,
where M is found in (3.4). For the term 13,2 we get that
13,2--Atff
<- At
At
luo(x)-vo(y)lo(x-y)dxdy
{ fr f
{ff luo(y+z)-uo(Y)llzl
luo(x)
uo(y)lw(x
y)dxdy +
Izlog(z) dzdy
ff
+ fr
luo(y)
vo(y)lw(x
luo(y)
vo(y)ldY
y)dxdy
}
}
Now using the definition of T V (u0) and property (vi) of 09 we obtain that
I3,2<_At{TV(uo)flzlw(z)dzq-Iluo-voll}
<_ At{TV(uo)e + Ilu0- v011}.
Inserting this expression and (3.29) into (3.28) we obtain that there exists a constant C3 such
that
limsup I3(e) <_ C3(At 2 + Atllu0- o011).
(3.30)
Finally, inserting (3.24), (3.25), (3.26), (3.27), and (3.30) into (3.17), we get (3.23) as e ----+ 0
[3
for a proper constant C.
We now use Lemma 3.3 to prove both Theorem 2.1 and Theorem 2.2.
3.1. Proof of Theorem 2.1. Recall that the operator splitting solution v at time T tv
is computed by performing N steps of the type considered in Lemma 3.3. From this lemma,
we have
Ilu(., T)
v(., T)II
.< (1 + CAt)llu(., t_x) w(., tN-1)l[ -+- CAt 2.
854
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
Hence, there is a finite constant K0 = e cr, independent of At, such that
Ilu(-, T)
v(., T)II _< K0(liu0
v01l + At),
which concludes the proof of Theorem 2.1.
3,2. Proof of Theorem 2.2. We want to bound the difference between the entropy solution u and the approximate operator splitting solution w. For the latter the solution operator
H of the homogeneous equation in the splitting procedure is approximated by the operator
H. This operator is supposed to generate a family of approximations converging toward the
entropy solution of the homogeneous problem; cf. (2.9).
For T = N At we consider
(3.31)
Ilu(., T)
w(., T)II _.< Ilu(., T)
v(., T)II-4-I1(’, T)
w(., T)II,
where v v(x, n At) is the operator splitting solution in which the homogeneous conservation
law is solved exactly. The first term on the right-hand side of (3.31) is bounded by the same
expression as derived in Theorem 2.1, except that we now have the freedom to choose v0 u0.
Hence, from Theorem 2.1, we have
(3.32)
Ilu(’, T)
v(., T)II _< tfoAt.
For the second term in (3.31), note that from the assumption that Hh is an approximation
of order ct of H we get that
Ill)(., tn)
w(., tn)ll
liB(At) E(At)v(., tn-1)
Hh(At) E(At)w(., t-)ll
<_ liE(At)v(’, tn-1)- E(At)w(., tn-)ll
_< (1
where Cg
+ AtCg)llv(’, t_)
+ KhaAt
w(., tn-1)ll-4- KxhAt,
SUpue8 Ig’(u)l. Using this expression, we easily get that
IIv(’, r)
w(., r)ll <_ e cr Ilu0
w011 +
b--h
Inserting this expression and (3.32) into (3.31) we achieve, for a suitable choice of constant
K, the desired result (2.11).
4. Nmedcal expedmen. The purpose of this section is to give some numerical illustrations of the convergence result discussed above. We will also apply the splitting procedure
to systems, and the results from these computations indicate that the rate of convergence derived for a scalar conservation law may also be valid for certain systems. But we are far from
having a rigorous proof of such a general result.
If we have a first-order accurate numerical method for homogeneous conservation laws
at our disposal, Theorem 2.2 states that by using operator splitting, it is possible to solve
problems involving source terms keeping the same order of accuracy.
The numerical method to be considered is a front-tracking scheme for one-dimensional
conservation laws first described by Risebro in 16]. For scalar problems it is quite similar
to the first-order method of Dafermos [3] if a proper approximate Riemann solver is applied.
For completeness, we first briefly describe the front-tracking method. A detailed algorithm is
found in 17],
Like the solution generated by Dafermos’ method, the front-tracking solution is piecewise
constant. For the former, this is achieved by a piecewise linear approximation of the flux
function and a piecewise constant approximation of the initial function. A direct generalization
CONVERGENCE OF OPERATOR SPLITTING
855
New fronts
jump defines the new
Riemann problem
This
FIG. 4.1. The discontinuities or fronts emanating from different Riemann problems initially will collide. This
gives rise to new Riemann problems. The resulting new fronts will again be tracked until they collide.
of Dafermos’ method to general systems of conservation laws seems to be hard to define. For
specific examples the idea of approximating the flux function by a piecewise linear function
can be applied; cf. Hedstrom’s work on the p-system [4].
In the front-tracking case there is generally no approximation of the flux function, but the
solution of the Riemann problem is approximated by a piecewise constant function. Approximating the solution in this way makes it easier to extend the method to systems of conservation
laws.
The first step in the front-tracking scheme is to approximate the initial function with
a piecewise constant function. This gives rise to local Riemann problems. The solution to
each of these Riemann problems is approximated by a piecewise constant function in x / t. The
resulting moving discontinuities or fronts are tracked until they collide and form new Riemann
problems. The algorithm consists of repeating this step. Figure 4.1 illustrates this procedure.
The extension of the front-tracking scheme to nonhomogeneous problems, using operator
splitting, is straightforward. The following algorithm describes the procedure for finding an
approximate solution of
(4.1)
at time
ut
+ f (U)x
g(u)
T.
U :=u0
t:=0
while < T
1. Solve ODE with forward Euler for time At
U := U + Atg(U)
2. Advance the solution using the front tracking scheme for
time At with initial function U
3. := + At
856
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
10-3
2.80
l
400.00
:’
1.60
1.20
200.00
0.80
100.00
0.00
0.00
0.50
1.00
0.50
0.00
2.00
1.50
1.00
1.50
2.00
2000 taken to be a
FIG. 4.2. Example 1. The left panel shows the solution at
1.5 computed with N
reference solution (solid line), and one using N 50. The right panel shows the front curves for 0 <__ < 3.
We now present three problems, one scalar and two involving systems. The time step At
in the operator splitting is related to the number of fronts N used in the approximation of the
initial function. The precise relation will be given in each example.
4.1. Example 1. We consider the problem
(4.2)
ut+
1
u 2}
=u(1-u),
with initial condition
(4.3)
uo(x)
O.l+O.lsin(2zrx),
0.1,
O<x < 1,
else.
The initial profile will steepen into a shock. But due to the source term, the left and right state
of the shock will increase and asymptotically reach u
1. As a consequence the jump will
decrease. From the Rankine Hugoniot condition we obtain that the shock speed will increase
with t, but will stay bounded by one.
In these computations, At 1/N. In the left panel of Figure 4.2, the solution is depicted
1.5 using a very accurate computation (N 2000). A less accurate computation using
at
N 50 is also shown. The right panel of the same figure shows the front curves. From this
panel it is easy to see the shock formation and the increasing shock velocity. Based on several
10, 25, 50, 75, 100, 150, 200, 300} we computed an estimate of the
computations using N
order of convergence. The N 2000 computation was used as a reference solution and the
error e was measured in an relative L 1-norm,
(4.4)
e
where u and U refer to the reference and the from tracking solution, respectively. We estimated
c in e
1.010, with an estimated
(.9(&t ) using standard regression analysis and obtained
error of 0.002.
857
CONVERGENCE OF OPERATOR SPLITTING
4.2. Example 2. In the previous example we estimated the order of convergence based
on a scalar problem and observed approximately first-order convergence as expected from
Theorem 2.2. We now consider a system of conservation laws with a source term and based on
computations using the front-tracking method to once again estimate the order of convergence.
As mentioned earlier, the front-tracking scheme is first-order accurate on homogeneous scalar
problems and numerical experiments indicate that the same order is achieved when applied to
homogeneous systems; cf. 12] and 18]. Even if Theorem 2.2 only deals with scalar problems,
we can still hope for first-order convergence in this case.
The problem to be considered is a system modeling the flow of water down an inclined
conduit with friction. Using dimensionless and scaled variables, the system may be written as
[ ]
h
hv
+
I hvl I
v:Zh +
h2
x
0
h
C
l+h
tan s
v2
1
where h is the height of the free surface and v is the averaged horizontal velocity. The friction
coefficient C is taken to be 0.1, while the inclination angle s zr/6. A detailed derivation of
this system is found in Kevorkian [8].
As an initial condition we use a perturbation of a uniform flow, in which the gravitational
and frictional forces balance. For the initial velocity we use
v0
1.699,
while the initial height of the free surface consists of a triangular perturbation of the uniform
flow level,
x+l.5,
-x+1.5,
1.0,
ho(x)=
-1/2<x<0,
0<x <_ 1/2,
else.
With no friction (C 0), two symmetrical waves will arise from this initial profile. The
introduction of friction not only slows down the velocity of these waves, but also changes the
shape. With C 0.1, one can still observe two waves, but the symmetry is lost.
In Figure 4.3, the solution is shown at
1.0 for two computations, one using N 1000
and one using N 50. The former is used as a reference solution for this problem. In these
computations, At 2/N. As a measure of the error we used
h
v
Ilhll
where (h, v) refers to the reference solution.
Based on computations using N E {25, 50, 75, 100, 150, 200, 300} we obtained an order
of convergence of ot 1.06, with an estimated error of 0.01.
4.3. Example 3. In the final example we consider the Euler equations with gravity, i.e., a
system of equations modeling ideal, inviscid gas under the influence of gravitation. We write
the system in such a way that the homogeneous part and the quantities used agree with the
standard formulation of the homogeneous Euler equations. As a consequence, the total energy
in the definition used does not include potential energy. Hence, the system is written as
(4.5)
,or
E
+
,or + p
v(E+p)
x
-,og
-gv
858
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
1.88
1.18
1.84
1.14
1.80
1.76
1.10
1.06
1.68
1.02
-1.00
0.00
FxG. 4.3. Example 2. The leftpanel shows the height of the free surface h at
(solid line) and N 50. The right panel shows the corresponding velocity v.
2.00
1.00
1.0 computed with N
1000
where p, v, E, and p denote the density, velocity, total energy, and pressure, respectively. When
using scaled and dimensionless constants, the gravitational constant is given by g
1/?,,
where y is a gas constant taken to be 5/3 in this example. We assume that the gas is polytropic,
hence the total energy E is related to the other quantities through the expression
E
1
-PvZ
p
+ y_ 1
The system (4.5) has a nonzero stationary solution
(4.6)
p0
ge -x
v0
0,
/9o
e -x
As an initial condition we use this solution and add a strong perturbation in the pressure and
density,
(4.7)
p(x, O)
Po + e -5(x-1)2,
v(x, O) = O,
p(x, O)
(1/g)p(x, 0).
Two sound waves will emanate from this initial condition. Due to the strong perturbation these
waves will, after a short time, develop shocks. In contrast to what is seen in the homogeneous
Euler equations, the velocity jump across the shock is larger for the shock moving upward (to
the right) than for the one moving downward. When considering density and pressure, the
situation is reversed. This is caused by conservation of energy and the decreasing pressure
and density in the stationary solution.
0.4 is shown for both N 50 and N 1000. In these
In Figure 4.4, the pressure at
computations we used At 2/N. As a measure of the error we used
(4.8)
e
lip-/311
Ilpll
+ IIvIlvllll +
lip 311
Ilpll
,
where (p, v, p) refers to the reference solution (N 1000) and (/3, t3) refers to the front10, 25, 50, 75, 100, 150, 200, 300} we
tracking solution. Based on computations using N
obtained an order of convergence of ot 1.013, with an estimated error of 0.009.
4.4. Some remarks. Example 1 confirms the result of Theorem 2.2, i.e., a first-order
method for the homogeneous equation together with operator splitting implies a first-
859
CONVERGENCE OF OPERATOR SPLITTING
X
10-3
85O.0O
750.00
650.00
550.00
450,00
350.00
250.00
150.00
50.00
0.00
0.50
FIG. 4.4. Example 3. The pressure at
1.00
0.4 computed with N
1.50
2.00
50 and N
1000 (solid line).
order method for the complete inhomogeneous scalar equation. Examples 2 and 3 indicate
that the same result may be valid for certain systems of equations.
The front-tracking method is computationally very efficient when applied to homogeneous problems 12]. The main reason for this pleasant behavior is the dynamical nature of
the method. Efforts are made only where there are wave interactions. In the extreme case
where there are no interactions between waves, the front-tracking method has the necessary
information to compute the solution for all time, while a conventional grid-based method is
limited by some time step condition.
The most obvious way to generalize the front-tracking method to inhomogeneous problems is to replace the homogeneous Riemann solver by an inhomogeneous one. However,
such an approach will lead to front curves which are locally nonlinear.
In the operator splitting technique we have used here, the shock curves remain piecewise
linear. However, even if the rate of convergence is preserved, the introduction of the time step
has the consequence that some of the computational efficiency of the front-tracking method
is lost.
5. Proof of Lemma 3.1. In this section we are going to prove Lemma 3.1. Since v is
the entropy solution of (3.5), Definition 1 states that for any smooth
(x, t) >_ 0 with
compact support and for any k I, we have
r
o
f Iv
+f
klcbt + sign(v k)(f (v)
Iv0 kick(x, O) -Iv(x, T)
f (k))cbx dxdt
klqb(x, T)dx >_ O.
860
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
Following the arguments given in Kruzkov [9, pp. 236, 240], this inequality implies the
E(v), then for any smooth
following. Given a smooth and convex entropy function E
and
0
with
k N,
compact support
4 4 (x, t) _>
r
o
f,
E(v
+
f
k)qbt
+ F(v, k)x dxdt
k)dp(x, O)
E(vo
E(v(x, T)
where F(v, k) is an entropy flux-associated E (v
OF
(v, k) =
Ov
k)(x, T)dx >_ O,
k), i.e., F satisfies
E’ (v k) f’ (v).
It turns out to be advantageous to define an entropy function based on the mollifier function
co introduced in 2.
For any 8 > 0, let
l)
(5.2)
ors(v)
-1
+2
cos(z) dz
and
Es(v) =
(5.3)
+
ors(z) dz.
,
,
We observe that ors (v) is an approximation to sign(v) and Es(v) approximates Ivl. To be more
Also note that IEs(v) Ivll <_
precise, Es(v) = Ivl and ors(v) sign(v) for Ivl _>
Furthermore, E’(v) ors(v) and E’(v) 2cos(v). A corresponding entropy flux is defined
by
(5.4)
crs(z
Fs(v, k)
k) f"(Z) dz.
Observe that
(5.5)
Fs(v, k)
sign(v- k)(f (v)
f (k)).
Let G (v, k) denote the function
Gs(v, k) =
--(v, k) = -2
cos(z
k)f’(z) dz.
Then the properties of cos imply that
(5,.6)
lim Gs(v k)
SO
= -sign(v k)f’(k).
We will use (5.1) to show that q(x, t) = v(x, t) + /(x, t) is the entropy solution of (3.6).
For a given test function q and a given constant k in (5.1), we define a family of test functions
cb(x, t)co(x y)coe(t r) and a family of constants k ap(y, r). Then, by integrating over
x [0, T] in the (y, r)-variables we obtain the following inequality:
(5.7)
I1 (e, 8) q-/2(8, 8) +/3(8, 8) d- 14(8, 8) _> 0.
861
CONVERGENCE OF OPERATOR SPLITTING
Here the terms I1 I4 are given by
Ii(e )
=
fo f fo f
E(v(x, t) + (y, z)
k)(qbt(x, t)w(x
+ qb(X, t)oO(x y)w’(t
(5.8)
y)w(t
r)) dxdtdydz
I1,1(e, ) + I1,2(e, 6).
F(v(x, t), k- @(y, r))(tPx(X, t)w(x
I2(e, 6)
+ dp(x, t)w’(x
(5.9)
y)w(t
r)
r)) dxdtdydz
y)w(t
12,1(e, 6) d- i2,2(8, 6).
E(vo(x) + (y, r)
(5.10) i3(e, 6) =
k)gp(x, O)w(x
Ea(v(x, T) + (y, r)
(5.11) 14(e, 6) =
where, as above, w =
By using the smootess of
Lemma 3.3, it follows that
lira I (e, )
E
y)w(r)dxdydz.
k)(x, T)w(x
y)w(T
z)dxdydZ,
and arguments similar to those used in the proof of
=
E(q(x, t)
k)t(x t) dxdt.
Hence, the dominated convergence theorem implies that
lira lira I,(e, )
o
kt(x, t) dxdt.
Iq(x, t)
For I,(e, ) integration by pas with respect to r yiel
{E(v(x, t) + O(Y, O)
i,(e, )
-E(v(x, t) + O(Y, T)
+
k)(x, t)m(x
k)(x, t)m(x
y)m(t
E;(v(x, t) + O(Y, r) k)Or(y, r)(x, t)m(x
Note that for T > e,
easily derive
lim I1,2(e, )=
-->o
+
- f re(t)dx f m(t
lf
E(q(x, 0)
E’(q
T)et
k)(x, O)dx
k)t dxdt.
y)m(t)
T)}dxdydt
y)m(t
r) dxdtdydr.
/. Hence, by arguing as above we
lf
E(q(x, T)
k)(x, T)dx
862
J.O. LANGSETH, A. TVEITO, AND R. WINTHER
Consequently, we get for I1 that
lim lim 11 (e, 3)
-0 e--+O
+
(5.12)
+
klqbt dxdt
[q
-
k)tdp dxdt
sign(q
Iq(x, O)
kldp(x, O)
Iq(x, T)
klcP(x, T)dx.
For the term 12 we proceed in the same way. From (5.5) we obtain that
lim lim 12,1 (8, )
8--+0 e--0
f(k- 7t))g)x dxdt
sign(q -k)(f(v)
k)(f (q, x, t)
sign(q
f (k, x, t))dpx dxdt.
For/2,2 integration by parts in y and application of (5.6) yields
Gs(v(x, t), k- (y, "t’))lpy(y, Z’)$(X, t)tO(X --y)co(t- r)dxdtdydr
I2,2
-
0
---+
8--+0
Gs(v, k
gr)Ox dxdt
k) f’ (k
sign(q
O)Ox4) dxdt
k) f (k, x, t) dxdt.
sign(q
Hence
r
lim lim/2 (e, 3)
8--0 e-+0
(5.13)
f
k)(](q, x, t)
sign(q(x, t)
f (k, x, t))CPx(X, t)dxdt
T
k)f(k, x, t)xcP dxdt.
sign(q
Finally, it is easily seen that
(5.14)
and
(5.15)
-
lim lim 13 (8, 8)
--,o -o
lim lim 14 (8, 3)
8--,oo
1
f
f
Iqo
Iq(x, T)
kl4 (x, 0) dx
kldp(x, T)dx.
863
CONVERGENCE OF OPERATOR SPLITTING
By inserting (5.12), (5.13), (5.14), and (5.15) into (5.7) we obtain that
r
fo f
iq -kl)t
+ sign(q -k)(f(g,x, t)- f (k, x, t))dpx dxdt
sign(q
>
k)f(k, x, t)x dxdt +
sign(q
Iq0
klqb(x, O)
Iq(x, T)
klqb(x, T) dx
k)zt(D dxdt
for any test function 4 and any constant k. According to Definition 1, q is the entropy solution
of (3.6). Hence, the lemma is proved.
ekledgment. We thank Hans J. Schroll for useful comments on a preliminary version of this manuscript.
REFERENCES
[1] A. J. CHOP,IN, T. J. R. HUGHES, M. E MCCRACKEN, AND J. E. MARSDEN, Product formulas and numerical
algorithms, Comm. Pure Appl. Math., 31 (1978), pp. 200-256.
[2] M. CRANDALL AND A. MAJDA, The method offractional stepsfor conservation laws, Numer. Math., 34 (1980),
pp. 285- 314.
[3] C.M. DAFERMOS, Polygonal approximations of solutions of the initial value problem for a conservation law,
J. Math. Anal. Appl., 38 (1972), pp. 33-41.
[4] G.W. HEDSTROM, Some numerical experiments with Dafermos’ method for nonlinear hyperbolic equations,
Lecture Notes in Math. 267, Springer-Vedag, Berlin, New York, 1972, pp. 117-138.
[5] H. HOLDEN, L. HOLDEN, AND R. HOEGH-KROHN, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Appl., 15 (1988), pp. 595-602.
[6] H. HOLDEN AND N. H. RISEBRO, A method offractional steps for scalar conservation laws without the CFL
condition, Math. Comp., 60 (1993), pp. 221-232
[7] -------, Conservation laws with a random source, Preprim Series, Institute of Mathematics, University of
Oslo, Norway, 1993.
[8] J. KEVORKIAN, Partial Differential Equations, Analytical Solution Techniques, The Wadsworth & Brooks/Cole
Mathematics Series, Pacific Grove, CA, 1990.
[9] N. KRUZKOV, First order quasilinear equations with several space variables, Math. USSR-Sb., 10 (1970),
pp. 217-243.
10] N.N. KUZNETSOV, Accuracy of some approximate methods for computing the weak solutions of a first order
quasilinear equation, USSR Comp. Math Math Phys, 16 (1976), pp. 105-119.
[11] R.J. LEVEQUE AND H. Y. YEE, A study of numerical methodsfor hyperbolic conservation laws with stiffsource
terms, J. Comp. Phys., 86 (1990), pp. 187-210.
[12] J.O. LANGSETH, N. H. RISEBRO, AND A. TVEITO, A conservative front tracking scheme for 1D hyperbolic conservation laws, Proceedings from the Fourth International Conference on Hyperbolic Problems, Donato,
Oliveri, eds., Taormina-Italy, 1992.
[13] B.J. LUCIER, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp., 46 (1986),
pp. 59-69.
[14] ., Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal., 22 (1985),
pp. 1074-1081.
[15] O. OLEINIK, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2,
26 (1963), pp. 95-172.
16] N.n. RISEBRO, A front tracking alternative to the Random Choice Method, Proc. Amer. Math. Soc., 117 (1993),
pp. 1125-1139.
17] N.H. RISEBRO AND A. TVEITO, Front tracking applied to a nonstrictly hyperbolic system of conservation laws,
SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1401-1419.
A front tracking method for conservation laws in one dimension, J. Comp. Phys., 101 (1992),
[18]
pp. 130-139.
[19] Z-H TENG, On the accuracy offractional step methods for conservation laws in two dimensions, SIAM J.
Numer. Anal., 31 (1994), pp. 43-63.
[20] H. WESTERBERGER AND J. BALLMANN, The homogeneous homoentropic compression or expansion--A test
case for analyzing Sod’s operator-splitting, in Notes on Numerical Fluid Mechanics 24, J. Ballmann and
R. Jeltsch, eds., 1988.