Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Initial Boundary Value Problems for Hyperbolic
Partial Differential Equations
Heinz-Otto Kreiss
1, Differential equations in one space dimension. The simplest hyperbolic differential equation is given by
(1.1)
du/dt = cdu/dx,
where c is a constant, Its general solution is u(x, t) — F(x + ci), i.e., it is constant
along the "characteristic lines" x + ct = const (see Figure 1). Therefore, if we
u(l,t) = g ( t )
u(0,t)*g(t
K = 0U<*.°,Sf<»>
K'.I
FIGURE 1
want to determine the solution of (1.1) in the region 0 ^ x ^ 1, t ja 0, we have
to describe initial conditions
(1.2)
u(x,0)=f(x),
for t = 0 and boundary conditions
(1.3)
u(l,t) = g(t)
<0,t) = g(t)
ifc>0,
ifc<0,
© 1975, Canadian Mathematical Congress
127
128
HEINZ-OTTO KREISS
for x = 1,0 respectively.
There is no difficulty in generalizing the above results to systems
(1.4)
du/dt = Adu/dx.
Here u(x9t) = (ua)(x, t), •••, w(w) (x, t))' denotes a vector function and A a constant
n x n matrix. Hyperbolicity implies that A can be transformed to real diagonal
form, i.e., there is a nonsingular transformation S such that
(1.5)
SAS-i = (£' 2„) =
where
0
«1
Ai**\
u0
o
w0
"a22 •
•0
/£i r+ i
<0,
,4" = j»0
0
lo
ar
'•
.... o
«flHr+*2„" — »o
•0
> 0
an
are definite diagonal matrices. We can thus introduce new variables
(1.6)
v = Su
and get
(1.7)
dv/dt = Ädvßx.
The last equation can also be written in partitioned form
(1.8)
dvljdt = AW/dx,
dvll/dt = iiWV&c,
where v1 = (v(1), •••, v(r))', v11 = (v (r+1) , •••, v(w))'. (1.5) represents n scalar equations. Therefore we can write down its general solution
(1.9)
v<»(x, 0 = v<»(* -I- a,t)9
j = 1, 2, ..., n,
which are constant along the characteristic lines x + a-t = const. The solution is
uniquely determined in the domain 0 <Z x £ 19 t t£ 0, and can be computed explicitly if we specify initial conditions
(1.10)
v(x,0) = / ( * ) ,
O^x^l,
and boundary conditions
(1.11)
v»(0, 0 = 2*0vn(0, 0 + gQ(t),
v"(l, 0 = R^(l,
t) +
gl(t).
Here R0, Ri are rectangular matrices and g0, gx are given vector functions. If we
consider wave propagation, then the boundary conditions describe how the waves
are reflected at the boundary.
Nothing essentially is changed if A = A(x, t) and Rj = Rj(t) are functions of
x, t. Now the characteristics are not straight lines but the solutions of the ordinary
differential equations dx/dt = aj(x, t). More general systems
(1.12)
dv/dt = A(x, t)dv/dx + B(x, t)v + F(x, t)
can be solved by the iteration
INITIAL BOUNDARY VALUE PROBLEMS
(1.13)
dp+n/dt = Afa t)dv^/dx
lf0
129
+ F™
m
where F «= Bfa t)v + F, Furthermore, it is no restriction to assume that Â
has diagonal form, If not, we can, by a change of dependent variables, achieve the
form (LIO).
We can therefore develop a rather complete theory for initial boundary value
problems by using characteristics. This has of course been known for a long time.
The only trouble is that this theory cannot be easily generalized to problems in
more than one space dimension, For difference approximations it is already inadequate in one space dimension,
2. The energy method. The main tool for proving the existence of solutions in
more than one space dimension consists of "a priori estimates", Once these estimates have been established the existence and uniqueness of solutions follow by
standard functional analytic arguments. The estimates are of the following type,
Consider a system of partial differential equations
(2.1)
du/dt *= Pfat,d/dx)u
in a domain Q with initial conditions
(2.2)
ufah)
=f(x)
at some time t = th and boundary condition
(2.3)
Rfa t)u « 0 on dQ.
FIGURE 2
The problem is called weakly well posed if
(2.4)
||K(X, * 2 )|| 0
g Kexp(a(h-tx))\\ufa
t2)\\0,p.
Here || • \\0 denotes the usual L2-norm over Q and || • \QiP thç i2-norm which also
contains all space derivatives up to order p. If p = 0 then we call the problem
strongly well posed.
There is a large class of problems for which the estimate (2.4) is immediate. This
is the class of problems for which P is semibounded, i.e., for every fixed t and all
sufficiently smooth w which fulfill the boundary conditions we have
(2.5)
(w, Pw) + (Pw9 w)a è 2a\\w\\l
130
HEINZ-OTTO KREISS
Here a is some constant independent of w. (2.5) implies for all sufficiently smooth
solutions
3|M|o ßt = (du/dt, u)0 + (u-du/dt)0 = (Pu, u)Q + (u, Pu)0 ^ 2oc\\u\\20.
Therefore
h fa h)\\o ^ ^pHt2-h))\\ufa
t£\\0.
For symmetric hyperbolic systems this theory has been developed by K.O.
Friedrichs [3]. As an example consider afirst-ordersystem
m
(2.6)
du/dt = Adu/dxi + £ Bjdu/dxj = P(d/dx)u
j—2
with constant coefficients for / ^ 0 and x e Q. Here fl denotes the half-space
0 ^ xi < oo, — oo < Xj < 00,7 = 2, •••, w. Furthermore >4 has the diagonal form
(1.5) and the Bj are symmetric matrices.
FIGURE 3
For t = 0 initial values
(2.7)
u(x,0)=f(x),
||/|0<oo,
and for x\ = 0 boundary conditions
(2.8)
w*(0, x_, 0 = jR0wn(0, x_, 0, * - = (x2, —, x j ,
are given.
Partial integration gives for all sufficiently smooth w e L2(Q) which fulfill the
boundary conditions
fa Pw)0 + (Pw, w)Q = - J w*Aw\x^o dxdo
Therefore the operator P is semibounded if R0 is such that A11 + R^A^Q ^ 0.
This is for example the case if \RQ\ is sufficiently small. The disadvantage of the
energy method is that it is a trick. When it works it is the most simple method to
derive existence theorems. But it does not give necessary and sufficient conditions.
INITIAL BOUNDARY VALUE PROBLEMS
131
We shall now discuss another technique based on the Laplace transform which
gives necessary and sufficient conditions,
3. Laplace transform. We consider again the problem (2.6)—(2.8) and assume
now that the system is either symmetric or strictly hyperbolic, i.e., the matrices A
and Bj are symmetric or the eigenvalues of the symbol
P(iœ) ;=
ì(Aù)I
+ S
CüJBX
O)V real,
S |û)v|2 ^ 0,
are all distinct and purely imaginary. Furthermore the matrix A has the form (1.5)
which is obviously no restriction.
In one space dimension the initial boundary value problem is always well posed.
This is not true in higher dimensions. Already S. Agmon [2] has observed
LEMMA
(3.1)
3,1 Assume that the problem (2,6)—(2.8) has a solution of the form
wfa t) = (j>(x) exj>(st + i(co-, x^)),
<ûJ_, X_> - 2 a)jXj9 <0j real9
j=2
2
2
where real s > 0 and \\<j>(xi)\\ = Jo°|^| dxx < oo.
Then the problem is ill posed.
PROOF,
If wfa t) is a solution then the same is true for
wTfa t) = exp (r(st + i(co-9 * - » ) <f>(zxx)
for all real numbers % > 0. Thus there are solutions which grow arbitrarily fast
with time.
We shall now derive algebraic conditions such that there are solutions of the
above form, Introducing (3,1) into (2.6) gives us
LEMMA
3,2. There is a solution of type (3.1) if and only if the eigenvalue problem
(3.2) s(j> = Ad(j>jdxx + iB(coJ)(/>9 B(œJ) *= S Bjù>j9 \\<f>\\ < oo, (j>\0) = *o0ir(O),
has an eigenvalue with real s > 0.
(3.2) is a system of ordinary differential equations which can also be written in
the form
(3.3)
d<j>ldxx = M<j>9
M = A-Ks - iB(o>-)).
For M we have
LEMMA 3.3. For real s > 0 the matrix M has no eigenvalues K with real K = 0.
The number of eigenvalues with real K < 0 is equal to r, the number of boundary
conditions.
Therefore the general solutions of (3.2) belonging to £ 2 can be written as
(3.4)
jj*y#/*).
Introducing (3.4) into the boundary conditions gives us a system of linear equations
132
HEINZ-OTTO KREISS
C(S)X = 0,
l^&u-tW.
Thus we can express our results also in the following form :
LEMMA 3.4. The problem (2.6)—(2.8) is not well posed if Det | C(s) | = Ofor some
s with real s > 0.
The main result of this section is (see [7], [14], [13]):
THEOREM 3.1. Assume that Det|C(^)| ^ 0 for real s ^ 0. Then the problem is
strongly well posed.
There is still the boundary case that Det | C(s) | = 0 for some s = ?'£, £ real. As R.
Hersh [5] has shown these are weakly well-posed problems. The main trouble is
that the generalization of these boundary cases to variable coefficients is very
difficult.
4. Problems with variable coefficients in general domains. Now we consider systems (2.6)—(2.8) with variable coefficients in a general domain Q x (0 g / ^ 7"),
FIGURE 4
Here we assume that the coefficients and the boundary dû are sufficiently smooth.
Connected with this problem there is a set of half-plane problems which we get in
the following way: Let P0 = (x0ï to), dû x (0 ^ t ^ T), be a boundary point and
let x = S(x), i = t — to with S(xo) = 0 be a smooth transformation which locally
transforms the boundary into the half-plane xx = 0. Apply this transformation to
the differential equations and the boundary conditions, freeze the coefficients at
x = t = 0 and consider the half-plane problem with constant coefficients. Then
we have
THEOREM 4.1. Assume that for all the half-plane problems the conditions of%2
hold, i.e., that all the operators connected with the half-plane problems are semibounded. Then the original problem is strongly well posed (see [3]).
THEOREM 4.2. If the system (2.6) is strictly hyperbolic and iffor all the half-plane
problems with frozen coefficients the determinant condition of Theorem 3.1 is fulfilled
then the original problem is strongly well posed (see [7], [14], [13]).
REMARKS. (1) It is not known whether the determinant condition guarantees
well-posedness for symmetric systems which are not strictly hyperbolic. This is a
rather disturbing gap in the theory.
(2) Quite a lot of progress has been made for the boundary case that Det | C(s) |
INITIAL BOUNDARY VALUE PROBLEMS
133
= 0 for some s = iÇ, £ real, The key is to consider not only the half-plane problem
for du/dt = Pu but also all perturbed problems du/dt « Pu + Bu where J? is a
constant matrix.
(3) It is assumed that A is nonsingular. However, progress has been made also
for the singular case (see [12]),
(4) If the boundary is not smooth then new serious problems arise. See for
example [10], [11].
5. Difference approximations in one space dimension. We start with an example
which explains most of our difficulties. Consider the differential equation
(5.1)
du/dt = du/dx
in the quarter-plane x ^ 0, t ^ 0, with initial values
(5.2)
u(x90)*=f(x).
From § 1 we know that no boundary conditions need to be specified for x = 0,
t ^ 0. We want to solve the above problem using the leap-frog scheme. For that
reason we introduce a time step At > 0 and a mesh with âx > 0 and divide the
#-axis into intervals of length Ax, Using the notation vy(t) <= v(xV9 t)9 xy = vAx9
t = tfi - [xAt9 we approximate (5.1), (5.2) by
(5.3)
vv(t + At) = vv(t - At) + 2AtD0vv(t)9
v = 1, 2, - ,
with initial values
(5.4)
v„(0) = ffa),
vv(At) = ffa) + Atffa)/dx.
Here D0vv = (vv+x - vv-X)/2Ax denotes the usual centered difference operator.
We assume that (5,3) is stable for the Cauchy problem, i,e,, 0 < At/Ax ^ 1.
It is obvious that the solution of (5.3), (5,4) is not yet uniquely determined.
We must give an additional equation for v0. For example
(5.5)
Vo = 0.
This relation is obviously not consistent. In general it will destroy the convergence.
Let/fa) ss 1. Then ufa t) = 0 and v„(0 = 1 + ( - 1 )*%(*)> where yv(t) is the solution of
^ ~
yv(t + At) *= yjj - At) - 2AtD0y,(t)9
yM = yJLàt) = o,
with boundary conditions
K
v = 1, 2, •••,
}
(5.7)
yo(t) = - 1.
(5.6) and (5.7) is an approximation to the problem dwjdt = - dw/dx, wfa 0) - 0,
w(0, 0 = - 1 , i.e.,
wfa t) = 0
for t < x9
= - 1 for t ^ x,
Therefore
134
HEINZ-OTTO KREISS
v„(0 ~ 1
for t < X,
~ 1 - ( - 1)' for t ^ x.
This behaviour is typical for all nondissipati ve centered schemes. Therefore one
needs to be very careful when overspecifying boundary conditions. The oscillation
decays if the approximation is dissipative. However, near the boundary the error is
as bad and, for systems, it can be propagated into the interior via the ingoing characteristics.
Now we replace (5.5) by an extrapolation rule
(5.8)
Vo(0 - 2vx(t) + v2(t) = 0,
which is the same as using for v = 1 the one-sided difference formula
(5.9)
vx(t + At) = vx(t - At) + (2At/Ax)(v2(t) - vx(t)).
The approximation is only useful if it is stable. If we choose
v„(0) = 1 for v = 0,
= n0 ffor v ^> A
0,
,,.
A
vx
v(dt)
' = °
v
„
f o r a11
v>
as initial values then an easy calculation shows that
|| v(0 \Ax = c o n s t a ) ,
I v I Ax = S | vy |2 Ax.
This growth rate is the worst possible and one might consider the approximation
to be useful. However, if we consider (5.1) in a finite interval 0 ^ x ^ 1 and add
the boundary condition
(5.10)
M(1, 0 = vN(t) = 0,
NAx = 1,
for both the differential equation and the difference approximation, then there are
solutions which grow like
(5.11)
| K 0 U = conrt(f/J0<,
which is not tolerable. This behaviour can be explained as follows : At the boundary
x = 0 a wave is created which grows like t/At. This wave is reflected at the boundary x = 1 and is increased by another factor t/At when it hits the boundary x = 0
again, and so on.
All these difficulties can be avoided by using, instead of (5.9), the one-sided approximation vi(f + At) = vx(t) + (At/Ax)(v2(t) - vx(t)) or
vx(t + At) = vx(t - At) + (At/Ax)(v2(t) - \(vx(t + At) + vx(t - At))).
One can also keep (5.8) if one replaces the leap-frog scheme by the Lax-Wendroff
approximation or any other dissipative approximation.
Let us discuss the general theory. (For details see [4], [7], [8].) We consider
general difference approximations
(5.12)
with boundary conditions
vy+x(t + At) = Qvv(t)
INITIAL BOUNDARY VALUE PROBLEMS
(5.13)
135
BVo = 0
such that the solution is uniquely determined by the initial values vy(0) = /,,,
The approximation is useful only if it is
(1) consistent, i.e., it converges formally to the continuous problem,
(2) stable (weakly or strongly) which is the difference analog of well-posedness.
There is never any problem in deriving consistent approximations. It is the
stability which causes the problem. Corresponding to the continuous problem there
are two methods to decide whether a given method is stable : Laplace transform
and energy method.
The theory based on Laplace transform is analogous to the theory for the continuous case. The stability is determined by the properties of the eigenvalue
problem
(5.14)
(z-Q)<j>y = 09 i t y o - 0 ,
\\<f>\\% = S | & | M x < o o .
Corresponding to Lemma 3,2 we have, under reasonable assumptions for Q:
LEMMA 5.1. Assume that (5,14) has an eigenvalue z = <z0 wiVA \zQ\ > 1. Then the
approximation is not stable,
This condition can also be expressed as a determinant condition Det | C(z0) | = 0
for some z = z0 with \zQ | > 1. Then, corresponding to Theorem 3,1, we have
THEOREM
5.1. The approximation is strongly stable if'Det | C(z) | ^ Ofor | z | ^ 1,
Now we turn to the energy method. Consider again the differential equation
(5.1), (5.2). The problem is well posed because there is an energy equality
(5.15)
(u, du/dx) + (du/dx9 u) = - |w(0)|2.
Therefore we want to construct approximations to d/dx which have the corresponding property.
We define a discrete norm
oo
(5.16)
(M, V) A = 0*AvAx + S u*Vy/lx.
v—r
Here ü = (w0, •••, wr-i)', v *= (v0, •••, vr„x)' denote thefirstr components of w, v and
A = A* is a positive definite r x r matrix. In [9] we have shown that one can construct accurate approximations Q for which (5,16) holds. The main trouble is that
the norm and the approximation near the boundary are very complicated. This
makes its generalisation to approximations in more than one space dimension on
general domains difficult. Furthermore, it is not known how to include dissipation
in the construction. However, it should be pointed out that this construction also
works in more than one space dimension provided the net follows the boundary.
6. Difference approximations in more than one space dimension. Nothing essentially
new needs to be added to derive the theory of difference approximations for halfplanes because Fourier transforms of the tangential variables x„ give us a set of onedimensional problems. The situation becomes much more complicated if we con-
136
HEINZ-OTTO KREISS
sider general domains with smooth boundaries. Observe that this is not the case
for the differential equations because we can always introduce a local coordinate
system, thus reducing the problem to a set of half-plane problems. This is not
possible for difference approximations, Once we have picked the net everything is
fixed. D. Schaeffer [15] has tried to handle this situation and has developed a beautiful theory. However, its practical importance is somewhat doubtful. Let us consider
a very simple example. We want to solve the differential equation
(6.1)
du/dt = - du/dx
in the two-dimensional domains 2y — x ^ 0. The initial values are
(6.2)
ufa y9 0) = ffa y) for 2y - x ^ 0, t = 0,
and the boundary conditions are given by
(6.3)
ufa y, t) = gfa y, t) for 2y - x = 0, t ^ 0.
We introduce gridpoints by x3- = jAx, yt- = iAy9 Ax = Ay.
FIGURE 5
Thus, there is a gridpoint on the boundary only on every second row. Now we
approximate (6.1) by the leap-frog scheme and the boundary conditions by
v,-.; = &•,;
if 2j= i,
2
?u + M.J = Si+i/2j if y = i + 1.
Here v,-,y = v(iAx, jAy9 t). Therefore we get two different solutions on two different meshes. As long as the solution of the differential equation is smooth the
solutions of the difference equation on these different meshesfittogether. However,
if for example/ = 0 and g = 1 then the solution of the differential equation is a
discontinuous wave propagating into the interior. Now the solutions of the difference approximation on the different nets do notfittogether,
We get oscillations in the tangential direction of the wave. There are two possible
methods for remedying the situation: (1) Add dissipation to smooth out the tangential oscillations. (2) Introduce curved meshes which follow the boundary. The
second procedure is much more accurate and should be preferred if possible. A
lot of progress has been made in this direction. See for example [1].
V
INITIAL BOUNDARY VALUE PROBLEMS
137
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