Gray Jennings.

Discrete traveling waves which approximate shocks online

. (page 1 of 2)
Online LibraryGray JenningsDiscrete traveling waves which approximate shocks → online text (page 1 of 2)
Font size
QR-code for this ebook




Courant Institute of
Mathematical Sciences

AEG Computing and Applied Mathematics Center

Discrete Traveling Waves
Which Approximate Shocks

Cray Jennings

AEC Research and Development Report

yV Mathematics and Computing

^>,-, f-t q\

^ ■ = — T^ + —7, - rTT — rf(U.,-,)-f(u.-,j] (loOJ

J 2 2 2Ax *■ ^ J+1 J-1

to approximate (1.7)0 This equation can be written in the form (1.6)

2 2
, , \ b-a Ax , a +b f-i r^\

g(a,b)=-^.^ + — ^ — . (1.9)

The right-hand side of (lo8) is a monotone increasing function

of each of u^,-, and u] -, so long as l-r-— u.l < 1 for all j. The
J+1 J-1 'AX J ' ^

bound on At/Ax is the Courant-Friedrichs-Lewy stability condition

for the linearized difference scheme.

In general, we require that the right-hand side of (lo6) be a

monotone increasing function of each of the variables u.,,,...,u. , .V

J+k j-k

We give the right-hand side of (1.6) a name rewriting the equation as

n+1 _, / n n n \ / -, -, ^\

Note that G(u, u, ...,u) = u.

We construct traveling wave solutions of (1.6) with limits u
and u at plus and minus infinity respectively. We require that f(«)
satisfy Condition E across the interval (u , u ). In the following,

-^ Jo

we assume u < u„. The case u > u„ can be reduced to this case by
r £ r ^

symmetry. A traveling wave u which moves with speed s satisfies
the difference equation

^x-s(At/Ax) = ^^^x+k^'^x+k-l'-'-'^x-k^ " ^^-^^^

The speed s is related to u and u. by the Rankine-Hugoniot relation.

v-^ That (1.9) does not depend explicitly on u . is a technical problem


that can be removed by iterating the equation once. The resulting

relation is of the form (1.6) and defines u] as a function of u^ „,

n n J ~

u., and u.,„. Leibniz's rule for calculating the partial derivatives

of a composite function implies that the iterated relation is a

n n n
monotone increasing function of each of u . „, u., and u.,„.

Condition E is the weakest condition that one can expect to
imply the existence of traveling waves for all values of At/Ax.
Consider the closely related problem of constructing traveling waves

when e > 0. Traveling waves exist and are continuous, Foy [4] . To
obtain the relation they satisfy write u(x, t) = v(x-st) and substitute
to obtain

-sv' + [f(v)]' = ev" .
Integrate this relation once,

V' = ^^' ^^^^ + constant . (1.12)


The boundary conditions u = u and u = u determine the

I (JO X ~" LU JO

constant as that value at which u^ and u are critical points

of (1.12), i.eo, the right side vanishes when v equals u^ or u »

The Rankine-Hugoniot condition implies the right side vanishes at u^

if and only if it vanishes at u . Note that no orbit running from

u toward u„ can get past the first zero of the right side of (l.ll),
r P,

that is, past a point where Condition E is violated.

The traveling waves we construct are fixed points of an opera-
tor T defined by

(TV)^ = G^^X+k-n'^X+k-l-Tl' - -^X-k-Tl


where r) = -sAt/Ax. In a subsequent paper, we shall show that these
fixed points are stable in the following strong sense. If u^ is a
function defined on the integers which takes on values for which
G{' ,',...,' ) is a monotone increasing function of each of its argu-
ments, then T% converges in L-j^ to a traveling wave.

2. Existence of Traveling Waves When ri is Rational

We fix a choice of u and u. across which f(*) satisfies

Condition E. The speed s is determined from the Rankine-Hugoniot

relation on u , u , and f(*)- The parameter r] is then a function of

6 = At/Ax. We construct in this paper a traveling wave for each

choice of 6 in the interval (0,0 ) where 6 is the largest value

^ max max

of 6 for which each of the partial derivatives -^ — (u,,u, -,, ...,u_, ),
k > j > -k, is nonnegative when all u., k _> j _> -k lie in the inter-
val [u^,u^] .

The construction of traveling waves is done in two parts.
First we construct waves when 6 is chosen so that r) = s0 is rational.
Traveling waves which satisfy (1.12) when s0 is irrational will be
constructed as the limits of traveling waves which satisfy (1.12)
with values of s0. rational which converge to s6 ,

In the remainder of this section, rj is rational. We seek
solutions of

^X+Tl = ^(^X+k'^X+k-l - -^X-k) ^2.1)

such that u, = u and u = u.. We iterate (2.1) once and obtain
+ oo r _ CO i ^ '


the equation

^x+2ti " ^ (^x+k'^^x+k'-l^ •••'^x-k' ^ •

G^^ is a monotone increasing function of each of its arguments as
G is. We may iterate (2.1) a finite number of times so that N is
an integer, maintaining the form of the equation after each itera-
tion. Hence, without loss we consider (2.1) when ri is an integer.
The Courant-Friedrichs-Lewy condition requires that |ti| < k. We fix
u , u„, s and f(0 and consider functions u^ defined on the integers.
We seek traveling waves as solutions of (2.1) with limits at plus

and minus infinity, ^^ qq = ^-^ ^i^^ ^_ od ^ ^H'
We prove the following

Theorem 1: {t\ rational) Let f(') satisfy Condition E across

fu ,u ) where u and u„ determine a shock solution of (l.l) which
^ r S,' r i

travels with speed s. Suppose that Q = At/Ax is chosen so that
sAt/Ax is rational and is smaller in magnitude than k. Moreover, we
suppose that the right-hand side of (2.2) is a monotone increasing
function of each of u _^,,-u. +v_i' • • • ^ ^x-k ^^^"^ ^^^ arguments take on
value in the interval [u jU ]. Then for each number u^ e {\i^,u^),
there is a unique function defined on the integers which takes on the

value u at X = and which satisfies

^X+Tl = ^X - 1^ fg(^X+k'^X+k-r' - ^X-k+l^

- S^^x+k-r^x+k-2" - '^x-k)^ •

The solution u is a monotone function of x and depends continuously
at each value of x on u .


The monotonicity of the right side of (2o2), which we have
named G( •,•,...,*) previously, is central to the proofs which follow
We collect the consequences of this assumption that we will need
later. We name the variables on which G depends writing it as
G(u,,u, -,,..., u , ). The partial derivatives with respect to each of

u,,,u, ., ...,u , are positive. Hence, writing g = g(^j^^^i^_]_^ • • • ' w_1^4-]_)

and 4^ = gi > it follows that
dw^ ^k'

5 . - e(g. -g., J > for k > j > -k

J J J -^

where 5 . = 1 if j = 0, and otherwise. Moreover, gi,^-| = S k " *-**
From g, , T =0, it follows that

-eg^ >

and by induction, -6g. > -6g.,n, 3 > implies

-eg. > for k _> j > . (gl)

At j = 0, we obtain (l -0g ) > -6g, > 0. moreover, by induction it
follows that

(1- eg.) > for -k < j < . (g2)


Additional inequalities can be obtained by considering first that
0g , > and then proceeding by induction for j > -k. We need only
the relations (gl) and (g2).

¥e give now an estimate for the difference

^(^x+k'^x+k-l-'-^x-k) -^^^x+k^^x+k-l - -^x-k)- ^^ ^^^ ^^^ "'^^'^
value theorem and write the difference as


/ 5i:|^(P v^ at a
single point, then u^ > v for all x.

Proof: By assumption

u , — Giu .-ijU ,, -i,o..,u ,
x+T] ^ x+k' x+k-1' ' x-k'


then by inequality (P)

Sum this inequality from x = -M to +M, and eliminate the common terms.

, /^ /^

As M — ► + 00, each of the terms goes to zero. We obtain

< .

This implies that Inequality (P) was an equality for each value of x.
Hence v^^^ - u^^^, u^^j^_^ - v^^^_^, . . . and v^_^ - u^_j^ are of the same
sign for each value of x. The sign may depend on x. The difference
V -u cannot go from positive to negative without being zero. If
u^ = V for a single value of x, then



= u-v = r |S (eu* +(l-0)v* ).(u*-v*)d0

X X J dj ^ X-TJ ^ X-T] X x'

but, as ^G/^j > 0, it must be that u, = v, for t e [x-i^-k, x-ri+k] .
Proceeding by induction, it follows that u = v if u = v at a


single point. Thus if u^ > v^ at a single point, u^ > v for all x.


This completes the proof.

Consider that u is a traveling wave, then u -, is also a
traveling wave. Hence, u^ -u^,-, is of the same sign for all x.

X X~r X

Therefore we have


Lemma 2.2. Let u be a traveling wave such that u e [u ,u.],
X X r i;

then u is a monotone function of x and lies in the interval (u ,u„).
X ^ T i'

¥e now proceed to the construction of traveling waves. ¥e
study the fixed points of the operator T defined on functions whose
domain is the integers greater than -(k+r)). For x+r| > we define

^V^x+Ti = ^^^x+k'^x+k-l - -^x-k) (2.3)

T is the identity on the integers x such that -k- t] < x j< 0. We
define a fixed point of T to be a function u defined on 1 , the
integers greater than -k-T], which satisfies

V = (T v)
X ^ r 'x

and has the limit u, = u .

+ 00 r

A one-parameter feunily of fixed points of T can be extended
to a traveling wave. We identify this family as those fixed points
of T which can be "patched" to a fixed point of T . T is analogous

-'■ Jo X/

to T and is defined on functions with domain, 1~ , the integers less

than k-T] by

T.w) , = G(w ,,,w ,, -,,...,w , )
■ i 'x+T) ^ x+k x+k-1 ' x-k'

when x+T] < and (T w) = w when < x < k-ri. We require a fixed

i/ X X —

point of T. to have the limit w = u„. We define the initial value
i - CO i

of a function with domain equal to I to be the values which it

assumes on the interval -k - ri < x _< 0. Here and in the following,

X assumes values in the integers. The objective of the following
lemmas is to establish


Lemma 2.3' Let ^ be an initial value on I such that
^r — ^x — ^9 ^°^ -k - T) < X < and u _< | < u , for x = 0, then
there exists a unique fixed point u of T with ^ as initial value.
For a fixed value of x, u is a continuous function of |. Moreover,


u < u < max P for x > 0.
r — X ^x

An analogous result is true for T,o The initial value of a
function defined in I", is its value on j< x < k-r^. We state

Lemma 2.4. Let C t)e an initial value on I~ such that

u < C ^ u. for < X < k-ri with u < C ^ u., at x = 0, then there

exists a unique fixed point w^ of T. with C^ a-s initial value. For

^ & X

a fixed value of x, w is a continuous function of C* Moreover


min C^ < w^ < u^.

With these two lemmas we can construct traveling waves when r\

is rational. We prove Theorem 1 now assuming Lemmas 2.3 and 2.4.

Fix u^ e (u^,u^) and let |^ satisfy u^ < |^ < u^, -k - ti < x < 0,

with I at X = equal to u . Then there exists a fixed point of T ,

u^d) such that u < u < u. for all x. The function u restricted
X r — X j> X

to the interval [0, k-ri) is an initial value of a fixed point w^ of T..

Moreover, ^y- "^ "^x — ^P ^'~^^ ^^'^ "^ ^"^"^ -^^^ -k - r) < x < in particular.
The fixed point w determined in this manner is a continuous function


of ^ . Moreover, for all ^, u < 2 < u, for -k - ri < x < 0, it
X r — X — a,

follows that u _< w (^) < u for -k- t] < x < 0. Hence the Brouwer
fixed point theorem implies there is a fixed point | . Define a
function v on I as u(|*) and on I~ as w {i* ) . As vi (^*) = ^*, the


definition is the same on I n I~ . Notice that v is a traveling wave


of the difference scheme and takes on the value u at x = 0, which
was chosen arbitrarily. We gather this into


Lemma 2.3 ■ Theorem 1 is a consequence of Lemmas 2.3 and

Proof : All that remains is to show that a traveling wave
depends continuously on the value it assumes at any point.

Let V be the traveling wave which takes on the value u. at
X = and suppose that u. -+ u . We may suppose that u < u. < u for
all i and that u and u~ e (u >u ). Then by Lemma 2.1,

- i +

u < V < u


- + - +

where u and u are traveling waves which u and u determine


respectively. Extract a subsequence, if necessary, so that v""" — ^ v .

^x+T = ^^^x+k'^x+k-r-'-^x-k)

and moreover, u=u7 "^v, to x+r] = M. Eliminate the
common terms from the sums on each side of the inequality and let
M -^ + OD. The result is

x 0. Moreover, by induction

_^ n _^ n+1 ^
u < w < w < u .
r — X — X — X

Thus the limit w as n — ► co exists which we write as w and satisfies


u < w < u
r — X — X

Hence w, = u , and the proof is complete.
+ 00 r' ^ ^

The continuous dependence of a fixed point upon its initial
value now follows.

Lemma 2.9. (Continuous Dependence on Initial Values) Let ^
be a sequence of initial values such that


r — ^x — ^x

where C is an initial value that determines a fixed point of T ,
then if l"^ -► | for each value of x, it follows that v (^"'') ->■ v (O


pointwise, where v i^'^) is the fixed point of T that ^ determines.

X -^

Proof : Because of the monotonicity of fixed points of T^, it
follows that

Passing to a subsequence if necessary, it follows that v (^ ) — ► w .



from which it follows that w = u o The uniqueness result implies

We define (f(a) to be a set of initial values, ^ , such that

u < ^ < a for -k-Ti < x <
r — ^x — '


u < i < a when x =
r — ^x

We are interested in the supremum of the set

1^ r

n n^n o

Then u = Tutu } which satisfies
X r X ' x^

^x+r| = ^^^x+k'^'x+k-l'-'-^^x-k^ •

Moreover as ? < a and ^ = a* for some x the maximum principle

implies that u < a* for all x > and thus u restricted to (0,k+Ti]

is an initial value ^ which lies in (^(a ) for i sufficiently large.

Hence, there exists a fixed point of T^, w , with initial value C«

r X

'^ o n

Set w = w /i , \» Then w > u for x > and as w bounds u on

X X— y K.~rT1 j X X XX

the interval (O, k+ri] it follows that u^ < w for x > 0. Hence

X -~ X

u < u < w and u, = u . This completes the proof,
r — X — X +00 r i- jr^

We now turn to characterizing a*.

Lemma 2.11 . Let a > u . If, there exists a fixed point of T
with initial value | = a, then {^i contains an interval about a.

Proof : We assume that there is a fixed point u of T with
initial value ^^ = a. As u is not a constant, it follows from the


maximum principle that u^ < a for x > 0. The restriction of u to


(0,k+ri] is an initial value ^ which lies in (f(a-e) for some e
positive and sufficiently small. The dimension of (5(a-e) is k+rjo
Consider the equation


^x+Ti = ^(^x+k'^x+k-r - -^x-k


as a nonlinear recursion of order 2k. The process of calculating

the value of v , as a function of "^j^+j^^ ^x+k-1' " * * '"^x-k+1 ■^^ ^

2k 2k
function Z.' in the usual way from R -> R , which is well defined


Online LibraryGray JenningsDiscrete traveling waves which approximate shocks → online text (page 1 of 2)