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COO-3077-20

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\

^ â–  = â€” T^ + â€”7, - rTT â€” rf(U.,-,)-f(u.-,j] (loOJ

J 2 2 2Ax *â–  ^ J+1 J-1

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

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

g(a,b)=-^.^ + â€” ^ â€” . (1.9)

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

of each of u^,-, and u] -, so long as l-r-â€” 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(Â«)
satisfy Condition E across the interval (u , u ). In the following,

-^ Jo

we assume u < uâ€ž. The case u > uâ€ž can be reduced to this case by
r ÂŁ 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

J

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^ â€ž,

n n J ~

u., and u.,â€ž. 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 . â€ž, u., and u.,â€ž.

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
of

when e > 0. Traveling waves exist and are continuous, Foy  . 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)

ÂŁ

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 Â»

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 uâ€ž 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 -^ â€” (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 ^ '

6

the equation

^x+2ti " ^ (^x+k'^^x+k'-l^ â€˘â€˘â€˘'^x-k' ^ â€˘

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 , uâ€ž, 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 uâ€ž 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' â€˘ â€˘ â€˘ ^ ^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
o

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

(2.2)
- S^^x+k-r^x+k-2" - '^x-k)^ â€˘

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

7

The monotonicity of the right side of (2o2), which we have
named G( â€˘,â€˘,...,*) 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^_]_^ â€˘ â€˘ â€˘ ' 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)

J

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).

ÂĄe 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

8

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

Proof: By assumption

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

and

then by inequality (P)

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

, /^ /^

As M â€” â–ş + 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

X X

1

= 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

XX XX

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

XX A A

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

10

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 ,uâ€ž).
X ^ T i'

ÂĄe now proceed to the construction of traveling waves. ÂĄe
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

-'â–  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 , )
â–  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 â€”

point of T. to have the limit w = uâ€ž. 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

11

Lemma 2.3' Let ^ be an initial value on I such that
^r â€” ^x â€” ^9 ^Â°^ -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,

X

u < u < max P for x > 0.
r â€” 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

X

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 â€” X j> X

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

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

X

of ^ . Moreover, for all ^, u < 2 < u, for -k - ri < x < 0, it
X r â€” X â€” 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

XX XX

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

X

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

12

Lemma 2.3 â–  Theorem 1 is a consequence of Lemmas 2.3 and
2.4.

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

XXX

- + - +

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

X X

respectively. Extract a subsequence, if necessary, so that v""" â€” ^ v .
Then

^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 â€” X â€” X â€” X

Thus the limit w as n â€” â–ş co exists which we write as w and satisfies

X X

u < w < u
r â€” X â€” 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

16

r â€” ^x â€” ^x

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

XX XX

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 (^ ) â€” â–ş w .

X X

But

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 â€” ^x â€” '

and

u < i < a when x =
r â€” ^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^ â€˘

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 CÂ«

r X

'^ o n

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

X Xâ€” 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 â€” X â€” 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

X X

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

X X

(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

18

^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 â– ^^ ^

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

1

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