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A Comparison of an Approximate Analytic Solution
With a Finite Difference Method

[^ Prepared under

Navy Contract N6ori-201 Task Order No. 1
\ By the

. Institute for Mathematics and Mechanics

Sj. New York University

October, 1947


The present report by Miss Anneli Leopold,
prepared under Navy Contract N6orl-201, Task
Order No. 1, supplements Dr. Friedrichs' paper
Formation and Decay of Shock Waves report
IMM-NYU 158. It discusses the range of validity
of the numerical approximation given in the
latter report for the treatment of decaying shock

Miss Leopold was greatly assisted in the
numerical computations by Mr. John Butler.

R. Courant


A Comparison of an Approximate Analytic Solution
with a Finite Difference Method.


In report IMM-NYU 158, Formation and Decay of
Shock Waves by K. 0. Friedrichs, an approximate method
was developed to determine the behavior of a decaying
shock wave in a gas, i.e. the behavior of a shock wave
weakened by a rarefaction wave overtaking it. This
method is expected to yield rather accurate results In
the case of weak shocks; it is of interest to deter-
mine for what range of shock strengths its use may be
Justified, and beyond what shock strength a more exact
solution is required.

In order to test the validity of the "approximate"
method by comparison with an "accurate" result, a corre-
sponding problem for surface waves in shallow water is
treated first by the "approximate" method of IMM-NYU 158,
and then by a finite difference scheme applicable to
shocks of any strength. The accuracy of the solution
obtained by the latter scheme is limited only by the size
of the mesh used. The discussion of a problem concerning
water wave phenomena rather than gas dynamics proper Is
advantageous; for, while the differential equations are
of the same form in both cases, there are no entropy
changes in water so that the finite difference scheme
is simplified considerably.

- 2 -

For every shock strength considered, a comparison

of the results obtained by both methods determines the

accuracy of the "approximate" method. The shock strengths

treated here. I.e. the ratios of height h^ of the

moving water to the height of h^ of the still water

(see fig. 1), are: 1.2, 1.5, 1.8, 3.0 respectively. The

results show a very good agreement of the two methods of

solution for all these cases, even for ^ = 3.0 which is


a stronger shock than those usually classed among weak


The formulation of the physical problem Is as follows;

A shock wave - or bore as It Is also called - moves with

constant speed U Into still water of uniform depth h^.

The height of the water behind the shock Is h^^, and Its

velocity Is u^ (see fig. 1).



Figure 1

At a certain time t = 0, a vertical wall Is suddenly In-
serted into the moving water at a point x = 0, so that
a depression (or rarefaction) wave is created which will
catch up with the shock, since such disturbances are propa-
gated with a speed greater than that of the shock (see
pages 5 and 6). Thus the shock will gradually be weakened.

* Such a bore may be created by pushing a rigid vertical
plate through the water at constant speed.

- 3 -

Figures 2 and 3 describe the physical situation at two
Instances after the insertion of the wall.


Plp^ure 2


Figure 3

In section 1 of this report, the "approximate" method
will be discussed and applied to the problem as formulated
here. In section 2, the finite difference method is de-
scribed and used. The graphs II and III show the results
of both methods.

A noteworthy result of these considerations is that
strong shocks weaken very rapidly at first. This suggests
the following treatment of strong shocks: One uses the
finite difference method initially but finds, after rela-
tively few steps, that the shock has weakened sufficiently
to justify the use of the "approximate" method in con-
tinuing the solution. Thus the method of IMM-NYU 158 is
an extremely useful one even in cases where shocks are

- 4 -

Let g be the acceleration due to gravity, h the
depth of the water, x the distance from the wall and t
the time after the Insertion of the wall. Let x^ be
the distance between the wall and the discontinuity at
the time the wall is inserted. I.e. at t = 0. We, use
the following definitions in order to work with dimension-
less units:

u "Vgh = particle velocity.

XX^ = X,

U Ygh = shock velocity.


= ygh so


= ^ and


= t.


Our initial conditions are:

h(x) = h, for < X < 1. h(x) = h^ for at > 1
1 o

u(x,0) = for X > 1

The problem is to find the strength, velocity and position
of the shock, as functions of time.

- 5 -

Section I « Approximate method (applicable to weak shocks) .

If P Is the density and p the pressure of a gas and
V the velocity relative to the shock, then the mechanical
shock conditions for a gas are:

) =i [ i^ + 2]

* This assumption is consistent with the "approximate"
theory of report IMM-NYU 158, because the change In the
Riemann invariant across the shock appears only in third
and higher order terms.

- 8 -

In order to find the shock curve U = f[x( 5 ),'t(l )],
a differential equation for T( 5 ) is set up and solved.
We have now, on the left side of the shock,

i-=-i- = u( S) + c(M or X = [u( !>) + c(S-)] ^ + § ;



dt . -rr. '

[u( 5 ) + c( S )] ^ + t[u ( 5 ) + c ( 5 )) + 1 ,

where the prime denotes differentiation with respect to 5 •


dx _ dx dt n ^^
d5 " dt dj - " dl

u II =hx{l )} || + "t [u (5) + c (S)] + 1

(11) or ^[y.(^(l)^ci$))]=:-tU\l)^c\l))-^l


Equation (S() is an ordinary differential equation in t( 1 ).

Setting r ^4^Lll£-^7Ti *i3 = ^ (3> ), the soluti

/ u-[u(S)+c(5)] -^





t( S ) = e

- i(M

U-.(u+c) ^^5



x( 5 ) = 5 + (u + c)t


yields a pair of parametric equations for the shock curve.

In order to obtain a simple explicit expression for

"t( S), an expansion of the quantities u and U in terms

of the parameter cr(5) =u(!>) + c(S) - 1 (similar to

- 9 -

that described In IMM-NYTJ 158) Is made; terms up to second
order are kept and equations (12) and (13) become:



1(5) = k

ff^(36/5 - ef
_ o''^(36/5 - a^)"^


:(5 ) = 5 + (OT + 1) t( !))

where S = 5(0) = u(0) + c(0) - 1 = u^ + c. - 1 .


In terms of the parameter 5 , these equations can be written
In the form ^

-2 1



t( !>) = k



(41/5 - 8)(^

x(5) = 3 + -^ t(!,)

- •^4^)(3 - 1)

K^ - 1)

- 1

where s = u. + c | or. In terms of the original coordinates
X and t.

t(3) = k

(41/5 - -^-^)(s - 1)
(41/5 - s) (jL^ - 1)


x(5 ) = ^^ t(3)

which is convenient for numerical work. Equations (17)
were used to compute the shock curves represented on the
graph by broken curves.

It Is interesting that only a part of the depression
wave interacts with the shock; the remaining part leaves
the shock unaffected. In the physical situation, the active

- 10 -

portion of the depression wave Is the forward moving part
whose height Is greater than h (see figure 5). In the
X, t-plane, the active region Is that In which the straight
characteristics satisfy the Inequality -^ = u + c > 1
(see figure 6). This behavior may be deduced by examining
equations (17). When ^^'^ = 1, x and t (the coordinates
of the point of Intersection of the shock with the charac-
teristic where 5 = J> = k - ^) become Infinite. Insert-

ing this value of


In (9 ) and (10 ), one has:

u( 3*) = 0, c( 5*) :* 1

This illustrates that only the forward moving part of the
water affects the shock (see figures 5 and 6), and that
it alone decays the shock for an infinltaly long time.

Figure 5


Figure 6

- 11 -

Suppose that. Instead of moving the wall with such a
velocity that cavitation occurs (i.e. the case in which
c varies from c. to across the rarefaction wave),
the wall Is given a constant velocity w greater than
u. - 2c. hut less than u,. Then the rarefaction wave
Is narrowed and a constant state appears to the left of
the variable zone. The particle velocity in this con-
stant state is just the wall velocity w. If w < 0,
nothing essentially different happens, because then, the
rarefaction wave is still wide enough to Include the
"active" portion which decays the shock for an infinitely
long time. If however, < w < u., there is an interaction
which ends after a finite time during which the shock de-
cays from one constant shock to a weaker constant shock,
(see figures 7 and 8). The final constant shock is, of
course, constant only within the approximation considered









Figure 7

- 12 -

Figure 8

In order to Investigate at which time a shock has

decreased from h. to an arbitrary height

2 ^
h = c ,(h < h < h^), one uses the equation

t [S (c*)] = k

(17/5 - e")(s - 1)
(41/5 - s)(c* - 1)


for t as a function of c = h •

(t = k represents the time at which the decay begins )>

Graph I represents the height as a function of
time in case of the following initial shock strengths:

2 ^1
c!' = Tr= = 1.2 , 1.5 , 1.8 , 3.0 , respectively.


It Is apparent that the height decreases rapidly at first
and then approaches Its asympotlc value slowly.

7o t

- 13 -

Sactlon II. Finite difference method.

The differential equations describing the flow In
this problem are:



u. + uu + 2cc =

t X X

2c. + cu + 2uc =
t X X

The corresponding characteristic equations are

(2) ^ = u*o

u + 2c = const.


along C characteristics


11 = '"- =

u - 2c = const

along C characteristics

Replacing the derivative ^ In (2) by the difference
quotient, one obtains, for any Interior point of the net
shown In figure 9:

J^~- )s.s

Figure 9

- 14 -

iA\ ^Ik " ^1-1, k _ 1 r„ ,, ,

- X

(5) Ik 1,^-1 = I [T + T ]

hk \,k-l '^ ^^ ^*^ ^

where S^^^ = ^x^^ + c^^^ and T^^^ = u^^^ - c^^^
The slBiultaneoua solution of (4) and (5) yields:

^, ^


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