Numerical methods for some nonlinear models for long, low water waves

Numerical methods for some nonlinear models for long, low

water waves

Citation for published version (APA):

van Ginneken, C. J. J. M., Jong, de, L. S., & Veltkamp, G. W. (1979). Numerical methods for some nonlinear models for long, low water waves. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 79-WSK-08). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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Eindhoven Nederland

Onderafdeling der Wiskunde

Eindhoven

The Netherlands

Department of Mathematics

Numerical methods for some nonlinear models

for long, low water waves

C.J.J.M. van Ginneken, L.S. de Jong and G.W. Veltkamp

T.H.-Report 79-08

This report deals with the numerical calculation of the solutions of three different systems of partial differential equations for gravity driven waves.

The systems are nonlinear and known as equations of Boussinesq type. It turns out that one of the systems is ill-posed. By regularizing this sys-tem it is possible to approximate the solution, corresponding with a very smooth initial condition, during a limited time. The systems are_solved, u81nq standard techniques for partial differential equations (Crank Nicolson and method of lines). k_number of numerical results are presented.

AMI Subject Classification Scheme 1979:

1. Introduction

1.1. Statement of the problem 1.2. Outline of contents

2. Background of the various models 2.1. Exact equations

2.2. Waves of small amplitude 2.3. Long waves of small amplitude 2.4. Boussinesq equations

3. Classical Boussinesq equations 3.1. The linearized equations

3.2. Regularization of the linearized equations 3.3. Numerical method

3.4. Numerical results 3.5. Figures

4. The stabilized Boussinesq equations 4.1. The linearized equations

4.2. The numerical method

4.2.1. The choice of boundary values 4.2.2. The staggered net method 4.2.3. The computation of Rv 4.2.4. The accuracy

4.3. The results

5. Discrete Boussinesq equations 5.1. The linearized equations 5.2. The numerical method 5.3. The results References .1 1 2 3 3 4 5 9 11 12 13 17 19 23 28 28 28 29 30 32 32 35 36 36 37 38 49

1. Introduction

1.1. Statement of the problem

The following problem was set by L.J.F. Broer. Given are the following three models to describe gravity driven waves on a water surface in two space dimensions:

( i) Classical Boussinesq equations 1 n t ;:; - (v + -3 v xx + llV) x (1.1.1) 1 2 -(n+'2 v )x (CB)

(ii) Stabilized Boussinesq equations (SB) n t = -(Rv + nv)x (1.1.2) with (1.1. 2a) 1 2 v = - ( n + - v ) t 2 x u := Rv satisfying 1 u - - u = v, u (i 0 ) , t) 3 xx 0,

(iii) Discrete Boussinesq equations (DB)

d n)

-1

(w + n w - w - n w )

dt n -2 n-l n-l n-l n+1 n+l n+l

(1.1.3)

The systems CB and SB are considered in the region _00 < x < 00 , t > O.

The set DB is considered for n

=

0, ±1,±2, •.• and t > O.

The quantity

n

is the (dimensionless) height of the wave, measured from the

equilibrium.

The quantities v and ware connected with the velocity at the water surface.

In the set (DB)

n

corresponds with n(n,t), whereas the following relation

n

between v and w should hold:

(1.1.4) v(n, t)

In order to be able to compare the three models, we were asked, given certain initial conditions, to compute the solutions numerically.

(1.1.5) (1.1.6) n (x,O) h/ (1 + AX ) 2 v(x,O) = 0 , 2 n (0)

=

h/(1 + An ) n w (0)

=

0, n (CB and SB) (DB)

where h :: 1/3 and It "" 1/20, 1/5, corresponding to a "flat" and a "steep" inital elevation.

The following quantities

+00 +00

(1.1.7) _H :=

J

ndx E :=

J

1 2 1 2 1 2

('2(

1 + n) v -

6"

v x +

'2

n } dx,

-""

where n and v are the solutions of (CB), should be constants

of the motion. The ~Ciuantity H ~=-C)r::-:~~_c)_ll~dS to the total mass C)f the s~~_::ID._, __ _ whereas E is an approximate expression for the energy in the system. Their

constan~i could serve as a check on the numerically computed

n

and v.

By analogy we have for (SB) and (DB) the constants of the motion

+00 +00

f

:=

J

1 1 2 1 2 (1.1.8) H ;= lldx E (2"RV + znv +

III )

dx (SB) _00 _00 and _{+00 00} 1 2) Y. :=

I

(1.(1 )w 2 H := _lln E ₂ + _nn + n

2l1

n (DB) n=-oo n=-oo respectively. 1.2. Outline of contents

In chapter 2 i t is shown how the three models can be derived from the

exact equations for irrotational, gravity waves on a layer of a (non-viscous, incompressible) fluid.

This treatment differs from the derivation of Broer [1,2J . It is a preastion that we are dealing with long, low waves. In literature, the preas sump-tions of either low or long waves is classical. In case of low waves, all non-linear terms are deleted from the exact equations; in case of long waves, every reference to vertical motion is deleted. The three models treated in this report should be considered as a mixture of these two cases.

The chapters 3, 4 and 5 are devoted to CB, SB and DB respectively. In chapter 3, i t is shown that CB is ill-posed. Only for initial functions n (x,D) of which the Fourier spectrum decays as fast as exp(-ak2) for some a :> 0, a sol-ution exists during a finite or infinite period of time; the solsol-ution does not depend continuously on the data (perturbations on the initial function may

grow beyond limits). Nevertheless, by regularizing eB, i t is possible to approximate the solution, be i t only during a limited time. It is striking that initial functions of the form

2

j

n(x,o) =£ exp(-x ) v(x,O) :: 0 ,

(0 < E < 1)

give rise to a solution for which n(O,t) increases for some time after t

=

O. The other models are well-posed. In both cases the solutions are computed during the time period (0 < t < 50). In case of the longer initial wave

n(x,O) :: h/(l + A,,2), (h = .333, Act .05) the solutions of SB and DB are in-distinguishable from each other. If h :: .333 and A

=

.2 however, the

sol-u~on of DB is more oscillatory, especially for the smaller values of x. The question how n and v behave for very large values of t is not considered here and would require further investigations.

2. ~ackqround of the various e~uations

In this section tile background of the various models given in 1.1 will be discussed.

In literature [1, 2,5J t.hese models are derived in another way. There an ex-pression for the total energy of the physical system is considered, a so-called Hamiltonian. By constructing approximate expressions for the Hamiltonian vari-ous models are found. In the following the models are derived from the exact differential equations for the wave motion, by approximating c.q. deleting certain terms. The errors introduced are estimated by considering a linear-ized theory for long, low waves.

2.1. Exact equations

The equations for irrotational, gravity driven wave motion in two space dimen-sions on a layer of a (non-viscous, incompressible) fluid, as indicated in Figure 2.1.1, are [4,ch.1] : y

t

~x y=O

///1//// /

_bottom

/

Figure 2.1.1.

(2.1.1) _{(jlxx + (jlyy} = 0 for - ~ < x <

=,

0 < y < hO + n(x,t) (2.1.2) _<ry :::: ₀ at y :::: ₀ (2.1.3) Il t :::: _{(jl - <rxllx} Y (2.1.4) (jlt = - gil - -(cp 1 2 + <p ) 2 2 x y at y :::: hO + n(x,t) , where

x,y space coordinates in horizontal, vertical direction respectively

t time

q>(x,y,t) velocity potential

n(x,t) free surface elevation

g acceleration of gravity.

The relation (2.1.3) expresses that fluid particles on the surface remain on it, whereas the equation (2.1.4) is the Bernoulli law.

Introducing the dimensionless quantities X' := X/hO

n := n/ho

y' := y/ha t'

t/¥

<p' :-'!p1 (h

a

Igh

a)

(2.1.1) to (2.1.4) transform into (deleting the primes) (2.1.5) _{<fIxx + <fIyy} :::: 0 for - 0) < x < 00, a < y < 1 + n (x, t)

(2.1.6) _q>y

=

₀ at y

₌

0 (2.1.7) _n

₌

_{(jly - <i>x"x}

I

at y -t 1 ( 2 2 1 + n(x,t)

.

(2.1.8) q> _t

= - n -

_'2

_q>x + <r ) y

2.2. Waves of small amPlitude (linearized theory)

Assuming that the flow is a small disturbance from the state of rest, we

find from (2.1.5) to (2.1 .8) the linearized equations:

(2.2.1) cp xx + 'yy = 0 for -0) < x <

00,

0 < y < 1 (2.2.2) !Py = 0 at y 0 (2.2.3) (2.2.4) llt =q> , y <i> _t = - n at y 1 .

Eliminating n from (2.2.3) and (2.2.4) we obtain the single condition on ~

(2.2.5) {jltt == - {jly at y = 1 .

The solution of (2.2.0, (2.2.2) and (2.2.5) can be written as a superposi-tion of the waves

(2.2.6) tp(x,y,t;k) eikx cosh (ky) [A(k)sin(w(k)t) + B(k)cos(~(k)t)J

with real k and

(2.2.7) W (k)

The coefficients A(k) and B(k) follow from the initial conditions.

2.3. Long waves of small amplitude

In this section we "derive" the equations, given in 1.1, in a heuristic way on the basis of a contemplation which seems to be reasonable for long waves of small amplitude.

Assume that the initial conditions are such that only long waves are of

in-ten~st.

Consider a long wave with wave number )J Ln (2.2.6) or

(2.3.1) cp(X,y,ti)l) e i)Jx cosh ()Jy) [A()l) sin(w(p)t) + B(p) cos(w()l)t)] , with

(2.3.2) Then we have

(2.3.3) q> = O(e)

From (2.2.7) i t follows that

(2.3.4) we).!) = 0()1) •

By differentiation of (2.3.1), using (2.3.3) and (2.3.4) we deduce the follow-ing relations, which we give for future reference

q> == 0 (JlL) _(jly O(Jl E:) 2 _{CPt '"} 0()1E:) x

(2.3.S) _!Pxx = O(j.l t) 2 CfJ 2 3

yy

=

O{p t) (jlxy =O()1t)

2 4 4

qJxt = O(lJ L) <jlxxy

=

O(p c) CjJ == 0 (Jl c)

(2.3.6) T'\ == O(]..Ie;) T'\ _x == O(J..I 2 d Tj = O(]..I 3 £} • xx

In the following we shall use the relations (2.3.5) and (2.3.6) to construct, starting from the exact equations, approximate equations for long waves of small ampli tude.

3 2

The nonlinear term q>

n

in (2.1.7) is of the order ]..I £: and the nonlinear

1 2 2 xx 22

term ?CfJ

x + CfJy ) in (2.1.8) is of the order ]..I C • We introduce the function

(2.3.7) 1>(x,t) := cp(x,l + n(x,t), t) ,

being the velocity potential at the free surface. Now we proceed to approxi-mate (2.1.7) and (2.1.8) in terms of Tj, ~ and their derivatives, neglecting

terms of the order higher than ]..I3e:2 in (2.1.7) and of the order higher than ]..12£2 in (2.1.8). Differentiating (2.3.7) we find (2.3.8) [m + m '" ] "'x "'y"x y==l+n (2.3.9) _<fit [cp T'\ + q> ] Y t t y:c1+n

Using(2.3.5) and (2.3.6) i t follows that (2.3.10)

~xl

=.

$x + O(114e:?} y=1+l'\ (2.3.11)

,..I

₌

'1

y=l+n 4 2 ~t + O{ 11 I": ) .

On substituting (2.3.10) and (2.3.11) in (2.1.7) and (2.1.8) we obtain

(2.3.12) l1 t =:: <P

I

-

1> n + 0(11 6 <::3) Y y=l+n x x (2.3.13)

~

= -

n -

44

2 + O(]..I4£:2) t 2 x

Now we have to express CfJ

I

in terms of

~

neglecting terms of the order

3 2 Y y=l+n

higher than 11 E • For that purpose we introduce the function

Then we can write

1

63

cp +ncp

I

+ O(ll c ),

Y y=1 YY y=l

or, using that cp satisfies Laplace's equation,

(2.3.15)

1

63

cp = cp

I -

ncp + 0 (lJ e: )"

y y=l+n y y=1 1xx

AS the function cp satisfies

CPxx + IP _YY = 0 IPy :: 0 at y = 0 cp = _{CP1 at} Y = 1, we have +00 (2.3.16)

=

--1--

I

eikXFk(IP1)COSh(kY)/COSh(k)dk,

Ii1T

where +00 F _{k "'1} (m ) :=

~

_{$ '}

I

e-ikx _CP1 ( t ) d _{x, x} _00

is the Fourier transform of the function CPl" From (2.3.16) we obtain (2.3.17) _IP

I

-Y y=l 1

-I2iT

Making use of the convolution theorem we can of the functions R, with Fk(R)

=

tanh (k)/k,

write cp

I

as Y y=1 and IP1 _xx *) so

the convolu"eion

*) The convolution of two functions f and g, denoted by fog, is defined

by _+""

(2.3.18)

Substituting (2.3.18) in (2.3.15) we obtain

(2.3.19) RO'l -'1'1(1)1 + O(ll 6 3 E: ) • .

xx xx

=

-Finally we have to express Iplxx in terms of ~.

By differentiation of (2.3.7) and making use of (2.3.5) and (2.3.6) it is easily seen that

(2.3.20) ~

=

Ml + O(ll 5 2 € ) . "'xx T xx

From (2.3.19) and (2.3.20) it follows that

(2.3.21) (I)

I=:-

RO~ -

'1'1'

+ 0 (\.I 5 2 e ).

y y=1+n xx xx

Inserting (2.3.21) in (2.3.12) we arrive at the approximate equations

'/'It

= -

Ro$ _xX - '1'1$ _xX - ~ _{"'x X}

n

(2.3.22)

1 2 4 2

't · -

T') -

2'x

(.+ O(lJ €

».

If we introduce the function

(2.3.23) v : - . I

X

5 2

(+ O(IJ e::

n

and differentiate the second equation in (2.3.22) to x we obtain, using Ro¢xx'" (RQI/lx)x'

(2.3.24)

These equations are the basis for the three different systems of Boussinesq

2.4. Boussinesq equations

The classical Boussinesq equations can be "derived" from (2.3.24) in the following way.

Consider the term

+00

(2.4.1) = -1

J

e ikx Fk(v)tanh(k)/kdk.

Since only long waves are of interest Rov is approximated by +IX> (2.4.2)

-

1

n;

f

eikXFk(V) (1 - ; k 2 )dk

=

-IX>

on the basis of the formula

tanh (k)/k

1 V+-V

3 xx

4

By dropping the term O{k ) the truncation error in the approximation for Rov

5

is of the order of v c O(~ c).

xxxx

Inserting the approximation (2.4.2) in (2.3.24) we find the classical Boussinesq equations (1.1.1).

The stabilized Boussinesq equations (1.1.2) are obtained by employing

tanh (k)/k

This leads to the approximation

+00 (2.4.3) u := - 1

J

e ikx F k(V)/(1

;2;

-"" 5

for Rov with a truncation error of O(~ &) From (2.4.3) it follows that

1

u -

3'

_uxx

=

v, u

(.:!:. "",

t) .. 0

The "derivation" of the discrete Boussinesq equations (1. 1. 3) is more complicated.

Since Fk(R)

=

tanh(k)/k > 0 for all k we can write R as

(2.4.4) R

=

GoG, with (2.4.5) We define +00 (2.4.6) _{w:= Gov}

₌

~

J

-00 Then we have Rov Gow

and (2.3.24) transforms into

Since Itanh (k)/k"" 1 + O(k2) it "follows" from (2.4.6) that

3

w '" v + 0 (v )

=

v + 0 (ll £). xx

Using that n

=

o (ll€:) and v = O(ll€:) we find

(2.4.8) (2.4.9) 4 2

nv

=

nw

+ O(ll £ ) 2 2 4 2 v

=

w + O(ll £ ) .

Inserting (2.4.8) and (2.4.9) in (2.4.7) we optain the approximate equations -(GOw + nw) x Furthermore we have leading to (2.4.10) -GO~w + nw) x

I f we approximate Fk (G) ;:: /tanh (k)/'k by 1 -

~

k2 I we find

a 1 a3 _nw)

j"t

;:: - ( - + -

3')

_{(w +} ax 6 (2.4.11) ax 3 a 1

_...L.)

(n 1 2 w_t ;:: - ( - + - +-w) ax 6 dX3 2

The discrete Boussinesq equations (1.1.3) arise by employing

(f(x+l) - f(x-l»/2 .. ft (x) +

1.

fll t (x) + o(lIf(S)II>,

6

which shows that replacing (2.4.11) by (1.1.3) introduces and error of the order 0 ().I6 e:) •

3. Classical Boussinesq equations

In this section we study the numerical calculation of the solution of the classical Boussinesq equations (CB)

(3.t) \ n t 0:: - (v V _t == - (n + _1,,2) 'v 2 x 1 + - v + nv) 3 xx x

with the initial conditions

I

n(X10}

=

f(x} (3.2)

v(x,O) = 0

3.1. The linearized equations

To obtain an impression how the solution of (3.1) and (3.2) behaves we neglect the nonlinear terms. It may be expected that this approximation is a good one in case n and v are small.

The linearized equations are

\n

t 1 = - (v + - v ) 3 xx x (3.1.1) \vt ==

-n

_x

with the initial conditions

(3.1.2)

v(x.O) .. O.

The solution of (3.1.1) and (3.1.2) can be (formally) written as +00

(3.1.3a) n(x,t)

:: -

1

J

( e ikx F(k)cos( ~2 - -1 4' k t)dk

I21i'

3

_00

+00

(3.1. 3b) v(x,t) :::: - -i

f

e ikx. KF(k)sin( y{2

v'27r

_-co

+co

(3.1.4) F(k)

= -

1

J

( f(x)e -ikx dx.

I2i

_{_co}

The integrand of (3.1.3a) behaves for large values of k as

This means that the integral converges for all t or at least for O<t <T with certain T > 0, only if F is sufficiently small for \kl ~ co.

For the initial conditions (1.1.5) we have

hence in this case the linearized problem has no solution or at best a generalized solution.

From this we see that only for a restricted class of very smooth initial conditions, which we shall call admissible initial conditions, the linearized equations have a solution for all t or during a finite time. This means that the linearized problem is ill-posed in the sense of Hadamard. Disregarding the question whether solving ill-posed physical problems is at all sensible, we will try to find a method for the numerical calculation of solutions corresponding to admissible initial conditions.

If an admissible initial condition is disturbed by an arbitrary, small disturbance, then this disturbance will at once dominate the solution

belonging to the admissible initial condition. By rounding errors in numeri-cal numeri-calculations a disturbance will, be inevitably induced.

Hence, if, given an admissible initial condition, we want to compute the solution numerically, we will have to build in a mechanism to suppress the influence of the induced disturbances.

3.2. Re2ulari~ation of the linearized equations

The idea 1s globally as follows [3J.

The linearized equations are replaced by a Itneighbouring" system, being a well-posed one. One then looks upon the solution of this system, in the case

of admissible initial conditions, as an approximate solution of the original system (3.1.1).

For instance:

consider the equations

1

n

t

=

-(v + -3 xx v + YO xxx x ) (3.2.0

v _t

= -

(n + yv _{xxx x} )

where y is a small positive parameter.

We call the system (3.2.1) the regularized system.

The solution of (3.2.1) with the initial conditions (3.1.2) can be written as

(3.2.2a)

n

(x,t,y) +"" = _1

I

eikXF(k)e-Yk4t 12 1 4' - cos(v'k -

'3

k t) dk

I21i

_{_00} (3.2.2b) v(x,t; y) +00 -i

I

ikx. -yk 4t

121"4'

.42 1 4'

= -

e JtF(k)e sin (Ik - - k t)! k - - k dk,

~ 3 3

-00

Remarks

(i) The .ystem (3.2.1) is well posed; its solution depends continuously

on

f.

(ii) The cli.ff'e.ren.ce-·.betw~n the.int&g:HftQa of t3.103) and (3.2.2) is the factor e'::Yk4t,. 'From this we see that for small'valU;;- of

k···---the integrands practically match, but that for large values of k the integrands of (3.2.2) are extremely small as compared with those of (3.1.3).

(iii) Suppose that the initial function f(x) is such that the integral +00

J

I

FCk)lek2T!13 dk converges.

Then f(x) is admissible and it follows from (3.1.3) and (3.2.2) that for all t, satisfying 0 <; t <; T,

n(x,t;y) == n(x,t;O) + O(y) y '" 0

v(x,t;y) == v(x,t; 0) + 0 (y) y ... 0,

Consequently, the difference between the solution of (3.1.3) and (3.2.2) approaches zero for y '"

o.

(iv) To obtain a more quantitative impression of the effect of the regularization we consider the initial conditions

v(x,O) '" O.

In this case we have

F (k) =

fi

e -';'/4 /2 I

which shows that these initial conditions are admissible. The solution of the linearized classical Boussinesq equations exists for all t > 0 satisfying -k2/4 + k2t/13 < 0 or 0 <; t < /3/4

~

0.433.

By calculating the integrals in (3.2.2) numerically the influence

of the regularization can be demonstrated.

Some results are illustrated in the Figures in section 3.5.

'In Figure 3.5.1 the full line is a graph of n(x,O. 3,y) with

O.~y<; 0.0001.

For all these values of y the graph does not depend on y Visibly. The broken line is a graph of

n

(x,D. 3; 0.004).

In the ~i9ures 3.5.2 and 3.5.3 the full line is a graph of

n(x,0.42,y) with 0 <; y <;0.000Q01. In Figure 3.5.2 the broken line represents n(x,0.42;O.OOOl) and in Figure 3.5.3 n(x,0.42iO.004).

We see that we have to choose y smaller as t approaches the

critical time t ~ 0.433, to obtain a reasonable agreement between the solutions n(x,t,O) and n(x,t,y ).

(v) For large values of k the modulus of the integrand in (3.2.2a) behaves as

The amplification factor

has a maximum of e t/ (12y) at k2 • 1./ (2y/!)

(vi) Let f be an admissible initial condition for the unregularized system and suppose that we want to compute the solution during

o

<; t <; T, working with the regularized system.

(3.2.3)

Let 5f be the numerically induced disturbance. Assume that

II

of

II ....

e::

II

f

II ,

where e:: is the machine precision.

In virtue of the foregoing remark (v) we expect that this disturbance (which contains all frequencies) will be amplified during a time T to

If we do not want this disturbance to influence our solution too much we can demand, for instance, that

e::

Ilf lIe

T/ 12Y <; O.Oll1f

II •

This implies a lower bound on y.

On the other hand it follows from the remarks (iii) and (tv) that y should be small in order to have a reasonable agreement between the solutions of the regularized system and those of the

3.3. Numerical method

We assume that we can handle the nonlinear system (3.1) in the same way as we did the linearized system (J.l.l).

Consider the regularized classical Boussinesq equations

-(v 1

I"t

=

+ - v + 3 xx (3.3.1)

=

- ('1'1 + yv ... v t xxx

with the initial conditions

(3.3.2)

j

T1(X,O)

=

v(x,O)

=

f(x)

o •

YT1 xxx + T1V) x 1 2) -v-2 x I

To solve this system numerically, we start by replacing (3.3.1) by a finite difference approximation (Crank Nicolson).

We assume, for simplicity, that f is an even function of x. Hence T1 is an even function of x and v an odd one.

Therefore it suffices to consider the region

R :

=

[O,co) )( (0 ,IX» •

We introduce a rectilinear grid with sides parallel to the x- and t-axis,

~x and ~t being the grid spacing in the x- and t-direction, respectively

t

1

The grid points are

where (x , t ), m n x =mAx, m t

=

nAt, n

< Ax

) - - - - -... x Figure 3. 3 • 1 • m == 0,1,2, ••• , n == 0,1,2, ••••

In the sequel we denote g(x ,t ) by gn •

m n m

1\ i';t

V

We consider the first equation of (3.3.1) at the points

ex ,

t~) ,

m

=

0,1,2, ••• M-1,

n

= 0,1,2, ••• ,

m

n'"2

and the second equation at the same points with exception of m

O.

We then replace the derivatives by central differences.

The central differences for the x-derivatives at the time t 1 are replaced

n-rz

thereupon by the averages of the values at

n+1 the times tn and t n+1•

The nonlinear term (nv) is replaced by

m

(nv) n + (nn+l _n n) _v n + _n n (vn+1 _{- vm)} n

₌

_nm n+1 n _{vm + nmvm} n n+1

m m m m m m

The other nonlinear terms are handled in the same way.

n n

- n

v •

As a consequence of the discretization also quantities arise for negative values of x.

These are expressed in quantities for positive values of x, using the fact that

n

is an even function of x and that v is an odd function of x.

n Since n and v tend to zero if x tends to infinity we take nm m ~ M and all n.

=

vn == 0 for m

In this way we obtain a system of 2M-1 linear equations from which, given

n

and v at the time tn'

n

and v at the time tn+1 can be computed. Remarks

(i) For the linearized regularized equations (3.2.1) it can be proved that the solution of the corresponding difference equations

converges to the solution of the linearized regularized equations. We assume that this is also the case for the nonlinear regularized equations.

(ii) From a Fourier-analysis of the difference equations corresponding to the linearized regularized equations it turns out that again the mode k, with k2 Ail

1/

(2 y

13),

of the spectrum of f is

t/(12y) maximally amplified, the amplification factor being e

(see also remark of section 3 2).

3.4. Numerical results

In this section we present some numerical results obtained by applying the numerical method to the regularized classical Boussinesq equations (3.3.1).

The values of Ax and At have always been choosen so small that the presented graphs can be considered as representing the exact solution of the differen-tial equations (3.3.1).

Firstly we consider the initial conditions

2 (3.4.1)

j

n(x,o) III e-x

From remark (vi) of section 3.2 we can roughly "deduce" a lower bound on y !f we wish to compute the solution during a time T.

-11

We work with the B7700 which has a machine precision of about 10 •

In this case formula (3.2.3) suggests the bound

or

y;> 0.004T •

In Figure 3.5.4 the full line is a graph of n(X,0.310.002) and the broken line is a graph of n(x,0.3;0.004).

The agreement between the two graphs suggests that it may be expected that they differ little from the solution of the unregularized system (compare also the graphs in Figure 3.5.1 for the linearized system).

In Figure 3.5.5 the same is done at t

=

0.42 (being almost the critical

time for the unregularized linearized system) with y

=

0.003 and y

=

0.004.

We see that the influence of the regularization parameter y is considerable.

-6

Figure 3.5.2 suggests to take y - 10 ,but this is only possible on a

-15000

machine with machine precision of 10 •

Nevertheless there are arguments to expect that our computed solutions

qualitatively match the solution for

y

=

0.

In particular we expect that the solution for y

=

0 has an oscillatory

behaviour. For, the larger the y the stronger the damping of the hiqher

Fourier-modes. Furthermore we saw in Figure 3.5.3 that in the linear case the shapes of the regularized solution and the unregularized one are similar.

In the sequel we will always work with y

=

0.004.

In Fiqure 3.5.6 graphs of n(x,O,O.004) and n(x,0.42;0.004) are given (full lines) and the broken line represents n(x,0.43,0.004).

(3.4.2)

n(x,O}

=

.:;e

-x

v(x,O)

=

0 , 2

where € > 0 is a parameter introduced to examine the influence of the

nonlinear terms. Indeed the functions

(3.4.3)

are solutions of the system

n =

-(v

+

1

~

+

yn

+

en;)

t 3 xx xxx x

(3.4.4)

with the initial conditions

n

(x,O)

2

-x

=

e

v(x,O)

=

0 ,

so the influence of the nonlinear termes increases as € becomes larger.

,....,

In figure 3.5.7 the full line is a graph of n(x,0.42iO.004,e) with

o

< & ~ 0.01. For all these values of & the graph does not change visibly. The broken line represents n(x,O.42;O.004,0.1).

In the Figures 3.5.8 and 3.5.9 the full line is again a graph of

,...;

n{x,O.42;O.004,e) with 0 < e ~ 0.01.

....

In Figure 3.5.8 the broken line represents n(x,0.42;0.004,0.5) and in Figure 3.5.9 n(x,O.42,O.004,1).

We see that as the influence of the nonlinear terms increases the solution

Finally a peculiar property of the classicai Boussinesq equations should

be mentioned. From numerical experiments it appears that, if e < 1, the

height of the wave at x

=

0 increases initially.

This phenomenon can also be verified analytically. Assuming that

n

and v

have a Taylor-series expansion with respect to t, we obtain the following

formula for n satisfying (3.3.1) and (3.4.2):

2 2 3

n(O,t;y)/€=1-12yt+(1+840y - d t +O(t),t-l-O.

Therefore, if Y

=

0 and € < 1, n(O,t) is initially an increasing function of t at x

=

o.

3.5. Figures n 1.5 1 .75 .5 .25 _

~n(x,0.3;y)

0 <; y <; 10-4 ... ~ ~ ~n(x,O.3,O.004) ~

o

~--~----~----~----~----~----~--~~--~

o

.25 .5 .75 1 1.25 1.5 1.75 2 Figure 3.5.1. 2.25 2 \ 1 .5 1.25 \-+TI(X,O.4210.0001) 1 .75 .5 .25

o

~

__

~

____

~

__

~~

__

- L _ _ _ _ ~ _ _ ~ _ _ _ _ - L _ _ ~

o

.25 .5 .75 1.25 1.5 1.75 2 Figure 3.5.2. x x

1 .5 1.25 1 .75

.S

.25

o

o

n 1.5 1 .75 .S .25

o

o

..

,

'"

"

\ \ \ .25 .25 '~n (x,0.42 ,0.004) \ \ \ \ .5 .5 \ \ .75 1 1.25 Figure 3.5,3. I\,.

"

,

'"

"'~n(x,0.310.004) , 1 1.S Figure 3.5.4. x 1.75 2 x 1.75 2

n 1.S 1.25 1 .75 n(x,O.42,O.004) .5

o

~--~----~--~~--~--~~--~----~--~ x

o

.5

.75

1.S Figure 3.5.5. n

1.25

1

_

...

--

_{~~n(x,O,43,O.004)} .76 ~ (x,O.42,O.004)

.6

.25

x .5 .75 1 Figure 3.5.6.

n 1.5 1.25 0<e:<0.01 1 .75 0.1) .5 .25

o

~--~--~--~~--~--~--~----~--~ 1.5 1 .75

.s

.25 0

o

.25 .5 1 1.25 Figure 3.5.7. lj --+ n(x,0.42,0.004,e:)

°

<

E

<

0.01 .... 0 .25 .... ....

,

,

.5 \ ,.., \-* n(x,O.42,O.004,0.5) \ ....

_...

---.,.

... .75 1 1.25 1 .S 1.15 Figure 3.5.8. 2 2 x x

" . Tl 1.5 1.25 1

.75

.5

o

o

-+ n(x,0.42,0.004,£) o

<

£

<:

0.01

--

...

,

.s

,

"

_"

\

,-+

.... n(x,0.42,0.004,1) \ \ \ .75 1 1.25 1.5 Fiqure 3.5.9. x 1.75 2

4. The stabilized Boussinesq equations 4.1.The linearized equations

In this chapter we treat the following initial value problem

a

l1 t "" - -

_ax

(Rv + nv)

a

1 2

- ax-

(n +

'2

v ) v III t n(x,O) = f(x) v(x,O) .. 0 R

=

(1

-t

a:~

)

-1}

;or _

~

< x <

~,

0

< t < • } for -

~

< x <

~.

We assume that the problem is well-posed, if the linearized problem,

1 2

obtained by deleting the terms. l1V and

'2

v , is well-posed. Formally, the linearized problem is solved by

co

11 (x, t) .. _1_

f

exp (ikx) F (k) cos ( k t ) dk

12".

-00 (l +

1

k 2 )

~

3

00

v (x, t) _{= _1}

J-

_{i exp (ikx) F (k)} (1 +

'3

1 k ) sin 2 ~ (---:1-'!':"2"""\~) kt dk

IiTI

-co

(1+'3

k )

where

00

F(k)

=

~

f

f(x)exp(-ikx)dx.

-00

This solution exists for the class of functions f(x) such that the inte-grals above

converge~(e.g.

the functions f(x) for which

f:

IF(k)k

I

dk < (0). It is also readily seen that the solution depends continuously on any f(x) from this class. In other words: the linearized initial value probleDL.is well-posed.

4.2. The numerical method

It is our. aim to compute a solution in case

( - 00 < x <: co)

of the initial wave.

We proceed as follows. First of all, the problem is for obvious reasons transformed into an initial value problem on a finite region. Subsequently, the system of partial differential equations is converted into a system of ordinary differential equations by discretizing the variable x (method of lines). Finally, the resulting syste"m is solved using the digital simulation language C.S.M.P. (Continuous SYstem Modelling Program). The reason to use another method than in chapter 2 springs mainly from the wish to obtain Some experience with C.S.M.P.

4.2.1. The choice of bounda!y conditions

In order to solve the problem on the finite region (-M!'> x !'> M; 0 ~ t !'> T) , the integral operator R has to be restricted to (-M S x S M). Moreover, boundary conditions have to be created at x

=

+ M.

We ~proximate R by ~ such that for RMU we have

at x

=

+ M

::; U , for all functions u

Regarding the boundary conditions we shall assume that

v(x,t)

o

, for x

=

~ M, O!'> t S T

From this condition and the definition of RM it follows that

are M ( 1 and Em :=

J

r(v, Rv + (M :=

J

ndx n 2 + nv )dx 2 -M -M constant in time: M M dH M

I

= dt n .. dx

=

f

(RMV + nv)xdx = -

(~v

+ nv) -M

...

-M M x "" M x

=

-M dE M

dt

f

~(V.RV

(after some computations)

-M

1 2 == -(n +

2

v ) (RMV + nv) 1 6 x "" M + x

=

-M x == M x

=

-M

Our assumption on v(~ M,t) being 0 means that in fact we use reflecting boundaries. Because with the given initial function the solution turns

o

out to be a travelling wave (with velocity ~ 1), it seems reasonable to

choose M and T such that the wave cannot reach the places x

=

+ M in the time period (0 s t S T). This !s the case i f M ,..., 1. 5T.

4.2.2. The staggered net method.

We now deal with the finite initial value / boundary value problem

a

n

_t

==--

_ax

, for -M < x < M, 0 < t < T

a

v

=

-t

ax

, for -M S; x S M v(x,O) == 0 v(+ M,t) == R v(+ M,t) == 0 - M - , for 0 S t S T I

On the x - axis we de~ine the eqUidistant points

fix

Let the distance between two sequential points be ~

Accordingly, we have

tlx == M/ (N +

1.)

2

Only in these points we are interested in the values of the various 1 variables. These values are denoted by attaching a subindex i or i + 2 to the variables.

The first partial differential equation 1s considered at the integer points xi (-N ~ i ~ N). rhe second one is considered at the intermediate points

xi+~(-N + 1 ~ i ~ N-1). First derivates with respect to x are replaced by

central differences:

The boundary conditions are used to simplify the differential equations at the boundary points. Doing so, we are left with the system of ordinarydif-ferential equational (1'l-N) t =

- -

_Ax 1 (~v + nV)_N+l:i ~ni) t

OX

(RMV + nv) i (-N < i <: N) = 6x (nN)t =

_t:'X

1 _{(~v + nV)N_l:i} 0 1 2 (Vi+l:i)t ...

_{- AX}

x (n +

2'

v ) i+, (-N < i < N)

with the initial values

(-N ~ i ~ N)

(-N <: i <: N)

Notice that the boundary conditions need not to be discretized.

If during the evaluation of the righthandsides, ni+l:i or Vi 1s required for some value of i(-N <: i < N), then

n

i +, or Vi is obtained by ~aking the mean of the values in Xi' xi +1 or xi _,' xi+l:i respectively.

The advantage of this method above one where both p.d.e.'s are considered in all points, is that the number of ordinary differential equations is halved whereas the discretization error is approximately of the same magnitude.

REMARK: with the given initial function it is easily seen that n(x,t)

=

n(-x,t),

v(x,t)= -v(-x,t), ~v(x,t)

=

-RMV(-x,t). Hence, it is sufficient to consider

4.2.3. The computation of Rv

Let 9 := RMV, then v termediate points and us with

2

== (1 -

t

~

)

g. Considering this equation in the

in-ax

replacing derivatives by central differences supplies

or in matrix notation v_N 1

-"'2

v N 3 1 01

-"'2

3(LlX)2 v 1 N-2' 2 with a = 1 + 3(Llx) -:1 -1 0 -1 0 (i = -N+l, ••• ,N-1)

o

o

1 -1 Ct 1 0 0 -1 1 g 1 -N- -₂ g -N+

.1

2 gN-

.1

2 gN+

.1

2

From the definition of Rm i t follows that gN+~

=

g-N-~ = O. Using this the system turns out to have 2 (N-l) equations in 2(N-l) unknowns with a regular matrix

of coefficients.

After a triangular (L.U) decomposition, one may find the gi+~(-N+l ~ i ~ N-l) in about 4N computations via 'a process of backsubs'ti tution.

3.2.4. The accuragy

If LlX approaches 0, the error behaves like O(6x2). This is verified by inte-gration over the region (0 :S t ~ T

=

7; 0 ~ x :S M = 12.5) with the (almost) halving values of x: 1, 0.5102, 0.2526, 0.1256. As initial functions we have

2 n(x,O) = h/{l + AX )

v(x,O) .. O.

Furthermore we check whether

ndx and E m M

I

-M 2 can be approximated up to 0 (llx )

The results show that the computed values at 6x = 1 are accurate to two figures. The exact values of H and E should be

m M

HM = 3.65519 •• , 2.07565 ••

EM = .382939 •• # .194288 ••

for A m .05, 0.2 respectively. The computed values correspond with these

n(O,t) (A=.05). ! t llx = 1 llx

=

0.5102 llx

=

0.2525 llx

=

0.1256 differential quotients 0 .33300 .33300 .33300 .33300

-

-1 .31504 .31457 .31444 .31444 3.6 4.3 2 .27123 .27035 .27013 .27007 4.0 3.7 3 .22065 .22011 .21997 .21994 3.9 4.7 4 .17580 .17576 .17574 .17574 2

-5 .14027 .14048 .14054 .14055 3.5 6 6 .11315 .11342 .11349 .11351 3.9 3.5 7 .092522 .092768 .092834 .092849 3.7 4.4 H H H H 0 3.6540 3.6549 3.6550 3.6550 9

-7 3.6537 3.6550 3.6551 3.6550 13 -1 1--E E E E 0 .38288 .38293 .38294 .38293 5 -1 7 .38041 .38232 .38279 .38289 4.1 4.7 n(O,t) (A .2) '--' 0 .33300 .33300 .33300 .33300

-

-1 .28209 .27880 .27798 .27778 4.0 4.1 2 .18402 .18249 .18221 .18214 5.5 4.0 3 .10955 .11162 .11194 .11202 6.5 4.0 4 .070488 .072057 .072373 .072448 5.0 4.2 5 .051163 .051180 .051354 .051398 0.1 4.0 6 .038736 .038771 .028874 .018900 0.3 4.0 7 .029352 .030069 .030131 .030146 11.6 4.1 H H H H 0 2.0753 2.0756 2.0756 2.0757

-

-7 2.0734 2.0754 2.0756 2.0756

-

-E E E E 0 .19429 .19429 .19429 .19429

-

-7 .19131 .19350 .19407 .19421 3.8 4.1

4.3. The results

In this experiment intended to integrate both the long as the steeper initial wave (A - .05 and .2) during a prolonged time, we choose ~x a 1, M

=

99.5

and T .. 49.

It app~ars that HM is only changing in its sixth and EM is only changing in ~.

its third figure. In the diagrams the dynamical behaviour of the wave height

has been recorded at the places x

=

0, x

=

10, x

=

20, x

=

30 and x - 40 (denoted by Yo' Yl' Y2' Y3 and Y4 respectively).

These diagrams are presented in subsection 5.3 together with the diagrams for the discrete equations in order to enable a comparison.

Contrary to the longer wave the steeper wave is not immediately at rest after passage of the wave front.

H99 •5

=

2.30580160, 4.54474730

(A = .2, .05)

5. Discrete Boussinesq equations

5.1. The linearized equations

The initial value problem considered in this chapter reads

dn 1 n

=

2"

(w n_1 dt dw 1 n

-

.., - (n dt 2 n-l " (0) _n == f _n

w

(0) =: 0 n w 1) + n 1 w _{n- n-l}

-

_wn+l

-

_{nn+l n+} 2 1 2 + w 1 - nn+1 _n-

-

2'

_wn+1) (n E Z, 0 < t < (0).

The variable n (t) corresponds with the n(n,t) of the previous chapters. Between

n

w (t) and v(x,t) of the previous chapters the following correspondance should

n

exist

III'n (t)

=

8w(n,t) - wen - 1,t) - wen + l,t). 6

As before we suppose that the well-posedness of the linearized problem holds

a quarantee for the well-posedness of the given problem. Formally, the lineari-zed problem 1s solved by

00 n (t) :: 1

f

F(k)exp(ikn)cos(t sink)dk n

I21T

_-00 00 w (t) -1

J

F(k)exp(ikn)i sin(t sink}dk

= -n

I2i

-<lO

where F(k) should be so chosen that

J

F(k)exp(lkn)dk

n=""

For the class {fn} _n= such that the integrals converge, the solution

_IlO

n=oo

exists. Moreover, if {f } is so chosen, then the solution depends

n n= -00

5.2. The numerical method

The initial values of interest are 2

TI _n (0) = h/(l +

An )

w (0)

=

0 n

(n €Z)

It is easy to show that with these initial values it holds that n (t)

=

n (t)

n -n

and that w (t) == -w (t) for (0

s

t < 00). Accordingly, we may restrict

our-n -n

selves to the computation of Tl

n and wn for n € [O,N] and t € [O,T], where N and T are sensibly chosen. Because only a finite part of the infinite system is considered, two extra equations for

to create the boundary N HN :=

I

l:'li= -N be constant in time. We obtain dH N 0 = = - == dt

o

=

-

dEN _dt conditions from TIn and n_N+1 and w_N+₁ are the demand that

~

1

EN :=

t

-{1 +

n=-N 2

Wi th these _{conditions the equations for ~} dTlN dWN a n d -dt dn N 1 [ (1 + + TlN)WN] "'"

-

n_N_₁) w N-1 + (1 dt 2 dW N 1 [ nN- 1 +

1.

w2

_lw2]

-

==

2'

dt _{2 N-l - TlN}

₂

_N needed. We choose 2 1 2 nn)W n +-n 2 n read

5.3. The results

We choose N

=

100, and T ~ 49 and integrate the equations with h

=

.333 and A

=

.05, .2 as parameters for the initial wave. During this time we find that HN and EN are constant in ten and four figures respectively. In the diagrams on the following pages the dynamical behaviour of the wave height is recorded at the places x.= 0, x

=

10, x = 20, x

=

30 and x = 40 (denoted by yO, yl, ••. , y4 respectively).

The upper diagrams correspond with the discrete model, the lower diagrams correspond with the stabilized model of chapter 4.

On the pages 39-43 the initial wave is: n(x,O) On the pages 44-48 the initial wave is: n(x,O)

2 .333/ (1 + .05x ).

2 .333/ (1 + .2x ).

It is seen that the models deliver indistinguishable results for the longer initial wave (A = .05). However, in case of the steeper initial wave (A

=

.2), the discrete model gives more OSCillatory results especially at x = 0 and x

=

10 (yO and yl).

Xl 0- 1 4.00 '(0 3.50 3.00 2.50 2·00

1 .50

1 .00 .50 .00

-.50

-1 .00 I I

.00

.98 1 .96 2.94 3.92 4.90

XI0

1

TIME

XlO'-\ 4.1]0 '(0 3.50. 3.00 2.50 2.00 1 .50, I '

no

.50 .00

-.50

··1.00_ --..,.----'," I I -,- I .00 .98 1 .96 2.94 3.92 4.90 Xl 0 1

TIME

XIO 1 4.00 Yl 3.50. 3.00 2.50 2.00 1 .50 1 .00

.50

.00

-.50

-1 . 00 .00 .98 1.96 2.94 3.92 4.90

X10

1

TIME

XlO-l 4.00 '( 1 3.50 3.00

2.50

2.00 1 .50 1 • 00 .50

.00

-.50 -1 . 00 _{i I I} .00 .98 1 .96 2.94 3.92 4.90

XI0

1

TIME

Y2 Y2 XI0- 1 4.00 3.50 3.00 2.50 2.00 1 .50 1 . 00 .50 .00

-.50

-1 . 00 .00 Xl0- 1 4.00 3.50 3.00 2.50. 2·00 J .50 1 • 00 .50 .00 -.50 I .98 I I I 1 .96 2.94 4.90

TIME

-1 . DO. --y----,-·· ... - - , - · · - , - -.... -·-.---.-...,.-·-.. 1 .00 .98 1.96 2.94 3·92 4.90 XIOl

TIME

Y3 Y3 XIO- 1 4.00 3.50 3.00 2.50 2.00 1 .50 1 . 00 .50 .oo~---.50 1 .00 ---.---·,.·--·T---,---T-~---,.I----r---.I--,___, .00 .98 1.96 2.94 3.92 4.90 XI0 1 4.00 3,50 3.00 2.50 2.00 1 .50 1 .00 .50 . 0 0 . ' --.. 50 -1.0o+-~r-~I---r~~I-.00 .98 1 .96 I 2.94 XI0 1

TIME

I \ 3.92 4.90 XIOI

T!ME

X 10-1 4.00 Y4 3.50 3.00 2.50 2.00 I .50 1 . 00 .50 .00 .50 -1 .00 _{I I I I I} .00 .98 1 .96 2.94 3.92 4.90 X10 l

TIME

X] 0- 1 4.00 Y4 3.50 3.00 2.50 2.00 I .50 1 .00 .50 .00 -.50 -1 .00 _{J I}

....-,

_{I I} .00 .98 1 .96 2.94 3.92 4.90 XIOI

TIME

X 10- 1 4.00 YO

3,501

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References

[1] ~, L.J.F.: On the Hamiltonian theory of surface waves. Appl. Sci.

Res. 30 (1974),430-446.

[2] ~, L.J.F.: Approximate equations for long water waves. Appl. Sci.

Res. 31 (1975), 377-395.

[3] Payne, L.E.: Some general remarks on improperly posed problems for

partial differential equations, in: Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Berlin etc., Springer-Verlag, 1973 (Lecture Notes in Mathematics, vol. 316), pp. 1-30.

[4] Stoker, J.J.: Water waves. New York, Interscience, 1957 (Pure and Applied Mathematics, vol. IV).

[5J Timmers, J.M.W.: Vergelijkingen voor lange, lage watergolven. (Master's

thesis), Eindhoven, Department of Physics, University of Technology Eindhoven, 1976.