IN229/Simulation: Lecture 4
Knut–Andreas Lie
Department of Informatics
University of Oslo
February 2003
We must specify boundary conditions on
initial conditions on
and
or
at
We call the equation a partial differential equation (PDE)
This is a time- and space-dependent problem
Mathematical model: the wave equation
Physical phenomenon: small vibrations on a string
The wave equation - our £rst PDE
and
IN229/Simulation: Lecture 4
1
Consider a function of several variables
The th partial derivative is de£ned as
Higher order derivatives are de£ned analogously.
and
For the wave equation we have
.
But £rst a digression: partial derivatives
IN229/Simulation: Lecture 4
2
IN229/Simulation: Lecture 4
Derivation of the model I
y
x
Physical assumptions:
the string = a line in 2D space
no gravity forces
up-down movement (i.e., only in -direction)
Physical quantities:
: position
: tension force (along the string)
: angle with horizontal direction
: density
3
IN229/Simulation: Lecture 4
Derivation of the model II
y
ρ
∆s
T(x+h/2)
u(x,t)
T(x-h/2)
x
h
Physical principle, Newton’s second law:
total mass acceleration
sum of forces
Total mass of line segment:
Acceleration:
The tension is a vector (with two components):
4
and in the limit
we get:
use geometrical considerations
eliminate -component of equation
Then we do some mathematical manipulations
A vector equation with two components
Newton’s law on a string element:
Derivation of the model III
IN229/Simulation: Lecture 4
5
) this simpli£es to:
String initially at rest:
String £xed at the ends:
Initial and boundary conditions:
For small vibrations (
Final equation
IN229/Simulation: Lecture 4
6
After a scaling, the equation becomes
The complete equation
For those interested, Glenn Terje will give a more thorough derivation of the equation on ..day
IN229/Simulation: Lecture 4
7
Introduce a grid in space-time
Finite difference approximation
Central difference approximations
IN229/Simulation: Lecture 4
8
is the CFL number
Here
. Then the difference equation reads
Solve for
Inserted into the equation:
Finite difference approximation...
IN229/Simulation: Lecture 4
9
Two conditions at
for all :
Initial conditions
The second condition inserted into the equation for
Two choices: either introduce a special formula for
£ctitious value
We use the second approach in the following.
, or a
IN229/Simulation: Lecture 4
10
IN229/Simulation: Lecture 4
Algorithm
De£ne storage
Set
,
for
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!#"$!%
Set initial conditions
De£ne
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Set
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Update all inner points (
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Set boundary conditions
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up, MyArray<double>& u,
um, double tstop, double C);
u0, MyArray<double>& um, double C);
u, double t);
}
// u at time level l+1
// u at time level l
// u at time level l-1
setIC(u, um, C);
timeLoop (up, u, um, tstop, C);
return 0;
cout << "Give Courant number: ";
double C; cin >> C;
cout << "Compute u(x,t) for t <= tstop, where tstop = ";
double tstop; cin >> tstop;
MyArray<double> up (n);
MyArray<double> u (n);
MyArray<double> um (n);
int main (int argc, const char* argv[])
{
cout << "Give number of intervals in (0,1): ";
int i; cin >> i; int n = i+1;
// forward declarations:
void timeLoop
(MyArray<double>&
MyArray<double>&
void setIC
(MyArray<double>&
void plotSolution (MyArray<double>&
#include ....
Straightforward F77/C style implementation
IN229/Simulation: Lecture 4
12
}
}
// initial displacement to file
plotSolution (up, t);
up(1) = 0; up(n) = 0;
um = u; u = up;
// plot displacement to file
// update boundary points:
// update data struct. for next step
// update inner points according to finite difference scheme:
for (i = 2; i <= n-1; i++)
up(i) = 2*u(i) - um(i) + Csq * (u(i+1) - 2*u(i) + u(i-1));
plotSolution (u, t);
while (t <= tstop)
{
t += dt; step_no++;
void timeLoop (MyArray<double>& up, MyArray<double>& u,
MyArray<double>& um, double tstop, double C)
{
int
n = u.size();
// length of the vector u (no of grid points)
double h = 1.0/(n-1); // length of grid intervals
double dt = C*h;
// time step, assumes unit wave velocity!!
double t = 0;
// time
double Csq = C*C;
// = (dt/dx)^2
int i;
// loop counter over grid points
int step_no = 0;
// current step number
The timeLoop function
IN229/Simulation: Lecture 4
13
}
// set the help variable um:
for (i = 2; i <= n-1; i++)
um(i) = u0(i) + 0.5*Csq * (u0(i+1) - 2*u0(i) + u0(i-1));
um(1) = 0; um(n) = 0;
// dummy values, not used in the scheme
// set the initial displacement u(x,0)
for (i = 1; i <= n; i++) {
x = (i-1)*h;
if (x < 0.7) u0(i) = (umax/0.7) * x;
else
u0(i) = (umax/0.3) * (1 - x);
}
void setIC (MyArray<double>& u0, MyArray<double>& um, double C)
{
int
n = u0.size();
// length of the vector u
double x;
// coordinate of a grid point
double h = 1.0/(n-1);
// length of grid intervals
double umax = 0.05;
// max string displacement
int
i;
// loop counter over grid points
double Csq=C*C;
The setIC function
IN229/Simulation: Lecture 4
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}
Here we have chosen to plot each time step in a separate
(hidden) £le with name .u.dat.<step number>
i++; sprintf(fn,".u.dat.%03d",i);
ofstream outfile(fn);
for (int i = 1; i <= n; i++)
outfile << h*(i-1) << " " << u(i) << endl;
void plotSolution (MyArray<double>& u, double t)
{
int
n = u.size();
// the number of unknowns
double h = 1.0/(n-1);
// length of grid intervals
char
fn[30];
static int i=-1;
The plotSolution function
IN229/Simulation: Lecture 4
15
visualisation = animated motion of string!
eleonoradus:~/in229/2000/prog/Wave1D$ ls .u.dat.*
.u.dat.000 .u.dat.012 .u.dat.024 .u.dat.036 .....
.u.dat.001
:
:
:
:
:
:
:
:
:
The simulation produces a set of 68 £les
eleonoradus:~/in229/2000/prog/Wave1D$ ./wave
Give number of intervals in (0,1): 20
Give Courant number: 0.9
Compute u(x,t) for t <= tstop, where tstop = 3
Running the simulator
IN229/Simulation: Lecture 4
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t=3.0; nr=68; incr=1;
j=0;
for i=0:incr:nr-1
%
% read simulation file number <i>
fn=sprintf(’.u.dat.%03d’,i);
fp = fopen(fn,’r’); [d,n]=fscanf(fp,’%f’,[2,inf]);
fclose(fp);
%
% plot the result
plot(d(1,:),d(2,:),’-o’); axis([0 1 -0.1 0.1]);
tittel = sprintf(’Time t=%.3f’, (t*i)/nr);
title(tittel);
%
% force drawing explicitly and wait 0.2 seconds
drawnow; pause(0.2);
end
Animation: in Matlab
IN229/Simulation: Lecture 4
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Run gnuplot afterwards: gnuplot gnuplt.com
}
ofstream outfile(fn);
:
i++; sprintf(fn,".u.dat.%03d",i);
comfile << "plot ’" << fn << "’ title ’u(t=" << t << ")’;";
if (i==0)
comfile << "pause 6;" << endl;
else
comfile << "pause 0;" << endl;
if (i<0) {
comfile << "set data style linespoints;" << endl;
comfile << "set yrange [-0.1:0.1];" << endl;
}
void plotSolution (MyArray<double>& u, double t)
{
:
static int i=-1;
static ofstream comfile("gnuplt.com");
Alternative: generate a Gnuplot command £le
IN229/Simulation: Lecture 4
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IN229/Simulation: Lecture 4
What about the parameter
?
How do we choose the parameter
Solution at time
DFEHG0IKJ
for
LME
?
NPO
G
0.08
0.015
0.06
0.01
0.04
0.005
0.02
0
0
−0.02
−0.005
−0.04
−0.01
−0.06
−0.015
−0.08
−0.02
0
0.1
0.2
0.3
0.4
QSR
0.5
0.6
0.7
0.8
0.9
1
−0.1
0
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
−0.01
−0.01
0.1
0.2
0.3
0.4
QSR
0.3
0.5
UYTVZ
0.4
QWR
0.015
0
0.2
+TVU
0.015
−0.015
0.1
0.6
0.7
0.8
0.9
1
−0.015
0
0.1
0.2
0.3
0.4
QSR
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
>TKUYX
0.5
UYTV[
19
IN229/Simulation: Lecture 4
Numerical stability and accuracy
We have two parameters,
related through
and
How do we choose
and
?
Too large values of
and
give
, that are
- too large numerical errors
- or in the worst case: unstable solutions
Too small
and
computing power
means too much
Simpli£ed problems can be analysed
theoretically
Guide to choosing
and
20
IN229/Simulation: Lecture 4
Example of stability analysis
The discrete solution can be written on the
form
\]
Inserted in the difference equation
^`_ba
^`_a
D
Aim: solve for
^dceagf
^dceagf
D
as a function of
In the real (continuous) problem: no damping,
which means is real
Want the same qualitative behavior in the
numerical scheme. This means is real,
which implies
, i.e.,
This is the stability criterion, often called the
CFL-condition.
21
IN229/Simulation: Lecture 4
Numerical example
Time t=0.0, σ=103
Time t=0.0, σ=1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
−10
−5
0
5
10
0
−10
Time t=15.0, CFL=1.0
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
−5
0
5
10
0
−10
Time t=15.0, CFL=0.8
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
−5
0
5
5
10
−5
0
5
10
Time t=15.0, CFL=0.8
1
0
−10
0
Time t=15.0, CFL=0.99
1
0
−10
−5
10
0
−10
−5
0
5
10
22
IN229/Simulation: Lecture 4
Large destructive water waves
The wave equations may also be used to
simulate large destructive waves
Waves in fjords, lakes, or the ocean,
generated by
- slides
- earthquakes
- subsea volcanos
- meteorittes
Human activity, like nuclear detonations, or
slides generated by oil drilling, may also
generate tsunamis
Propagation over large distances
Wave amplitude increases near shore
Run-up at the coasts may result in severe
damage
We will return to this topic later in the next
lecture..
23
m
idjlk
h
is the still-water depth (typically obtained from an electronic map).
For now: equation in 1-D
n
Here
The simplest (but also widely used) model for propagation of
water waves is the wave equation
The mathematical model
IN229/Simulation: Lecture 4
24
The term
phenomena
o
o
o
o
Two-step discretization, £rst outer operator
is common for many models of physical
Discretization of variable coef£cients
r
q
p
p
r
q
p
And the overall discretization reads
Then inner operator
IN229/Simulation: Lecture 4
r
p
s
s
s
25
Geometric mean:
Harmonic mean:
Arithmetic mean:
Often the function
is only given in the grid points, e.g., from
measurements. Thus we need to de£ne the value at the midpoint
Discretization of variable coef£cients...
IN229/Simulation: Lecture 4
s
26
IN229/Simulation: Lecture 4
Algorithm: arithmetic mean
De£ne storage
Set
Set
t
,
,
for
,
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t
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De£ne
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!
Update all inner points (
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Set boundary conditions
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@
Initialize for next step
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27
We are going to study:
“Oblig”: Tsunami due to a slide
IN229/Simulation: Lecture 4
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