Tutorial

This tutorial is a step by step description on how to setup, compile and run your first Basilisk simulation.

What you need:

  • You need to have at least a minimal understanding of shell commands. If you don’t know what I am talking about, you may want to start with one of the many online tutorials on this subject, such as Learning the shell.
  • You will also need a good text editor to write (C) programs in Basilisk. If you do not have a favourite one already, I recommend using emacs. You can install it easily on Debian-like systems by copying and pasting the following command in your shell:
sudo apt-get install emacs
  • Because Basilisk programs are written in a variant of the C language, any prior knowledge of C programming you may have will be very useful. If you have never seen a C program, you may want to read up on the topic, for example:

  • You then need to follow the installation instructions to setup basilisk on your system.

Getting started

You first need to open both a terminal and a text editor. On my system (Debian), the terminal is hidden in the “Activities -> Applications -> Accessories -> Terminal” menu. You can then start the text editor in the background by typing:

emacs &

To check that Basilisk is installed properly, do:

qcc --version

which returns on my system

gcc (Debian 4.7.2-5) 4.7.2
Copyright (C) 2012 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

If instead you get an error message, you need to go back to the installation instructions and check that you did everything correctly.

The simulation

The model problem we will study is a numerical approximation of the solution of the Saint-Venant equations in a square box with reflecting boundaries.

To create the Basilisk program, use the “File -> Visit New File” menu in emacs and type bump.c in the Name field. Then type the following line …

#include "saint-venant.h"

… and save the file using the “File -> Save” menu (or the Control-X Control-S keyboard shortcut). This C preprocessor command, includes the saint-venant.h file into our program. This file defines all the variables and functions which are required to run the Saint-Venant solver of Basilisk. If you follow the link above, you will see that the corresponding code is documented. Now is a good time to read the first two sections (up to Time integration) to remind yourself of what the equations, variables and parameters are…

Minimal program

We can now try to compile our program using (in the shell)

qcc bump.c

which produces something like

(.text+0x20): undefined reference to `main'
/tmp/ccueSq1Z.o: In function `normf':
bump.c:(.text+0x262d2): undefined reference to `sqrt'
...

The compiler (or more precisely the linker), complains that some functions are used but not defined. The sqrt function for example is defined in the standard math library which needs to be linked with the program. The main function however needs to be defined by us. Note that all this is not specific to Basilisk, it is just standard C.

To fix this problem, we need to add

#include "saint-venant.h"

int main() {
  run();
}

to bump.c (using the text editor). If you have some notions of C (i.e. you have done the homework given above…), you will recognise the definition of a C function called main which takes no parameters and just calls the function run. The run() function is defined by the Saint-Venant solver (i.e. is included in saint-venant.h).

We can now save the file and recompile using (in the shell)

qcc bump.c -lm

If we now type ls in the shell, we will see that this produced a file called a.out which is the compiled program, which we can launch using

./a.out

which gives

# Quadtree, 0 steps, 0 CPU, 0.001075 real, 0 points.step/s, 5 var

Amazing! we have done our first Basilisk simulation!

If you have compiled C programs before, you will see that all we have done here is very standard C, very little is specific to Basilisk. For example the -lm option is the standard way to link the math library (which defines the sqrt function i.e. square-root and other math functions). More generally, because the qcc command calls the C compiler in the background, you can use all the options supported by the C compiler. For example, if you are using GCC, it is a good idea to use

qcc -O2 -Wall bump.c -o bump -lm

where -O2 turns optimisation on (which makes the code faster), -Wall turns all compilation warnings on (which allows you to catch potential bugs in your program) and -o bump renames the compiled program to bump (rather than a.out).

A more interesting program

Of course our program does not do much yet. So why did we start with such a simple program? Because it is always a good idea to start simple and add complexity step-by-step. If a problem occurs in a single line of code, which you have just added, it is trivial to correct it. If this same line is buried within ten other (correct) lines, finding the problem can take (much) longer. This rule is true whether you are an experienced programmer or a beginner.

To make the program more interesting, it needs to produce some outputs at specified times. To do so, we can use events; like this for example:

#include "saint-venant.h"

event end (i = 10) {
  printf ("i = %d t = %g\n", i, t);
}

int main() {
  run();
}

What we have done here is told the solver to do 10 timesteps and then print the number of timesteps and the physical time it reached after 10 timesteps. If we now recompile using

qcc -O2 -Wall bump.c -o bump -lm

(note that you do not need to retype this command, you can just use the up and down keyboard arrows to go through the history of previous commands), and run using

./bump

we get something like

i = 10 t = 1e+11
# Quadtree, 10 steps, 0.04 CPU, 0.06103 real, 6.71e+05 points.step/s, 24 var

The first line comes from our code and the second line is the default output of basilisk. It gives the number of timesteps performed, the CPU time used for the computation, the real time elapsed, the corresponding computation speed (based on the real time) and the total number of fields allocated by the solver.

The syntax of events is specific to Basilisk (it is not standard C), however the body of the event works just like a standard C function. To learn more about events, have a look at the Basilisk C reference manual. Here we have used the standard C printf function to format the output.

Note that you can access the manual page for this function using either the link above or directly using the man command in the shell or in emacs (use the “Help -> More Manuals -> Read Man Page” menu and type “3 printf”).

man 3 printf

This can also be used to access the documentation for (almost) any command or program (for example man ls, man cp etc…) and is a good way to learn.

Initial conditions

For the moment, the solver just uses the default initial conditions of the Saint-Venant solver. We need to replace them with our own initial conditions.

By default the domain on which the equations are solved is a square box with reflective boundaries (i.e. symmetry conditions on scalar, vector and tensor fields). The origin of the coordinate system is the lower-left corner of the box and the box length is one. We can change this using the origin() and size() functions. For example, we can do:

...

int main() {
  origin (-0.5, -0.5);
  run();
}

which will center our box on the origin of the coordinate system.

Initial conditions are setup using the init event, like this:

#include "saint-venant.h"

event init (t = 0) {
  foreach()
    h[] = 0.1 + 1.*exp(-200.*(x*x + y*y));
}

event end (i = 10) {
  printf ("i = %d t = %g\n", i, t);
}

int main() {
  origin (-0.5, -0.5);
  run();
}

The init event will happen only at the beginning of the simulation (t=0). Within the body of the event, we use the Basilisk-specific foreach iterator to set the values of field h (the depth of the liquid layer as defined and documented in the Saint-Venant solver). We use a Gaussian bump of characteristic radius 1./200 and amplitude one on top of a layer of constant depth 0.1. The x and y coordinates are double values defined implicitly by the foreach operator.

If we now recompile and rerun, we get

i = 10 t = 0.0701117
# Quadtree, 10 steps, 0.06 CPU, 0.09314 real, 4.4e+05 points.step/s, 24 var

More outputs

For the moment, we don’t see much. Some graphical output would be nice. To generate simple images, we can use the output_ppm() function like this

...

event images (i++) {
  output_ppm (h);
}

event end (i = 10)
...

Recompiling and running, we get something like

...
�m��m��m��m��m��m�
�# Quadtree, 11 steps, 0.25 CPU, 0.5423 real, 8.31e+04 points.step/s, 24 var

What is all this garbage? By default, output_ppm() writes images on standard output (see Learning the shell if you don’t understand what this means). Standard output is the shell (i.e. the screen) by default. The strange characters we see on the screen are a translation of the binary contents of the images generated (at each timestep) by output_ppm(). We can change the standard output to a file rather than the screen using

./bump > out.ppm

where we used the .ppm extension to indicate that this file should contain images in PPM format. This format is recognised by many image processing tools. ImageMagick in particular provides nice command-line tools to manipulate these images. If you have not done so already, now is a good time to install ImageMagick and other graphics tools.

sudo apt-get install gnuplot imagemagick libav-tools smpeg-plaympeg

We can now try to use the display command of ImageMagick to display the images which should be in out.ppm.

display out.ppm

which should open a small window looking like this

h

h

We can right- or left-click on the window to access various menus. Using the space bar of the keyboard, we can cycle through the 10 images (one for each timestep) contained in out.ppm. If you look carefully, you will see that the radius of the red disk is slowly increasing with time. What we are looking at is a color-coded representation of the depth field h in the square box. Dark red is the maximum (one at the start) and light green is the minimum.

Why is the image so small? By default output_ppm() creates images with one pixel per grid point. If you right-click on the image and select the “Image Info” menu, you will see that the “geometry” is “64x64” i.e. the grid on which the equations are discretised is 64x64 grid points. This is the default in Basilisk. We will change it later.

To make things a bit more interesting, we will increase the number of timesteps with

...
event end (i = 300) {
...

Then re-compile and run using

qcc -O2 -Wall bump.c -o bump -lm
./bump > out.ppm

We could use display to view each of the 300 images we have now generated, but this would be a bit tedious. We will use another ImageMagick command instead

animate out.ppm

What we then see is all the images in quick succession. Once the last image is reached, the animation loops back to the beginning. This is a bit fast, we can use the menu or the ‘>’ keyboard shortcut to increase the delay between successive images (i.e. slow things down). We can then follow the evolution of the initial Gaussian bump. The circular wave propagates until it reaches the reflective walls, bounces back, refocuses in the center, bounces off and so on.

Simple measurements and graphs

Let’s assume we are interested in the evolution of the minimum and maximum depths as functions of time. We can record these values using for example

event graphs (i++) {
  stats s = statsf (h);
  fprintf (stderr, "%g %g %g\n", t, s.min, s.max);
}

event images (i++) {
...

In the first line, we call the statsf() function of Basilisk which fills the structure s with statistics on field h. In the second line, we use the standard C function fprintf() to write the time, minimum and maximum (of h) in the standard C file stderr. This stands for “standard error” which by default is the screen.

If we recompile and rerun, we get

...
3.77319 0.087509 0.195247
3.78692 0.0869611 0.192627
3.80072 0.0864178 0.189687

We can use file redirection to write these numbers to a file rather than on the screen. For example

./bump > out.ppm 2> log

To check what is in log we can use

more log

(use the space bar and the q key to scroll down or quit more).

We can do better than this if we use a plotting tool, gnuplot for example. Gnuplot is itself a command-line tool which has its own set of commands. You can start gnuplot with

gnuplot

You will then get a prompt looking like

gnuplot>

i.e. you are now within gnuplot (not within the shell anymore). To quit gnuplot and go back to the shell, just type ‘quit’. To display a graph of the min and max of h you can then do

gnuplot> set xlabel 'Time'
gnuplot> set ylabel 'Depth'
gnuplot> plot 'log' using 1:2 with lines title 'min', 'log' using 1:3 with lines title 'max' 

which should produce

Min and max

Min and max

Read one of the many tutorials if you want to know more about gnuplot.

Increasing the resolution

For the moment our mesh is only 64x64. As we can see in the animation and graphs of the maximum depth, the Saint-Venant equations easily develop non-linear shocks i.e. sharp discontinuities in the depth profile. These shocks are probably not well described on this relatively coarse grid.

To increase the resolution, we can simply do

...
int main() {
  origin (-0.5, -0.5);
  init_grid (256);
  run();
}

which will use a 256x256 grid. Why a power of two? Because by default, Basilisk uses a quadtree grid, which restricts the resolution to powers of two. We will see later that this is not the case for other grids.

Before we rerun the simulation, we will save the data we produced at low resolution using

mv log log.64
mv out.ppm out.64.ppm

We can now recompile and rerun the simulation. The first thing we note is that it is much slower. This is not surprising since the number of grid points has been multiplied by 16 and we can expect the simulation to be proportionately slower.

If we redo

animate out.ppm

we get a larger (and sharper) picture of the wave, however it does not propagate as far as before (it barely touches the walls of the box). Since we do the same number of timesteps, it must mean that each timestep is smaller than before. Indeed, the timestep is controled by the CFL condition for the Saint-Venant system and is thus proportional to the grid spacing. In our case we have decreased the grid spacing by a factor of four, so that we would need four times as many timesteps to reach the same time as in the previous simulation. Putting this together with the increase in number of grid points, we can expect the total runtime to be 16×4=64 times larger than for the previous simulation…

How can we make things faster without loosing the accuracy of the finer grid?

We could use a faster computer and/or use more processors (use parallel computing).

Changing the grid

With Basilisk, we also have the choice of the type of grid used to discretise the equations. Simpler grid structures usually run faster.

To have an idea of how long the simulation took, we can do

tail -n1 out.ppm

which gives on my machine

...
# Quadtree, 301 steps, 47.85 CPU, 53.98 real, 3.65e+05 points.step/s, 24 var

i.e. the program can do one timestep for 365 000 grid points in one second (The tail -n1 command displays the last line of file out.ppm; you can do man tail to learn more about this). You also see that this was running on a quadtree grid implementation (but at constant spatial resolution).

To change the grid used by Basilisk, we can edit the code as

#include "grid/cartesian.h"
#include "saint-venant.h"
...

which will force Basilisk to use a pure Cartesian grid implementation. If we now recompile, rerun and recover the last line with

qcc -O2 -Wall bump.c -o bump -lm
./bump > out.ppm 2> log
tail -n1 out.ppm

we get

...
# Cartesian, 301 steps, 22.05 CPU, 25.14 real, 7.85e+05 points.step/s, 24 var

i.e. the Cartesian grid implementation is about twice as fast as the quadtree implementation (for the same result).

Setting time intervals

Because the timestep is controlled by the spatial resolution, it is generally not a good idea to output physical results at regular intervals expressed in number of timesteps. It makes more sense to output results at intervals expressed in physical time units.

For example, in our case, we know that our initial simulation ran to a time of about t=4 (see the graph above) and that the 300 images we generated were sufficient to get a nice animation of the wave propagation. If we want to reproduce these results at higher resolution, it thus makes sense to modify our program like this

...
event images (t += 4./300.) {
  output_ppm (h);
}

event end (t = 4) {
  printf ("i = %d t = %g\n", i, t);
}
...

where the output intervals are now specified in units of physical time.

We can now recompile, rerun etc… Since this is going to take a while, it would be nice to be able to follow where the simulation is at. To do this, you can open a new terminal (for example using the “File -> Open Tab” menu), and type in the new terminal

tail -f log

In this case the tail command displays what is being written in file log. The left column is the time (which needs to reach 4). You can exit from tail using the Ctrl-C key.

Another way to follow the simulation is to open gnuplot in another terminal and display the graphs for h as we did before. Using the ‘replot’ command, or clicking on the blue circular arrow in the graph window, or hitting the ‘e’ key in the graph window, will refresh the curves as the simulation progresses.

We can also use animate on out.ppm while the simulation is running.

A rough check for convergence

How do we know if we need to increase the resolution further? A good way to estimate the numerical accuracy of the solution is to compare results obtained at different resolutions. We can do this “visually” with gnuplot

gnuplot> set xlabel 'Time'
gnuplot> set ylabel 'Depth'
gnuplot> plot 'log.64' using 1:2 with lines title 'min (64)', \
              'log.64' using 1:3 with lines title 'max (64)', \
              'log' using 1:2 with lines title 'min (256)', \
              'log' using 1:3 with lines title 'max (256)'

which gives

Convergence

Convergence

We see that the peaks and discontinuities are definitely sharper at higher resolution. Although other parts of the graphs are reasonably close, we may want to try an even higher resolution to see if the amplitudes of the peaks converge.

Using adaptive grid refinement

From the animation and graphs, we intuitively get the sense that the characteristic spatial scales of the waves we are studying are not constant. Some areas are very smooth with no significant features, while other areas include fine details (interacting shocks for example). Clearly, high resolution is not needed everywhere and the computation could probably be made faster if the resolution was adapted to the solution. This variable resolution also needs to evolve in time to follow the moving details.

Basilisk uses quadtrees to allow efficient adaptive grid refinement. The first thing we need to do is to remove the line setting the grid to Cartesian i.e.

#include "grid/cartesian.h"
...

We can then add

...
event adapt (i++) {
  adapt_wavelet ({h}, (double []){4e-3}, maxlevel = 8);
}

int main() {
...

We have just told Basilisk to adapt the resolution according to the (wavelet-estimated) discretisation error of field h. This adaptation is done at each timestep (i++). Whenever the discretisation error is larger than 4×103, the mesh is refined, down to a maximum of 8 quadtree levels (i.e. 28=256 points per dimension).

Before we rerun the simulation, we first save the previous results

mv out.ppm out.256.ppm
mv log log.256

and then recompile and rerun.

The first thing we note is that the simulation runs significantly faster. Which is confirmed by

tail -n1 out.ppm

which gives

# Quadtree, 1474 steps, 52.15 CPU, 61.65 real, 1.6e+05 points.step/s, 24 var

compared to (tail -n1 out.256.ppm)

# Cartesian, 1485 steps, 96.76 CPU, 113.9 real, 8.54e+05 points.step/s, 24 var

on the regular Cartesian grid i.e. roughly twice as fast with adaptivity.

This looks good at first, however the animation reveals that things are a bit more complex. In particular, the animation does not look as smooth; as shown on the following frame

First-order image interpolation

First-order image interpolation

What is happening is that because the resolution varies, there isn’t a one-to-one mapping between grid points and pixels in the image generated by output_ppm(). So interpolation is required. By default output_ppm() uses only first-order interpolation: all the pixels within a quadtree cell encode the same value so have the same color. The animation now shows information about both the field (the color) and the adaptive quadtree grid (the sharp changes in color).

That’s interesting but does not look very good. To use bilinear interpolation instead, we need to call output_ppm() like this

...
event images (t += 4./300.) {
  output_ppm (h, linear = true);
}
...

then recompile, rerun etc… The animation now looks good.

Are the results close to that obtained when using the regular Cartesian grid? As for the convergence study above, we can use gnuplot to find out

gnuplot> set xlabel 'Time'
gnuplot> set ylabel 'Depth'
gnuplot> plot 'log.256' using 1:2 with lines title 'min', \
              'log.256' using 1:3 with lines title 'max', \
              'log' using 1:2 with lines title 'min (adaptive)', \
              'log' using 1:3 with lines title 'max (adaptive)'

which gives

Comparison between adaptive and Cartesian simulations

Comparison between adaptive and Cartesian simulations

The results are close but not identical. Finding the right balance between computing time and accuracy is an important part of setting up numerical simulations.

Displaying the grid

Although our animation now looks good, we lost the information about grid size which was (partly) encoded in the first animation. Can we generate an animation of the grid being used during the simulation?

One way to do this is to animate a field containing the level of the quadtree cells rather than the depth h. Such a field does not exist by default (even though we can access the value level, as well as the coordinates x and y, within foreach loops). We thus need to allocate a new field and fill it with the values of level. We will then be able to generate the corresponding animation using output_ppm() with this field.

We can do all this with the following code

...
event images (t += 4./300.) {
  ...
  scalar l[];
  foreach()
    l[] = level;
  static FILE * fp = fopen ("grid.ppm", "w");
  output_ppm (l, fp, min = 0, max = 8);
}
...

We first declare and allocate a new scalar field l. This field is a local, automatic variable i.e. it will be accessible only within the images event. The memory necessary to store the field values will be automatically freed when the code leaves this function.

We then loop over all the cells and set the values of l to the level of the cell.

The next line declares a static variable i.e. a variable which is kept in memory between calls to images (in contrast to automatic variables). This variable is set only once, the first time images is called, and points to a new file called grid.ppm in which we will write (“w”) things. This is done using the standard C function fopen().

We can now call output_ppm() using l as the field to display. Rather than writing images to the standard output, we use our file pointer fp. We also set the minimum and maximum values of the colorscale to avoid changes in color during the animation.

After recompiling and re-running, we can now do

animate grid.ppm

which gives an animation looking like

Adaptive grid

Adaptive grid

Using Makefiles

By now, you are probably tired of typing

qcc -O2 -Wall bump.c -o bump -lm
./bump > out.ppm 2> log

You may also have made the mistake of doing only the second command (running the code) and wondered why nothing changed in the ouput (while you had edited bump.c).

Makefiles are a very useful tool to automate such a processing chain (i.e. log and out.ppm both depend on bump which in turns depends on bump.c).

Basilisk comes with a predefined Makefile which you can reuse for your own computations. You just need to create a new text file called Makefile (e.g. using the “File -> Visit New File” menu in emacs) and type

CFLAGS += -O2
include $(BASILISK)/Makefile.defs

Save the file and type in the shell

rm bump
make bump.tst

This should produce something like

.../Makefile.defs:9: Makefile.tests: No such file or directory
.../Makefile.defs:93: Makefile.deps: No such file or directory
Updating Makefile.deps
sh /home/popinet/basilisk/src/tests.sh
updating Makefile.tests
.../Makefile.defs:93: Makefile.deps: No such file or directory
.../qcc -MD -o bump.s.d bump.c
Updating Makefile.deps
qcc -O2 -Wall -o bump/bump bump.c -lm
[bump.tst]

Do not worry about the first lines, they come from the initial setup and will not be repeated when you invoke make again.

If you now do ls in the shell, you will see that a new directory called bump has been created. This directory contains both the executable (also called bump) and all the files produced when running the program in this directory.

ls bump/*

The standard output has been redirected to bump/out and the standard error to bump/log. As before, we can then run the animation using

animate bump/out

and display bump/log with gnuplot. Note that it is a good idea to open a new terminal, cd to bump and leave gnuplot running in this terminal. You will then be able to re-run the simulation using make bump.tst in one terminal and display the results (while the simulation is running) using replot in the other (gnuplot) terminal.

If you now redo

make bump.tst

you will get

make: `bump.tst' is up to date.

The Makefile detected that nothing was modified which required recompiling and/or rerunning the simulation.

The default Makefile in Basilisk does much more than this. Read “Running and creating test cases (and examples)” if you want to know more.

Using macros

I mentioned above that it would be a good idea to study the numerical convergence of our example a bit more seriously. To do this, we need to run the same code while varying the resolution. We could edit the code by hand, changing each reference to resolution, recompile, rerun etc… but that would be quite tedious and error-prone. A better way to do this is to use standard C macros (which you should have encountered already during your preparatory work).

If we look at our code, we see that the resolution or level of refinement occur three times. Once as an argument to output_ppm(), once as an argument to adapt_wavelet() and once as an argument of init_grid() in the main() function.

Rather than changing these three values manually, we can write instead

#include "saint-venant.h"

#define LEVEL 8
...
  output_ppm (l, fp, min = 0, max = LEVEL);
...
  adapt_wavelet ({h}, (double []){4e-3}, maxlevel = LEVEL);
...
  init_grid (1 << LEVEL);
...

That is, the macro LEVEL will be replaced by 8 in all three places. The << operator in C is a bit-shifting operation. All we need to know here is that 1 << LEVEL is identical to 2LEVEL. If we want to change the resolution of the simulation, all we need to do now is change the single value at the top of the file.

Wrapping up

If you have followed the tutorial, your bump.c file should look like

#include "saint-venant.h"

#define LEVEL 8

event init (t = 0) {
  foreach()
    h[] = 0.1 + 1.*exp(-200.*(x*x + y*y));
}

event graphs (i++) {
  stats s = statsf (h);
  fprintf (stderr, "%g %g %g\n", t, s.min, s.max);
}

event images (t += 4./300.) {
  output_ppm (h, linear = true);

  scalar l[];
  foreach()
    l[] = level;
  static FILE * fp = fopen ("grid.ppm", "w");
  output_ppm (l, fp, min = 0, max = LEVEL);
}

event end (t = 4) {
  printf ("i = %d t = %g\n", i, t);
}

event adapt (i++) {
  adapt_wavelet ({h}, (double []){4e-3}, maxlevel = LEVEL);
}

int main() {
  origin (-0.5, -0.5);
  init_grid (1 << LEVEL);
  run();
}

Further reading

I have made several references to the Basilisk C manual. Now is a good time to go through the manual and get more familiar with the concepts, keywords and functions which are specific to Basilisk. Note that you won’t need all the features of the language if you just want to use pre-defined solvers. The most important concepts and keywords have been covered in this tutorial.

For more examples of applications, post-processing, graphs etc…, you can also look at the

Note also that in these examples, as in the pieces of code above, various keywords will be linked either to the documentation for standard C functions or to the documentation of Basilisk keywords and functions. You can learn a lot by following these links.

If you want to know what Basilisk can be used for, have a look at the various solvers available. Again, if you just want to use these pre-defined solvers, reading only the first few sections should be sufficient (as we did for saint-venant.h), but feel free to dig deeper if you are interested. Note also that the “Usage” section at the end of each page contains links to various applications of the solvers.