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Commit 7b4c3f2d authored by Martin Bauer's avatar Martin Bauer
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Refactoring of plotting and stencil plotting 

- stencil plotting & transformation now in ps.stencil
- additional documentation & notebooks
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max-line-length=120
exclude=pystencils/jupytersetup.py,
pystencils/plot2d.py
exclude=pystencils/jupyter.py,
pystencils/plot.py
pystencils/session.py
ignore = W293 W503 W291
doc/notebooks/01_tutorial_getting_started.ipynb
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%% Cell type:code id: tags:
``` python
from pystencils.session import *
```
%% Cell type:markdown id: tags:
# Tutorial 01: Getting Started
## Overview
*pystencils* is a package that can speed up computations on *numpy* arrays. All computations are carried out fully parallel on CPUs (single node with OpenMP, multiple nodes with MPI) or on GPUs.
It is suited for applications that run the same operation on *numpy* arrays multiple times. It can be used to accelerate computations on images or voxel fields. Its main application, however, are numerical simulations using finite differences, finite volumes, or lattice Boltzmann methods.
There already exist a variety of packages to speed up numeric Python code. One could use pure numpy or solutions that compile your code, like *Cython* and *numba*. See [this page](demo_benchmark.ipynb) for a comparison of these tools.
![Stencil](../img/pystencils_stencil_four_points_with_arrows.svg)
As the name suggests, *pystencils* was developed for **stencil codes**, i.e. operations that update array elements using only a local neighborhood.
It generates C code, compiles it behind the scenes, and lets you call the compiled C function as if it was a native Python function.
But lets not dive too deep into the concepts of *pystencils* here, they are covered in detail in the following tutorials. Let's instead look at a simple example, that computes the average neighbor values of a *numpy* array. Therefor we first create two rather large arrays for input and output:
%% Cell type:code id: tags:
``` python
input_arr = np.random.rand(1024, 1024)
output_arr = np.zeros_like(input_arr)
```
%% Cell type:markdown id: tags:
We first implement a version of this algorithm using pure numpy and benchmark it.
%% Cell type:code id: tags:
``` python
def numpy_kernel():
output_arr[1:-1, 1:-1] = input_arr[2:, 1:-1] + input_arr[:-2, 1:-1] + \
input_arr[1:-1, 2:] + input_arr[1:-1, :-2]
```
%% Cell type:code id: tags:
``` python
%%timeit
numpy_kernel()
```
%% Output
3.84 ms ± 36.2 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
3.93 ms ± 40 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
%% Cell type:markdown id: tags:
Now lets see how to run the same algorithm with *pystencils*.
%% Cell type:code id: tags:
``` python
src, dst = ps.fields(src=input_arr, dst=output_arr)
symbolic_description = ps.Assignment(dst[0,0],
(src[1, 0] + src[-1, 0] + src[0, 1] + src[0, -1]) / 4)
symbolic_description
```
%% Output
$\displaystyle {{dst}_{(0,0)}} \leftarrow \frac{{{src}_{(-1,0)}}}{4} + \frac{{{src}_{(0,-1)}}}{4} + \frac{{{src}_{(0,1)}}}{4} + \frac{{{src}_{(1,0)}}}{4}$
src_W src_S src_N src_E
dst_C := ───── + ───── + ───── + ─────
4 4 4 4
%% Cell type:code id: tags:
``` python
plt.figure(figsize=(3,3))
ps.visualize_stencil_expression(symbolic_description.rhs)
ps.stencil.plot_expression(symbolic_description.rhs)
```
%% Output
%% Cell type:markdown id: tags:
Here we first have created a symbolic notation of the stencil itself. This representation is built on top of *sympy* and is explained in detail in the next section.
This description is then compiled and loaded as a Python function.
%% Cell type:code id: tags:
``` python
kernel = ps.create_kernel(symbolic_description).compile()
```
%% Cell type:markdown id: tags:
This whole process might seem overly complicated. We have already spent more lines of code than we needed for the *numpy* implementation and don't have anything running yet! However, this multi-stage process of formulating the algorithm symbolically, and just in the end actually running it, is what makes *pystencils* faster and more flexible than other approaches.
Now finally lets benchmark the *pystencils* kernel.
%% Cell type:code id: tags:
``` python
def pystencils_kernel():
kernel(src=input_arr, dst=output_arr)
```
%% Cell type:code id: tags:
``` python
%%timeit
pystencils_kernel()
```
%% Output
639 µs ± 35 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
643 µs ± 8.66 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
%% Cell type:markdown id: tags:
This benchmark shows that *pystencils* is a lot faster than pure *numpy*, especially for large arrays.
If you are interested in performance details and comparison to other packages like Cython, have a look at [this page](demo_benchmark.ipynb).
%% Cell type:markdown id: tags:
## Short *sympy* introduction
In this tutorial we continue with a short *sympy* introduction, since the symbolic kernel definition is built on top of this package. If you already know *sympy* you can skip this section.
You can also read the full [sympy documentation here](http://docs.sympy.org/latest/index.html).
%% Cell type:code id: tags:
``` python
import sympy as sp
sp.init_printing() # enable nice LaTeX output
```
%% Cell type:markdown id: tags:
*sympy* is a package for symbolic calculation. So first we need some symbols:
%% Cell type:code id: tags:
``` python
x = sp.Symbol("x")
y = sp.Symbol("y")
type(x)
```
%% Output
sympy.core.symbol.Symbol
%% Cell type:markdown id: tags:
The usual mathematical operations are defined for symbols:
%% Cell type:code id: tags:
``` python
expr = x**2 * ( y + x + 5) + x**2
expr
```
%% Output
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
2 2
x ⋅(x + y + 5) + x
%% Cell type:markdown id: tags:
Now we can do all sorts of operations on these expressions: expand them, factor them, substitute variables:
%% Cell type:code id: tags:
``` python
expr.expand()
```
%% Output
$\displaystyle x^{3} + x^{2} y + 6 x^{2}$
3 2 2
x + x ⋅y + 6⋅x
%% Cell type:code id: tags:
``` python
expr.factor()
```
%% Output
$\displaystyle x^{2} \left(x + y + 6\right)$
2
x ⋅(x + y + 6)
%% Cell type:code id: tags:
``` python
expr.subs(y, sp.cos(x))
```
%% Output
$\displaystyle x^{2} \left(x + \cos{\left(x \right)} + 5\right) + x^{2}$
2 2
x ⋅(x + cos(x) + 5) + x
%% Cell type:markdown id: tags:
We can also built equations and solve them
%% Cell type:code id: tags:
``` python
eq = sp.Eq(expr, 1)
eq
```
%% Output
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2} = 1$
2 2
x ⋅(x + y + 5) + x = 1
%% Cell type:code id: tags:
``` python
sp.solve(sp.Eq(expr, 1), y)
```
%% Output
$\displaystyle \left[ - x - 6 + \frac{1}{x^{2}}\right]$
⎡ 1 ⎤
⎢-x - 6 + ──⎥
⎢ 2⎥
⎣ x ⎦
%% Cell type:markdown id: tags:
A *sympy* expression is represented by an abstract syntax tree (AST), which can be inspected and modified.
%% Cell type:code id: tags:
``` python
expr
```
%% Output
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
2 2
x ⋅(x + y + 5) + x
%% Cell type:code id: tags:
``` python
ps.to_dot(expr, graph_style={'size': "9.5,12.5"} )
```
%% Output
<graphviz.files.Source at 0x7fc7dc51b2e8>
<graphviz.files.Source at 0x7ff8a018e7f0>
%% Cell type:markdown id: tags:
Programatically the children node type is acessible as ``expr.func`` and its children as ``expr.args``.
With these members a tree can be traversed and modified.
%% Cell type:code id: tags:
``` python
expr.func
```
%% Output
sympy.core.add.Add
%% Cell type:code id: tags:
``` python
expr.args
```
%% Output
$\displaystyle \left( x^{2}, \ x^{2} \left(x + y + 5\right)\right)$
⎛ 2 2 ⎞
⎝x , x ⋅(x + y + 5)⎠
%% Cell type:markdown id: tags:
## Using *pystencils*
### Fields
*pystencils* is a module to generate code for stencil operations.
One has to specify an update rule for each element of an array, with optional dependencies to neighbors.
This is done use pure *sympy* with one addition: **Fields**.
Fields represent a multidimensional array, where some dimensions are considered *spatial*, and some as *index* dimensions. Spatial coordinates are given relative (i.e. one can specify "the current cell" and "the left neighbor") whereas index dimensions are used to index multiple values per cell.
%% Cell type:code id: tags:
``` python
my_field = ps.fields("f(3) : double[2D]")
```
%% Cell type:markdown id: tags:
Neighbors are labeled according to points on a compass where the first coordinate is west/east, second coordinate north/south and third coordinate top/bottom.
%% Cell type:code id: tags:
``` python
field_access = my_field[1, 0](1)
field_access
```
%% Output
$\displaystyle {{f}_{(1,0)}^{1}}$
f_E__1
%% Cell type:markdown id: tags:
The result of indexing a field is an instance of ``Field.Access``. This class is a subclass of a *sympy* Symbol and thus can be used whereever normal symbols can be used. It is just like a normal symbol with some additional information attached to it.
%% Cell type:code id: tags:
``` python
isinstance(field_access, sp.Symbol)
```
%% Output
True
%% Cell type:markdown id: tags:
### Building our first stencil kernel
Lets start by building a simple filter kernel. We create a field representing an image, then define a edge detection filter on the third pixel component which is blue for an RGB image.
%% Cell type:code id: tags:
``` python
img_field = ps.fields("img(4): [2D]")
```
%% Cell type:code id: tags:
``` python
w1, w2 = sp.symbols("w_1 w_2")
color = 2
sobel_x = (-w2 * img_field[-1,0](color) - w1 * img_field[-1,-1](color) - w1 * img_field[-1, +1](color) \
+w2 * img_field[+1,0](color) + w1 * img_field[+1,-1](color) - w1 * img_field[+1, +1](color))**2
sobel_x
```
%% Output
$\displaystyle \left(- {{img}_{(-1,-1)}^{2}} w_{1} - {{img}_{(-1,0)}^{2}} w_{2} - {{img}_{(-1,1)}^{2}} w_{1} + {{img}_{(1,-1)}^{2}} w_{1} + {{img}_{(1,0)}^{2}} w_{2} - {{img}_{(1,1)}^{2}} w_{1}\right)^{2}$
(-img_SW__2⋅w₁ - img_W__2⋅w₂ - img_NW__2⋅w₁ + img_SE__2⋅w₁ + img_E__2⋅w₂ - img
2
_NE__2⋅w₁)
%% Cell type:markdown id: tags:
We have mixed some standard *sympy* symbols into this expression to possibly give the different directions different weights. The complete expression is still a valid *sympy* expression, so all features of *sympy* work on it. Lets for example now fix one weight by substituting it with a constant.
%% Cell type:code id: tags:
``` python
sobel_x = sobel_x.subs(w1, 0.5)
sobel_x
```
%% Output
$\displaystyle \left(- 0.5 {{img}_{(-1,-1)}^{2}} - {{img}_{(-1,0)}^{2}} w_{2} - 0.5 {{img}_{(-1,1)}^{2}} + 0.5 {{img}_{(1,-1)}^{2}} + {{img}_{(1,0)}^{2}} w_{2} - 0.5 {{img}_{(1,1)}^{2}}\right)^{2}$
(-0.5⋅img_SW__2 - img_W__2⋅w₂ - 0.5⋅img_NW__2 + 0.5⋅img_SE__2 + img_E__2⋅w₂ -
2
0.5⋅img_NE__2)
%% Cell type:markdown id: tags:
Now lets built an executable kernel out of it, which writes the result to a second field. Assignments are created using *pystencils* `Assignment` class, that gets the left- and right hand side of the assignment.
%% Cell type:code id: tags:
``` python
dst_field = ps.fields('dst: [2D]' )
update_rule = ps.Assignment(dst_field[0,0], sobel_x)
update_rule
```
%% Output
$\displaystyle {{dst}_{(0,0)}} \leftarrow \left(- 0.5 {{img}_{(-1,-1)}^{2}} - {{img}_{(-1,0)}^{2}} w_{2} - 0.5 {{img}_{(-1,1)}^{2}} + 0.5 {{img}_{(1,-1)}^{2}} + {{img}_{(1,0)}^{2}} w_{2} - 0.5 {{img}_{(1,1)}^{2}}\right)^{2}$
dst_C := (-0.5⋅img_SW__2 - img_W__2⋅w₂ - 0.5⋅img_NW__2 + 0.5⋅img_SE__2 + img_E
2
__2⋅w₂ - 0.5⋅img_NE__2)
%% Cell type:markdown id: tags:
Next we can see *pystencils* in action which creates a kernel for us.
%% Cell type:code id: tags:
``` python
from pystencils import create_kernel
ast = create_kernel(update_rule, cpu_openmp=False)
compiled_kernel = ast.compile()
```
%% Cell type:markdown id: tags:
This compiled kernel is now just an ordinary Python function.
Now lets grab an image to apply this filter to:
%% Cell type:code id: tags:
``` python
import requests
import imageio
from io import BytesIO
response = requests.get("https://www.python.org/static/img/python-logo.png")
img = imageio.imread(BytesIO(response.content)).astype(np.double)
img /= img.max()
plt.imshow(img);
```
%% Output
%% Cell type:code id: tags:
``` python
filtered_image = np.zeros_like(img[..., 0])
# here we call the compiled stencil function
compiled_kernel(img=img, dst=filtered_image, w_2=0.5)
plt.imshow(filtered_image, cmap='gray');
```
%% Output
%% Cell type:markdown id: tags:
### Digging into *pystencils*
On our way we have created an ``ast``-object. We can inspect this, to see what *pystencils* actually does.
%% Cell type:code id: tags:
``` python
ps.to_dot(ast, graph_style={'size': "9.5,12.5"})
```
%% Output
<graphviz.files.Source at 0x7fc798393fd0>
<graphviz.files.Source at 0x7ff84a432e10>
%% Cell type:markdown id: tags:
*pystencils* also builds a tree structure of the program, where each `Assignment` node internally again has a *sympy* AST which is not printed here. Out of this representation *C* code can be generated:
%% Cell type:code id: tags:
``` python
ps.show_code(ast)
```
%% Output
FUNC_PREFIX void kernel(double * RESTRICT _data_dst, double * RESTRICT const _data_img, int64_t const _size_dst_0, int64_t const _size_dst_1, int64_t const _stride_dst_0, int64_t const _stride_dst_1, int64_t const _stride_img_0, int64_t const _stride_img_1, int64_t const _stride_img_2, double w_2)
{
double * RESTRICT const _data_img_22 = _data_img + 2*_stride_img_2;
for (int ctr_0 = 1; ctr_0 < _size_dst_0 - 1; ctr_0 += 1)
{
double * RESTRICT _data_dst_00 = _data_dst + _stride_dst_0*ctr_0;
double * RESTRICT const _data_img_22_01 = _stride_img_0*ctr_0 + _stride_img_0 + _data_img_22;
double * RESTRICT const _data_img_22_0m1 = _stride_img_0*ctr_0 - _stride_img_0 + _data_img_22;
for (int ctr_1 = 1; ctr_1 < _size_dst_1 - 1; ctr_1 += 1)
{
_data_dst_00[_stride_dst_1*ctr_1] = ((w_2*_data_img_22_01[_stride_img_1*ctr_1] - w_2*_data_img_22_0m1[_stride_img_1*ctr_1] - 0.5*_data_img_22_01[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 - _stride_img_1] + 0.5*_data_img_22_01[_stride_img_1*ctr_1 - _stride_img_1])*(w_2*_data_img_22_01[_stride_img_1*ctr_1] - w_2*_data_img_22_0m1[_stride_img_1*ctr_1] - 0.5*_data_img_22_01[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 - _stride_img_1] + 0.5*_data_img_22_01[_stride_img_1*ctr_1 - _stride_img_1]));
}
}
}
%% Cell type:markdown id: tags:
Behind the scenes this code is compiled into a shared library and made available as a Python function. Before compiling this function we can modify the AST object, for example to parallelize it with OpenMP.
%% Cell type:code id: tags:
``` python
ast = ps.create_kernel(update_rule)
ps.cpu.add_openmp(ast, num_threads=2)
ps.show_code(ast)
```
%% Output
FUNC_PREFIX void kernel(double * RESTRICT _data_dst, double * RESTRICT const _data_img, int64_t const _size_dst_0, int64_t const _size_dst_1, int64_t const _stride_dst_0, int64_t const _stride_dst_1, int64_t const _stride_img_0, int64_t const _stride_img_1, int64_t const _stride_img_2, double w_2)
{
#pragma omp parallel num_threads(2)
{
double * RESTRICT const _data_img_22 = _data_img + 2*_stride_img_2;
#pragma omp for schedule(static)
for (int ctr_0 = 1; ctr_0 < _size_dst_0 - 1; ctr_0 += 1)
{
double * RESTRICT _data_dst_00 = _data_dst + _stride_dst_0*ctr_0;
double * RESTRICT const _data_img_22_01 = _stride_img_0*ctr_0 + _stride_img_0 + _data_img_22;
double * RESTRICT const _data_img_22_0m1 = _stride_img_0*ctr_0 - _stride_img_0 + _data_img_22;
for (int ctr_1 = 1; ctr_1 < _size_dst_1 - 1; ctr_1 += 1)
{
_data_dst_00[_stride_dst_1*ctr_1] = ((w_2*_data_img_22_01[_stride_img_1*ctr_1] - w_2*_data_img_22_0m1[_stride_img_1*ctr_1] - 0.5*_data_img_22_01[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 - _stride_img_1] + 0.5*_data_img_22_01[_stride_img_1*ctr_1 - _stride_img_1])*(w_2*_data_img_22_01[_stride_img_1*ctr_1] - w_2*_data_img_22_0m1[_stride_img_1*ctr_1] - 0.5*_data_img_22_01[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 + _stride_img_1] - 0.5*_data_img_22_0m1[_stride_img_1*ctr_1 - _stride_img_1] + 0.5*_data_img_22_01[_stride_img_1*ctr_1 - _stride_img_1]));
}
}
}
}
%% Cell type:code id: tags:
``` python
loops = list(ast.atoms(ps.astnodes.LoopOverCoordinate))
l1 = loops[0]
l1.prefix_lines.append("#pragma someting")
l1.is_outermost_loop
```
%% Output
False
%% Cell type:markdown id: tags:
### Fixed array sizes
Since we already know the arrays to which the kernel should be applied, we can
create *Field* objects with fixed size, based on a numpy array:
%% Cell type:code id: tags:
``` python
img_field, dst_field = ps.fields("I(4), dst : [2D]", I=img.astype(np.double), dst=filtered_image)
sobel_x = -2 * img_field[-1,0](1) - img_field[-1,-1](1) - img_field[-1, +1](1) \
+2 * img_field[+1,0](1) + img_field[+1,-1](1) - img_field[+1, +1](1)
update_rule = ps.Assignment(dst_field[0,0], sobel_x)
ast = create_kernel(update_rule)
ps.show_code(ast)
```
%% Output
FUNC_PREFIX void kernel(double * RESTRICT const _data_I, double * RESTRICT _data_dst)
{
double * RESTRICT const _data_I_21 = _data_I + 1;
for (int ctr_0 = 1; ctr_0 < 81; ctr_0 += 1)
{
double * RESTRICT _data_dst_00 = _data_dst + 290*ctr_0;
double * RESTRICT const _data_I_21_01 = _data_I_21 + 1160*ctr_0 + 1160;
double * RESTRICT const _data_I_21_0m1 = _data_I_21 + 1160*ctr_0 - 1160;
for (int ctr_1 = 1; ctr_1 < 289; ctr_1 += 1)
{
_data_dst_00[ctr_1] = -2*_data_I_21_0m1[4*ctr_1] + 2*_data_I_21_01[4*ctr_1] - _data_I_21_01[4*ctr_1 + 4] + _data_I_21_01[4*ctr_1 - 4] - _data_I_21_0m1[4*ctr_1 + 4] - _data_I_21_0m1[4*ctr_1 - 4];
}
}
}
%% Cell type:markdown id: tags:
Compare this code to the version above. In this code the loop bounds and array offsets are constants, which usually leads to faster kernels.
%% Cell type:markdown id: tags:
### Running on GPU
If you have a CUDA enabled graphics card and [pycuda](https://mathema.tician.de/software/pycuda/) installed, *pystencils* can run your kernel on the GPU as well. You can find more details about this in the GPU tutorial.
%% Cell type:code id: tags:
``` python
gpu_ast = create_kernel(update_rule, target='gpu', gpu_indexing=ps.gpucuda.indexing.BlockIndexing,
gpu_indexing_params={'blockSize': (8,8,4)})
```
%% Cell type:code id: tags:
``` python
ps.show_code(gpu_ast)
```
%% Output
FUNC_PREFIX __launch_bounds__(256) void kernel(double * RESTRICT const _data_I, double * RESTRICT _data_dst)
{
if (blockDim.x*blockIdx.x + threadIdx.x + 1 < 81 && blockDim.y*blockIdx.y + threadIdx.y + 1 < 289)
{
const int64_t ctr_0 = blockDim.x*blockIdx.x + threadIdx.x + 1;
const int64_t ctr_1 = blockDim.y*blockIdx.y + threadIdx.y + 1;
double * RESTRICT _data_dst_10 = _data_dst + ctr_1;
double * RESTRICT const _data_I_11_21 = _data_I + 4*ctr_1 + 5;
double * RESTRICT const _data_I_1m1_21 = _data_I + 4*ctr_1 - 3;
double * RESTRICT const _data_I_10_21 = _data_I + 4*ctr_1 + 1;
_data_dst_10[290*ctr_0] = -2*_data_I_10_21[1160*ctr_0 - 1160] + 2*_data_I_10_21[1160*ctr_0 + 1160] - _data_I_11_21[1160*ctr_0 + 1160] - _data_I_11_21[1160*ctr_0 - 1160] + _data_I_1m1_21[1160*ctr_0 + 1160] - _data_I_1m1_21[1160*ctr_0 - 1160];
}
}
......
doc/notebooks/04_tutorial_advection_diffusion.ipynb
View file @ 7b4c3f2d
%% Cell type:code id: tags:
``` python
from pystencils.session import *
```
%% Cell type:markdown id: tags:
# Tutorial 04: Advection Diffusion - Simple finite differences discretization
In this tutorial we demonstrate how to use the discretization layer on top of *pystencils*, that defines how continuous differential operators are discretized. The result of this discretization layer are stencil equations which are used to generated C or CUDA code.
We are going to discretize the [advection diffusion equation](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation) without reaction terms:
%% Cell type:code id: tags:
``` python
domain_size = (200, 80)
dim = len(domain_size)
# create arrays
c_arr = np.zeros(domain_size)
v_arr = np.zeros(domain_size + (dim,))
# create fields
c, v, c_next = ps.fields("c, v(2), c_next: [2d]", c=c_arr, v=v_arr, c_next=c_arr)
# write down advection diffusion pde
# the equation is represented by a single term and an implicit "=0" is assumed.
adv_diff_pde = ps.fd.transient(c) - ps.fd.diffusion(c, sp.Symbol("D")) + ps.fd.advection(c, v)
adv_diff_pde
```
%% Output
$\displaystyle \nabla \cdot(v c) - div(D \nabla c) + \partial_t c_{C}$
Advection(c_C, v_C__0) - Diffusion(c_C, D) + Transient(c_C)
%% Cell type:markdown id: tags:
It describes how the concentration $c$ of a passive substance, which does not influence the flow field, behaves in a fluid with given velocity $v$.
To illustrate the two effects here, image we release a constant stream of dye at some point in a river. The dye is transported with the flow (advection) but also the trace gets wider and wider normal to the flow direction due to diffusion.
%% Cell type:code id: tags:
``` python
discretize = ps.fd.Discretization2ndOrder(1, 0.01)
discretization = discretize(adv_diff_pde)
discretization.subs(sp.Symbol("D"),1)
```
%% Output
!
$\displaystyle 0.005 {{c}_{(-1,0)}} {{v}_{(-1,0)}} + 0.01 {{c}_{(-1,0)}} + 0.005 {{c}_{(0,-1)}} {{v}_{(0,-1)}^{1}} + 0.01 {{c}_{(0,-1)}} + 0.96 {{c}_{(0,0)}} - 0.005 {{c}_{(0,1)}} {{v}_{(0,1)}^{1}} + 0.01 {{c}_{(0,1)}} - 0.005 {{c}_{(1,0)}} {{v}_{(1,0)}} + 0.01 {{c}_{(1,0)}}$
0.005⋅c_W⋅v_W__0 + 0.01⋅c_W + 0.005⋅c_S⋅v_S__1 + 0.01⋅c_S + 0.96⋅c_C - 0.005⋅c
_N⋅v_N__1 + 0.01⋅c_N - 0.005⋅c_E⋅v_E__0 + 0.01⋅c_E
%% Cell type:code id: tags:
``` python
ast = ps.create_kernel([ps.Assignment(c_next.center(), discretization.subs(sp.Symbol("D"), 1))])
kernel = ast.compile()
```
%% Cell type:code id: tags:
``` python
y = np.linspace(0, 1, v_arr.shape[1])
v_arr[:, :, 0] = -y * (y - 1.0) * 5
plt.vector_field(v_arr);
```
%% Output