Toshiki Teramura

Posted on

stencil; abstract stencil calculation

I am developing a library for stencil calculation in Rust.

https://github.com/termoshtt/stencil
Abstract stencil calculation

stencil v0.2 is published on crates.io, and you can use it through cargo:

``````[dependencies]
stencil = "0.2"
``````

The full document is available on docs.rs.

StencilArray trait

This library abstract a stencil calculation through `StencilArray` trait:

``````/// Array with stencil calculations
pub trait StencilArray<St>: NdArray
where
St: Stencil<Elem = Self::Elem, Dim = Self::Dim>,
{
/// Execute a stencil calculation
fn stencil_map<Output, Func>(&self, out: &mut Output, Func)
where
Output: NdArray<Dim = Self::Dim>,
Func: Fn(St) -> Output::Elem;
}
``````

`stencil_map` function is the core of this crate. Basic example is here:

``````let mut a = torus::Torus::<f64, Ix1>::zeros(n);
let mut b = a.clone();
a.coordinate_fill(|x| x.sin());
let dx = a.dx();
a.stencil_map(&mut b, |n: N1D1<f64>| (n.r - n.l) / (2.0 * dx));
``````

This crate defines `Torus<A, D>`, which represents N-dimensional torus. `a` denotes one-dimensional torus, we can regard it as a region with periodic boundary condition. The second argument of `stencil_map`, a lambda function `|n: N1D1<f64>| (n.r - n.l) / (2.0 * dx)` denotes a central difference. `N1D1` means 1-neighbor on 1-dimensional space. So, this example describes a central difference of `sin(x)` function, i.e. `cos(x)` is set `b`.

`N1D1<A>` is a small struct:

``````/// one-neighbor, one-dimensional stencil
#[derive(Clone, Copy)]
pub struct N1D1<A: LinalgScalar> {
/// left
pub l: A,
/// right
pub r: A,
/// center
pub c: A,
}

``````

and implements the trait `Stencil`. It has values around "what we focused on"; when we update `b[i]` using `a`, `n` has `a[i]` as `n.c` and its neighbors `a[i-1]` as `n.l` and `a[i+1]` as `n.r`. This crate abstracts the stencil calculation in this way.
The implementation of `StencilArray` for the `Torus<A, D>` describes how to get `N1D1` from the array `a`. Currently, there are three `Stencil`s: `N1D1<A>`, `N2D1<A>`, and `N1D2<A>`, and two `StencilArray`s: `Torus<A, D>` and `Line<A, P: Padding, E: Edge>`. The `Line` has a padding at the each edge of line, and it can be regarded as a fixed boundary condition.

Manifold trait

stencil crate introduces one more useful trait, `Manifold`:

``````/// Uniformly coordinated array
pub trait Manifold: NdArray {
/// Type of coordinate
type Coordinate;

/// Increment of coordinate
fn dx(&self) -> Self::Coordinate;

/// Fill manifold by a function
fn coordinate_fill<F>(&mut self, F)
where
F: Fn(Self::Coordinate) -> Self::Elem;

/// Map values on manifold using a function
fn coordinate_map<F>(&mut self, F)
where
F: Fn(Self::Coordinate, Self::Elem) -> Self::Elem;
}
``````

This trait gives array a coordinate. This is useful to set value of array by mathematical function as you see above example

``````a.coordinate_fill(|x| x.sin());
``````

`x` is set by the implementation of `Manifold` for each objects.

Example1: Diffusion equation

Let us solve the diffusion equation for both periodic and fixed boundary conditions:

``````fn periodic(r: Sender<Vec<f64>>) {
let dt = 1e-3;
let mut t = torus::Torus::<f64, Ix1>::zeros(32);
t.coordinate_fill(|x| x.sin());
let dx = t.dx();
let mut s = t.clone();
for _step in 0..3000 {
t.stencil_map(&mut s, |n: N1D1<f64>| {
let d2 = (n.l + n.r - 2.0 * n.c) / (dx.powi(2));
n.c + dt * d2
});
r.send(s.as_view().to_vec()).unwrap();
swap(&mut t, &mut s);
}
}
``````

``````fn fixed(r: Sender<Vec<f64>>) {
let dt = 1e-3;
let mut t = Line::<f64, P1, Closed>::new(64, 0.0, 1.0, 2.5 * PI);
t.coordinate_fill(|x| x.sin());
let dx = t.dx();
let mut s = t.clone();
for _step in 0..10000 {
t.stencil_map(&mut s, |n: N1D1<f64>| {
let d2 = (n.l + n.r - 2.0 * n.c) / (dx.powi(2));
n.c + dt * d2
});
r.send(s.as_view().to_vec()).unwrap();
swap(&mut t, &mut s);
}
}

``````

This example uses asink for storing data. Please see the last post.

Example2: KdV equation

At last, we solve KdV equation using the ZK-scheme:

``````fn kdv(r: Sender<Vec<f64>>) {
let dt = 1e-5;
let mut t = torus::Torus::<f64, Ix1>::zeros(128);
t.coordinate_fill(|x| x.sin());
let dx = t.dx();
let mut b = t.clone();
let mut n = t.clone();
for step in 0..1_000_000 {
t.stencil_map(&mut n, |n: N2D1<f64>| {
// ZK scheme
let uux = (n.l + n.c + n.r) * (n.r - n.l) / (3.0 * dx);
let u3 = (n.rr - 2.0 * n.r + 2.0 * n.l - n.ll) / dx.powi(3);
uux + 0.01 * u3
});
azip!(mut n, b in { *n = b - dt * *n });
if step % 1000 == 0 {
r.send(n.as_view().to_vec()).unwrap();
}
swap(&mut t, &mut b);
swap(&mut t, &mut n);
}
}
``````

Leap-frog algorithm can also be written easily.

Please see the examples/ directory for full examples.
I'd like to get your feedback. Thanks.