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stencil; abstract stencil calculation

termoshtt profile image Toshiki Teramura ・4 min read

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 Stencils: N1D1<A>, N2D1<A>, and N1D2<A>, and two StencilArrays: 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);
    }
}

heat-periodic

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);
    }
}

heat-fixed

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);
    }
}

kdv

Leap-frog algorithm can also be written easily.

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

Posted on by:

termoshtt profile

Toshiki Teramura

@termoshtt

Ph.D. in Turbulence, Dynamical Systems / Data-Assimilation / Scientific-Rust

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