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Move integrate to separate crate

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Magnus Ulimoen
2021-03-22 17:49:35 +01:00
parent be1330ec02
commit 7aadda3de9
16 changed files with 187 additions and 118 deletions
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utils/integrate/src/lib.rs Normal file
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//! Integration of explicit PDEs using different Butcher Tableaus
//!
//! Integration can be performed on all systems that can be represented
//! as using a transform into an [`ndarray::ArrayView`] for both the state
//! and the state difference.
//!
//! The integration functions are memory efficient, and relies
//! on the `k` parameter to hold the system state differences.
//! This parameter is tied to the Butcher Tableau
use float::Float;
/// The Butcher Tableau, with the state transitions described as
/// [on wikipedia](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Explicit_Runge%E2%80%93Kutta_methods).
pub trait ButcherTableau {
/// This bound should not be overridden
const S: usize = Self::B.len();
/// Only the lower triangle will be used (explicit integration)
const A: &'static [&'static [Float]];
const B: &'static [Float];
const C: &'static [Float];
}
pub trait EmbeddedButcherTableau: ButcherTableau {
const BSTAR: &'static [Float];
}
pub struct Rk4;
impl ButcherTableau for Rk4 {
const A: &'static [&'static [Float]] = &[&[0.5], &[0.0, 0.5], &[0.0, 0.0, 1.0]];
const B: &'static [Float] = &[1.0 / 6.0, 1.0 / 3.0, 1.0 / 3.0, 1.0 / 6.0];
const C: &'static [Float] = &[0.5, 0.5, 1.0];
}
pub struct Rk4_38;
impl ButcherTableau for Rk4_38 {
const A: &'static [&'static [Float]] = &[&[1.0 / 3.0], &[-1.0 / 3.0, 1.0], &[1.0, -1.0, 1.0]];
const B: &'static [Float] = &[1.0 / 8.0, 3.0 / 8.0, 3.0 / 8.0, 1.0 / 8.0];
const C: &'static [Float] = &[1.0 / 3.0, 2.0 / 3.0, 1.0];
}
pub struct EulerMethod;
impl ButcherTableau for EulerMethod {
const A: &'static [&'static [Float]] = &[&[]];
const B: &'static [Float] = &[1.0];
const C: &'static [Float] = &[];
}
pub struct MidpointMethod;
impl ButcherTableau for MidpointMethod {
const A: &'static [&'static [Float]] = &[&[0.5]];
const B: &'static [Float] = &[0.0, 1.0];
const C: &'static [Float] = &[1.0 / 2.0];
}
/// Bit excessive...
#[allow(clippy::excessive_precision)]
#[allow(clippy::unreadable_literal)]
const SQRT_5: Float = 2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275;
pub struct Rk6;
impl ButcherTableau for Rk6 {
const A: &'static [&'static [Float]] = &[
&[4.0 / 7.0],
&[115.0 / 112.0, -5.0 / 16.0],
&[589.0 / 630.0, 5.0 / 18.0, -16.0 / 45.0],
&[
229.0 / 1200.0 - 29.0 / 6000.0 * SQRT_5,
119.0 / 240.0 - 187.0 / 1200.0 * SQRT_5,
-14.0 / 75.0 + 34.0 / 375.0 * SQRT_5,
-3.0 / 100.0 * SQRT_5,
],
&[
71.0 / 2400.0 - 587.0 / 12000.0 * SQRT_5,
187.0 / 480.0 - 391.0 / 2400.0 * SQRT_5,
-38.0 / 75.0 + 26.0 / 375.0 * SQRT_5,
27.0 / 80.0 - 3.0 / 400.0 * SQRT_5,
(1.0 + SQRT_5) / 4.0,
],
&[
-49.0 / 480.0 + 43.0 / 160.0 * SQRT_5,
-425.0 / 96.0 + 51.0 / 32.0 * SQRT_5,
52.0 / 15.0 - 4.0 / 5.0 * SQRT_5,
-27.0 / 16.0 + 3.0 / 16.0 * SQRT_5,
5.0 / 4.0 - 3.0 / 4.0 * SQRT_5,
5.0 / 2.0 - 1.0 / 2.0 * SQRT_5,
],
];
const B: &'static [Float] = &[
1.0 / 12.0,
0.0,
0.0,
0.0,
5.0 / 12.0,
5.0 / 12.0,
1.0 / 12.0,
];
const C: &'static [Float] = &[
4.0 / 7.0,
5.0 / 7.0,
6.0 / 7.0,
(5.0 - SQRT_5) / 10.0,
(5.0 + SQRT_5) / 10.0,
1.0,
];
}
pub struct Fehlberg;
impl ButcherTableau for Fehlberg {
#[rustfmt::skip]
const A: &'static [&'static [Float]] = &[
&[1.0 / 4.0],
&[3.0 / 32.0, 9.0 / 32.0],
&[1932.0 / 2197.0, -7200.0 / 2197.0, 7296.0 / 2197.0],
&[439.0 / 216.0, -8.0, 3680.0 / 513.0, -845.0 / 4104.0],
&[-8.0 / 27.0, 2.0, -3544.0 / 2565.0, 1859.0 / 4104.0, -11.0 / 40.0],
];
#[rustfmt::skip]
const B: &'static [Float] = &[
16.0 / 135.0, 0.0, 6656.0 / 12825.0, 28561.0 / 56430.0, -9.0 / 50.0, 2.0 / 55.0,
];
const C: &'static [Float] = &[0.0, 1.0 / 4.0, 3.0 / 8.0, 12.0 / 13.0, 1.0, 1.0 / 2.0];
}
impl EmbeddedButcherTableau for Fehlberg {
const BSTAR: &'static [Float] = &[
25.0 / 216.0,
0.0,
1408.0 / 2565.0,
2197.0 / 4104.0,
-1.0 / 5.0,
0.0,
];
}
pub struct BogackiShampine;
impl ButcherTableau for BogackiShampine {
const A: &'static [&'static [Float]] = &[
&[1.0 / 2.0],
&[0.0, 3.0 / 4.0],
&[2.0 / 9.0, 1.0 / 3.0, 4.0 / 9.0],
];
const B: &'static [Float] = &[2.0 / 9.0, 1.0 / 3.0, 4.0 / 9.0, 0.0];
const C: &'static [Float] = &[0.0, 1.0 / 2.0, 3.0 / 4.0, 1.0];
}
impl EmbeddedButcherTableau for BogackiShampine {
const BSTAR: &'static [Float] = &[7.0 / 24.0, 1.0 / 4.0, 1.0 / 3.0, 1.0 / 8.0];
}
pub trait Integrable {
type State;
type Diff;
fn assign(s: &mut Self::State, o: &Self::State);
fn scaled_add(s: &mut Self::State, o: &Self::Diff, scale: Float);
}
#[allow(clippy::too_many_arguments)]
/// Integrates using the [`ButcherTableau`] specified. `rhs` should be the result
/// of the right hand side of $u_t = rhs$
///
/// rhs takes the old state and the current time, and outputs the state difference
/// in the first parameter
///
/// Should be called as
/// ```rust,ignore
/// integrate::<Rk4, System, _>(...)
/// ```
pub fn integrate<BTableau: ButcherTableau, F: Integrable, RHS>(
mut rhs: RHS,
prev: &F::State,
fut: &mut F::State,
time: &mut Float,
dt: Float,
k: &mut [F::Diff],
) where
RHS: FnMut(&mut F::Diff, &F::State, Float),
{
assert!(k.len() >= BTableau::S);
for i in 0.. {
let simtime;
match i {
0 => {
F::assign(fut, prev);
simtime = *time;
}
i if i < BTableau::S => {
F::assign(fut, prev);
for (&a, k) in BTableau::A[i - 1].iter().zip(k.iter()) {
if a == 0.0 {
continue;
}
F::scaled_add(fut, k, a * dt);
}
simtime = *time + dt * BTableau::C[i - 1];
}
_ if i == BTableau::S => {
F::assign(fut, prev);
for (&b, k) in BTableau::B.iter().zip(k.iter()) {
if b == 0.0 {
continue;
}
F::scaled_add(fut, k, b * dt);
}
*time += dt;
return;
}
_ => {
unreachable!();
}
};
rhs(&mut k[i], &fut, simtime);
}
}
#[allow(clippy::too_many_arguments)]
/// Integrate using an [`EmbeddedButcherTableau`], else similar to [`integrate`]
///
/// This produces two results, the most accurate result in `fut`, and the less accurate
/// result in `fut2`. This can be used for convergence testing and adaptive timesteps.
pub fn integrate_embedded_rk<BTableau: EmbeddedButcherTableau, F: Integrable, RHS>(
rhs: RHS,
prev: &F::State,
fut: &mut F::State,
fut2: &mut F::State,
time: &mut Float,
dt: Float,
k: &mut [F::Diff],
) where
RHS: FnMut(&mut F::Diff, &F::State, Float),
{
integrate::<BTableau, F, RHS>(rhs, prev, fut, time, dt, k);
F::assign(fut2, prev);
for (&b, k) in BTableau::BSTAR.iter().zip(k.iter()) {
if b == 0.0 {
continue;
}
F::scaled_add(fut2, k, b * dt);
}
}
#[cfg(feature = "rayon")]
#[allow(clippy::too_many_arguments)]
/// Integrates a multigrid problem, much the same as [`integrate`],
/// using a `rayon` threadpool for parallelisation.
///
/// note that `rhs` accepts the full system state, and is responsible
/// for computing the full state difference.
/// `rhs` can be a mutable closure, so buffers can be used
/// and mutated inside the closure.
///
/// This function requires the `rayon` feature, and is not callable in
/// a `wasm` context.
pub fn integrate_multigrid<BTableau: ButcherTableau, F: Integrable, RHS>(
mut rhs: RHS,
prev: &[F::State],
fut: &mut [F::State],
time: &mut Float,
dt: Float,
k: &mut [&mut [F::Diff]],
pool: &rayon::ThreadPool,
) where
RHS: FnMut(&mut [F::Diff], &[F::State], Float),
F::State: Send + Sync,
F::Diff: Send + Sync,
{
for i in 0.. {
let simtime;
match i {
0 => {
pool.scope(|s| {
assert!(k.len() >= BTableau::S);
for (prev, fut) in prev.iter().zip(fut.iter_mut()) {
s.spawn(move |_| {
F::assign(fut, prev);
});
}
});
simtime = *time;
}
i if i < BTableau::S => {
pool.scope(|s| {
for (ig, (prev, fut)) in prev.iter().zip(fut.iter_mut()).enumerate() {
let k = &k;
s.spawn(move |_| {
F::assign(fut, prev);
for (ik, &a) in BTableau::A[i - 1].iter().enumerate() {
if a == 0.0 {
continue;
}
F::scaled_add(fut, &k[ik][ig], a * dt);
}
});
}
});
simtime = *time + dt * BTableau::C[i - 1];
}
_ if i == BTableau::S => {
pool.scope(|s| {
for (ig, (prev, fut)) in prev.iter().zip(fut.iter_mut()).enumerate() {
let k = &k;
s.spawn(move |_| {
F::assign(fut, prev);
for (ik, &b) in BTableau::B.iter().enumerate() {
if b == 0.0 {
continue;
}
F::scaled_add(fut, &k[ik][ig], b * dt);
}
});
}
});
*time += dt;
return;
}
_ => {
unreachable!();
}
};
rhs(&mut k[i], &fut, simtime);
}
}
#[test]
/// Solving a second order PDE
fn ballistic() {
#[derive(Clone, Debug)]
struct Ball {
z: Float,
v: Float,
}
impl Integrable for Ball {
type State = Ball;
type Diff = (Float, Float);
fn assign(s: &mut Self::State, o: &Self::State) {
s.z = o.z;
s.v = o.v;
}
fn scaled_add(s: &mut Self::State, o: &Self::Diff, sc: Float) {
s.z += o.0 * sc;
s.v += o.1 * sc;
}
}
let mut t = 0.0;
let dt = 0.001;
let initial = Ball { z: 0.0, v: 10.0 };
let g = -9.81;
let mut k = [(0.0, 0.0); 4];
let gravity = |d: &mut (Float, Float), s: &Ball, _time: Float| {
d.1 = g;
d.0 = s.v
};
let mut next = initial.clone();
//while next.z >= 0.0 {
while t < 1.0 {
let mut next2 = next.clone();
integrate::<EulerMethod, Ball, _>(gravity, &next, &mut next2, &mut t, dt, &mut k);
std::mem::swap(&mut next, &mut next2);
}
let expected_vel = initial.v + g * t;
assert!((next.v - expected_vel).abs() < 1e-3);
let expected_pos = initial.z + initial.v * t + g / 2.0 * t.powi(2);
assert!((next.z - expected_pos).abs() < 1e-2);
}