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SummationByParts
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working??

This commit is contained in:
Magnus Ulimoen 2020-01-26 15:46:54 +01:00
parent 4673fbfbf7
commit b700aeae94
3 changed files with 212 additions and 86 deletions
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23
main.js
View File

@ -8,6 +8,7 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
const wasm = await init("./maxwell_bg.wasm"); const wasm = await init("./maxwell_bg.wasm");
setPanicHook(); setPanicHook();
const DIAMOND = false; const DIAMOND = false;
const UPWIND = true;
const canvas = document.getElementById("glCanvas"); const canvas = document.getElementById("glCanvas");
@ -103,7 +104,7 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
for (let j = 0; j < height; j += 1) { for (let j = 0; j < height; j += 1) {
for (let i = 0; i < width; i += 1) { for (let i = 0; i < width; i += 1) {
const n = width*j + i; const n = width*j + i;
x[n] = 10.0*(i / (width - 1.0) - 0.5); x[n] = 20.0*(i / (width - 1.0));
y[n] = 20.0*(j / (height - 1.0)); y[n] = 20.0*(j / (height - 1.0));
@ -201,7 +202,7 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
}; };
chosenField.cycle(); chosenField.cycle();
universe.init(0, 10); universe.init(10, 10);
/** /**
* Integrates and draws the next iteration * Integrates and draws the next iteration
@ -237,8 +238,8 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
}; };
const field = new Float32Array(wasm.memory.buffer, fieldPtr, width*height); const field = new Float32Array(wasm.memory.buffer, fieldPtr, width*height);
gl.bufferData(gl.ARRAY_BUFFER, field, gl.DYNAMIC_DRAW); gl.bufferData(gl.ARRAY_BUFFER, field, gl.DYNAMIC_DRAW);
console.log(field.reduce((min, v) => v < min ? v : min)); // console.log(field.reduce((min, v) => v < min ? v : min));
console.log(field.reduce((max, v) => v > max ? v : max)); // console.log(field.reduce((max, v) => v > max ? v : max));
{ {
const offset = 0; const offset = 0;
@ -247,8 +248,13 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
gl.drawElements(gl.TRIANGLES, vertexCount, type, offset); gl.drawElements(gl.TRIANGLES, vertexCount, type, offset);
} }
if (UPWIND) {
universe.advance_upwind(dt/2);
universe.advance_upwind(dt/2);
} else {
universe.advance(dt/2); universe.advance(dt/2);
universe.advance(dt/2); universe.advance(dt/2);
}
window.requestAnimationFrame(drawMe); window.requestAnimationFrame(drawMe);
} }
@ -271,7 +277,14 @@ import { EulerUniverse, Universe, default as init, set_panic_hook as setPanicHoo
// Must adjust for bbox and transformations for x/y // Must adjust for bbox and transformations for x/y
const mousex = event.clientX / window.innerWidth; const mousex = event.clientX / window.innerWidth;
const mousey = event.clientY / window.innerHeight; const mousey = event.clientY / window.innerHeight;
universe.init(10*(mousex-0.5), 20.0*(1.0 - mousey));
const normx = mousex;
const normy = 1.0 - mousey;
universe.init(
(bbox[1] - bbox[0])*normx + bbox[0],
(bbox[3] - bbox[2])*normy + bbox[2],
);
}, {"passive": true}); }, {"passive": true});
resizeCanvas(); resizeCanvas();

225
src/euler.rs
View File

@ -121,7 +121,6 @@ impl Field {
} }
} }
#[allow(unused)]
pub(crate) fn advance_upwind<UO>( pub(crate) fn advance_upwind<UO>(
prev: &Field, prev: &Field,
fut: &mut Field, fut: &mut Field,
@ -176,12 +175,6 @@ pub(crate) fn advance_upwind<UO>(
.apply(|y1, &y0, &k1, &k2, &k3, &k4| *y1 = y0 + dt / 6.0 * (k1 + 2.0 * k2 + 2.0 * k3 + k4)); .apply(|y1, &y0, &k1, &k2, &k3, &k4| *y1 = y0 + dt / 6.0 * (k1 + 2.0 * k2 + 2.0 * k3 + k4));
} }
/// Solving (Au)_x + (Bu)_y
/// with:
/// A B
/// [ 0, 0, 0] [ 0, 1, 0]
/// [ 0, 0, -1] [ 1, 0, 0]
/// [ 0, -1, 0] [ 0, 0, 0]
pub(crate) fn advance<SBP>( pub(crate) fn advance<SBP>(
prev: &Field, prev: &Field,
fut: &mut Field, fut: &mut Field,
@ -241,42 +234,32 @@ fn pressure(gamma: f32, rho: f32, rhou: f32, rhov: f32, e: f32) -> f32 {
} }
#[allow(non_snake_case)] #[allow(non_snake_case)]
/// This flux is rotated by the grid metrics
/// (Au)_x + (Bu)_y = 1/J [
/// (J xi_x Au)_xi + (J eta_x Au)_eta
/// (J xi_y Bu)_xi + (J eta_y Bu)_eta
/// ]
/// where J is the grid determinant
///
/// This is used both in fluxes and SAT terms
fn RHS<SBP: SbpOperator>( fn RHS<SBP: SbpOperator>(
k: &mut Field, k: &mut Field,
y: &Field, y: &Field,
grid: &Grid<SBP>, grid: &Grid<SBP>,
boundaries: &BoundaryTerms, boundaries: &BoundaryTerms,
tmp: &mut (Field, Field, Field, Field), tmp: &mut (Field, Field, Field, Field, Field, Field),
) { ) {
let ehat = &mut tmp.0; let ehat = &mut tmp.0;
let fhat = &mut tmp.1; let fhat = &mut tmp.1;
fluxes([ehat, fhat], y, grid); fluxes((ehat, fhat), y, grid);
let de = &mut tmp.2; let dE = &mut tmp.2;
let df = &mut tmp.3; let dF = &mut tmp.3;
SBP::diffxi(ehat.rho(), de.rho_mut()); SBP::diffxi(ehat.rho(), dE.rho_mut());
SBP::diffxi(ehat.rhou(), de.rhou_mut()); SBP::diffxi(ehat.rhou(), dE.rhou_mut());
SBP::diffxi(ehat.rhov(), de.rhov_mut()); SBP::diffxi(ehat.rhov(), dE.rhov_mut());
SBP::diffxi(ehat.e(), de.e_mut()); SBP::diffxi(ehat.e(), dE.e_mut());
SBP::diffeta(fhat.rho(), df.rho_mut()); SBP::diffeta(fhat.rho(), dF.rho_mut());
SBP::diffeta(fhat.rhou(), df.rhou_mut()); SBP::diffeta(fhat.rhou(), dF.rhou_mut());
SBP::diffeta(fhat.rhov(), df.rhov_mut()); SBP::diffeta(fhat.rhov(), dF.rhov_mut());
SBP::diffeta(fhat.e(), df.e_mut()); SBP::diffeta(fhat.e(), dF.e_mut());
// And dissipation... azip!((out in &mut k.0,
eflux in &dE.0,
ndarray::azip!((out in &mut k.0, fflux in &dF.0,
eflux in &de.0,
fflux in &df.0,
detj in &grid.detj.broadcast((4, y.ny(), y.nx())).unwrap()) { detj in &grid.detj.broadcast((4, y.ny(), y.nx())).unwrap()) {
*out = (-eflux - fflux)/detj *out = (-eflux - fflux)/detj
}); });
@ -285,26 +268,100 @@ fn RHS<SBP: SbpOperator>(
} }
#[allow(non_snake_case)] #[allow(non_snake_case)]
#[allow(unused)]
fn RHS_upwind<UO: UpwindOperator>( fn RHS_upwind<UO: UpwindOperator>(
k: &mut Field, k: &mut Field,
y: &Field, y: &Field,
grid: &Grid<UO>, grid: &Grid<UO>,
boundaries: &BoundaryTerms, boundaries: &BoundaryTerms,
tmp: &mut (Field, Field, Field, Field), tmp: &mut (Field, Field, Field, Field, Field, Field),
) { ) {
// fluxes(k, y, grid, tmp); let ehat = &mut tmp.0;
// dissipation(k, y, grid, tmp); let fhat = &mut tmp.1;
fluxes((ehat, fhat), y, grid);
let dE = &mut tmp.2;
let dF = &mut tmp.3;
UO::diffxi(ehat.rho(), dE.rho_mut());
UO::diffxi(ehat.rhou(), dE.rhou_mut());
UO::diffxi(ehat.rhov(), dE.rhov_mut());
UO::diffxi(ehat.e(), dE.e_mut());
UO::diffeta(fhat.rho(), dF.rho_mut());
UO::diffeta(fhat.rhou(), dF.rhou_mut());
UO::diffeta(fhat.rhov(), dF.rhov_mut());
UO::diffeta(fhat.e(), dF.e_mut());
let ad_xi = &mut tmp.4;
let ad_eta = &mut tmp.5;
upwind_dissipation((ad_xi, ad_eta), y, grid, (&mut tmp.0, &mut tmp.1));
azip!((out in &mut k.0,
eflux in &dE.0,
fflux in &dF.0,
ad_xi in &ad_xi.0,
ad_eta in &ad_eta.0,
detj in &grid.detj.broadcast((4, y.ny(), y.nx())).unwrap()) {
*out = (-eflux - fflux + ad_xi + ad_eta)/detj
});
SAT_characteristics(k, y, grid, boundaries); SAT_characteristics(k, y, grid, boundaries);
azip!((k in &mut k.0,
&detj in &grid.detj.broadcast((3, y.ny(), y.nx())).unwrap()) {
*k /= detj;
});
} }
fn fluxes<SBP: SbpOperator>(k: [&mut Field; 2], y: &Field, grid: &Grid<SBP>) { fn upwind_dissipation<UO: UpwindOperator>(
k: (&mut Field, &mut Field),
y: &Field,
grid: &Grid<UO>,
tmp: (&mut Field, &mut Field),
) {
for j in 0..y.ny() {
for i in 0..y.nx() {
let rho = y[(0, j, i)];
assert!(rho > 0.0);
let rhou = y[(1, j, i)];
let rhov = y[(2, j, i)];
let e = y[(3, j, i)];
let u = rhou / rho;
let v = rhov / rho;
let uhat = grid.detj_dxi_dx[(j, i)] / grid.detj[(j, i)] * u
+ grid.detj_dxi_dy[(j, i)] / grid.detj[(j, i)] * v;
let vhat = grid.detj_deta_dx[(j, i)] / grid.detj[(j, i)] * u
+ grid.detj_deta_dy[(j, i)] / grid.detj[(j, i)] * v;
let p = pressure(GAMMA, rho, rhou, rhov, e);
assert!(p > 0.0);
let c = (GAMMA * p / rho).sqrt();
let alpha_u = uhat.abs() + c;
let alpha_v = vhat.abs() + c;
tmp.0[(0, j, i)] = alpha_u * rho * grid.detj[(j, i)];
tmp.1[(0, j, i)] = alpha_v * rho * grid.detj[(j, i)];
tmp.0[(1, j, i)] = alpha_u * rhou * grid.detj[(j, i)];
tmp.1[(1, j, i)] = alpha_v * rhou * grid.detj[(j, i)];
tmp.0[(2, j, i)] = alpha_u * rhov * grid.detj[(j, i)];
tmp.1[(2, j, i)] = alpha_v * rhov * grid.detj[(j, i)];
tmp.0[(3, j, i)] = alpha_u * e * grid.detj[(j, i)];
tmp.1[(3, j, i)] = alpha_v * e * grid.detj[(j, i)];
}
}
UO::dissxi(tmp.0.rho(), k.0.rho_mut());
UO::dissxi(tmp.0.rhou(), k.0.rhou_mut());
UO::dissxi(tmp.0.rhov(), k.0.rhov_mut());
UO::dissxi(tmp.0.e(), k.0.e_mut());
UO::disseta(tmp.1.rho(), k.1.rho_mut());
UO::disseta(tmp.1.rhou(), k.1.rhou_mut());
UO::disseta(tmp.1.rhov(), k.1.rhov_mut());
UO::disseta(tmp.1.e(), k.1.e_mut());
}
fn fluxes<SBP: SbpOperator>(k: (&mut Field, &mut Field), y: &Field, grid: &Grid<SBP>) {
let j_dxi_dx = grid.detj_dxi_dx.view(); let j_dxi_dx = grid.detj_dxi_dx.view();
let j_dxi_dy = grid.detj_dxi_dy.view(); let j_dxi_dy = grid.detj_dxi_dy.view();
let j_deta_dx = grid.detj_deta_dx.view(); let j_deta_dx = grid.detj_deta_dx.view();
@ -318,11 +375,14 @@ fn fluxes<SBP: SbpOperator>(k: [&mut Field; 2], y: &Field, grid: &Grid<SBP>) {
for j in 0..y.ny() { for j in 0..y.ny() {
for i in 0..y.nx() { for i in 0..y.nx() {
let rho = rho[(j, i)]; let rho = rho[(j, i)];
assert!(rho > 0.0);
let rhou = rhou[(j, i)]; let rhou = rhou[(j, i)];
let rhov = rhov[(j, i)]; let rhov = rhov[(j, i)];
let e = e[(j, i)]; let e = e[(j, i)];
let p = pressure(GAMMA, rho, rhou, rhov, e); let p = pressure(GAMMA, rho, rhou, rhov, e);
assert!(p > 0.0);
let ef = [ let ef = [
rhou, rhou,
rhou * rhou / rho + p, rhou * rhou / rho + p,
@ -340,23 +400,13 @@ fn fluxes<SBP: SbpOperator>(k: [&mut Field; 2], y: &Field, grid: &Grid<SBP>) {
let eflux = j_dxi_dx[(j, i)] * ef[comp] + j_dxi_dy[(j, i)] * ff[comp]; let eflux = j_dxi_dx[(j, i)] * ef[comp] + j_dxi_dy[(j, i)] * ff[comp];
let fflux = j_deta_dx[(j, i)] * ef[comp] + j_deta_dy[(j, i)] * ff[comp]; let fflux = j_deta_dx[(j, i)] * ef[comp] + j_deta_dy[(j, i)] * ff[comp];
k[0][(comp, j, i)] = eflux; k.0[(comp, j, i)] = eflux;
k[1][(comp, j, i)] = fflux; k.1[(comp, j, i)] = fflux;
} }
} }
} }
} }
#[allow(unused)]
fn dissipation<UO: UpwindOperator>(
k: &mut Field,
y: &Field,
grid: &Grid<UO>,
tmp: &mut (Array2<f32>, Array2<f32>, Array2<f32>, Array2<f32>),
) {
todo!()
}
#[derive(Clone, Debug)] #[derive(Clone, Debug)]
pub enum Boundary { pub enum Boundary {
This, This,
@ -378,6 +428,16 @@ fn SAT_characteristics<SBP: SbpOperator>(
grid: &Grid<SBP>, grid: &Grid<SBP>,
_boundaries: &BoundaryTerms, _boundaries: &BoundaryTerms,
) { ) {
/* // Whean using infinite boundaries, use this...
let steady_v = [1.0, 1.0, 0.0, {
let M = 0.1;
let p_inf = 1.0 / (GAMMA * M * M);
p_inf / (GAMMA - 1.0) + 0.5
}];
let steady_a = ndarray::Array1::from(steady_v.to_vec());
let steady = steady_a.broadcast((k.nx(), 4)).unwrap().reversed_axes();
assert_eq!(steady.shape(), [4, k.nx()]);
*/
// North boundary // North boundary
{ {
let hi = (k.ny() - 1) as f32 * SBP::h()[0]; let hi = (k.ny() - 1) as f32 * SBP::h()[0];
@ -388,6 +448,7 @@ fn SAT_characteristics<SBP: SbpOperator>(
k.north_mut(), k.north_mut(),
y.north(), y.north(),
y.south(), // Self South y.south(), // Self South
//steady.view(),
hi, hi,
sign, sign,
tau, tau,
@ -406,6 +467,7 @@ fn SAT_characteristics<SBP: SbpOperator>(
k.south_mut(), k.south_mut(),
y.south(), y.south(),
y.north(), // Self North y.north(), // Self North
//steady.view(),
hi, hi,
sign, sign,
tau, tau,
@ -414,17 +476,28 @@ fn SAT_characteristics<SBP: SbpOperator>(
grid.detj_deta_dy.slice(slice), grid.detj_deta_dy.slice(slice),
); );
} }
/*let steady = ndarray::Array2::from_shape_fn((4, k.ny()), |(k, _)| match k {
0 => 1.0,
1 => 1.0,
2 => 0.0,
3 => {
let M = 0.1;
let p_inf = 1.0 / (GAMMA * M * M);
p_inf / (GAMMA - 1.0) + 0.5
}
_ => unreachable!(),
});*/
// West Boundary // West Boundary
{ {
let hi = (k.nx() - 1) as f32 * SBP::h()[0]; let hi = (k.nx() - 1) as f32 * SBP::h()[0];
let sign = 1.0; let sign = 1.0;
let tau = -1.0; let tau = -1.0;
let slice = s![.., 0]; let slice = s![.., 0];
println!("{:?}", slice);
SAT_characteristic( SAT_characteristic(
k.west_mut(), k.west_mut(),
y.west(), y.west(),
y.east(), // Self East y.east(), // Self East
//steady.view(),
hi, hi,
sign, sign,
tau, tau,
@ -443,6 +516,7 @@ fn SAT_characteristics<SBP: SbpOperator>(
k.east_mut(), k.east_mut(),
y.east(), y.east(),
y.west(), // Self West y.west(), // Self West
//steady.view(),
hi, hi,
sign, sign,
tau, tau,
@ -472,14 +546,21 @@ fn SAT_characteristic(
assert_eq!(y.shape()[0], 4); assert_eq!(y.shape()[0], 4);
assert_eq!(y.shape()[1], detj.shape()[0]); assert_eq!(y.shape()[1], detj.shape()[0]);
for i in 0..z.shape()[1] { for (((((mut k, y), z), detj), detj_d_dx), detj_d_dy) in k
let rho = y[(0, i)]; .axis_iter_mut(ndarray::Axis(1))
let rhou = y[(1, i)]; .zip(y.axis_iter(ndarray::Axis(1)))
let rhov = y[(2, i)]; .zip(z.axis_iter(ndarray::Axis(1)))
let e = y[(3, i)]; .zip(detj.iter())
.zip(detj_d_dx.iter())
.zip(detj_d_dy.iter())
{
let rho = y[0];
let rhou = y[1];
let rhov = y[2];
let e = y[3];
let kx_ = detj_d_dx[i] / detj[i]; let kx_ = detj_d_dx / detj;
let ky_ = detj_d_dy[i] / detj[i]; let ky_ = detj_d_dy / detj;
let (kx, ky) = { let (kx, ky) = {
let r = f32::hypot(kx_, ky_); let r = f32::hypot(kx_, ky_);
@ -538,12 +619,7 @@ fn SAT_characteristic(
], ],
]; ];
let res = [ let res = [rho - z[0], rhou - z[1], rhov - z[2], e - z[3]];
rho - z[(0, i)],
rhou - z[(1, i)],
rhov - z[(2, i)],
e - z[(3, i)],
];
let mut TIres = [0.0; 4]; let mut TIres = [0.0; 4];
for row in 0..4 { for row in 0..4 {
for col in 0..4 { for col in 0..4 {
@ -566,7 +642,7 @@ fn SAT_characteristic(
} }
for comp in 0..4 { for comp in 0..4 {
k[(comp, i)] += hi * tau * TLTIres[comp]; k[comp] += hi * tau * TLTIres[comp];
} }
} }
} }
@ -574,7 +650,7 @@ fn SAT_characteristic(
pub struct WorkBuffers { pub struct WorkBuffers {
y: Field, y: Field,
buf: [Field; 4], buf: [Field; 4],
tmp: (Field, Field, Field, Field), tmp: (Field, Field, Field, Field, Field, Field),
} }
impl WorkBuffers { impl WorkBuffers {
@ -583,7 +659,14 @@ impl WorkBuffers {
Self { Self {
y: arr3.clone(), y: arr3.clone(),
buf: [arr3.clone(), arr3.clone(), arr3.clone(), arr3.clone()], buf: [arr3.clone(), arr3.clone(), arr3.clone(), arr3.clone()],
tmp: (arr3.clone(), arr3.clone(), arr3.clone(), arr3), tmp: (
arr3.clone(),
arr3.clone(),
arr3.clone(),
arr3.clone(),
arr3.clone(),
arr3,
),
} }
} }
} }

46
src/lib.rs
View File

@ -147,10 +147,10 @@ impl EulerUniverse {
pub fn init(&mut self, x0: f32, y0: f32) { pub fn init(&mut self, x0: f32, y0: f32) {
// Should parametrise such that we have radius, drop in pressure at center, etc // Should parametrise such that we have radius, drop in pressure at center, etc
let rstar = 0.5; let rstar = 1.0;
let eps = 1.0; let eps = 3.0;
#[allow(non_snake_case)] #[allow(non_snake_case)]
let M = 0.1; let M = 0.5;
let p_inf = 1.0 / (euler::GAMMA * M * M); let p_inf = 1.0 / (euler::GAMMA * M * M);
let t = 0.0; let t = 0.0;
@ -164,7 +164,7 @@ impl EulerUniverse {
let y = self.0.grid.y[(j, i)]; let y = self.0.grid.y[(j, i)];
let dx = (x - x0) - t; let dx = (x - x0) - t;
let dy = (y - y0) - t; let dy = y - y0;
let f = (1.0 - (dx * dx + dy * dy)) / (rstar * rstar); let f = (1.0 - (dx * dx + dy * dy)) / (rstar * rstar);
use euler::GAMMA; use euler::GAMMA;
@ -180,7 +180,7 @@ impl EulerUniverse {
1.0 / (GAMMA - 1.0), 1.0 / (GAMMA - 1.0),
); );
assert!(rho > 0.0); assert!(rho > 0.0);
let p = rho.powf(GAMMA) * p_inf; let p = p_inf * rho.powf(GAMMA);
assert!(p > 0.0); assert!(p > 0.0);
let e = p / (GAMMA - 1.0) + rho * (u * u + v * v) / 2.0; let e = p / (GAMMA - 1.0) + rho * (u * u + v * v) / 2.0;
assert!(e > 0.0); assert!(e > 0.0);
@ -197,8 +197,8 @@ impl EulerUniverse {
self.0.advance(dt) self.0.advance(dt)
} }
pub fn advance_upwind(&mut self, _dt: f32) { pub fn advance_upwind(&mut self, dt: f32) {
todo!() self.0.advance_upwind(dt)
} }
pub fn get_rho_ptr(&self) -> *const u8 { pub fn get_rho_ptr(&self) -> *const u8 {
@ -241,11 +241,41 @@ impl<SBP: operators::SbpOperator> EulerSystem<SBP> {
} }
} }
impl<SBP: operators::UpwindOperator> EulerSystem<SBP> {
pub fn advance_upwind(&mut self, dt: f32) {
euler::advance_upwind(
&self.sys.0,
&mut self.sys.1,
dt,
&self.grid,
Some(&mut self.wb),
);
std::mem::swap(&mut self.sys.0, &mut self.sys.1);
}
}
#[test] #[test]
fn start_and_advance_euler() { fn start_and_advance_euler() {
let x = ndarray::Array2::from_shape_fn((20, 20), |(_j, i)| {
5.0 * 2.0 * ((i as f32 / (20 - 1) as f32) - 0.5)
});
let y = ndarray::Array2::from_shape_fn((20, 20), |(j, _i)| {
5.0 * 2.0 * ((j as f32 / (20 - 1) as f32) - 0.5)
});
let mut universe = EulerUniverse::new(x, y);
universe.init(-1.0, 0.0);
for _ in 0..50 {
universe.advance(0.01);
}
}
#[test]
fn start_and_advance_upwind_euler() {
let x = ndarray::Array2::from_shape_fn((20, 10), |(_j, i)| i as f32 / (10 - 1) as f32); let x = ndarray::Array2::from_shape_fn((20, 10), |(_j, i)| i as f32 / (10 - 1) as f32);
let y = ndarray::Array2::from_shape_fn((20, 10), |(j, _i)| j as f32 / (20 - 1) as f32); let y = ndarray::Array2::from_shape_fn((20, 10), |(j, _i)| j as f32 / (20 - 1) as f32);
let mut universe = EulerUniverse::new(x, y); let mut universe = EulerUniverse::new(x, y);
universe.init(0.5, 0.5); universe.init(0.5, 0.5);
universe.advance(0.01); for _ in 0..50 {
universe.advance_upwind(0.01);
}
} }