SummationByParts/sbp/src/euler.rs

use super::grid::{Grid, Metrics};
use super::integrate;
use super::operators::{InterpolationOperator, SbpOperator2d, UpwindOperator2d};
use super::Float;
use ndarray::azip;
use ndarray::prelude::*;

pub const GAMMA: Float = 1.4;

// A collection of buffers that allows one to efficiently
// move to the next state
#[derive(Debug)]
pub struct System<SBP: SbpOperator2d> {
    sys: (Field, Field),
    k: [Field; 4],
    wb: WorkBuffers,
    grid: (Grid, Metrics),
    op: SBP,
}

impl<SBP: SbpOperator2d> System<SBP> {
    pub fn new(x: ndarray::Array2<Float>, y: ndarray::Array2<Float>, op: SBP) -> Self {
        let grid = Grid::new(x, y).expect(
            "Could not create grid. Different number of elements compared to width*height?",
        );
        let metrics = grid.metrics(&op).unwrap();
        let nx = grid.nx();
        let ny = grid.ny();
        Self {
            sys: (Field::new(ny, nx), Field::new(ny, nx)),
            grid: (grid, metrics),
            k: [
                Field::new(ny, nx),
                Field::new(ny, nx),
                Field::new(ny, nx),
                Field::new(ny, nx),
            ],
            wb: WorkBuffers::new(ny, nx),
            op,
        }
    }

    pub fn advance(&mut self, dt: Float) {
        let bc = BoundaryCharacteristics {
            north: BoundaryCharacteristic::This,
            south: BoundaryCharacteristic::This,
            east: BoundaryCharacteristic::This,
            west: BoundaryCharacteristic::This,
        };
        let op = &self.op;
        let rhs_trad = |k: &mut Field, y: &Field, _time: Float, gm: &(_, _), wb: &mut _| {
            let (grid, metrics) = gm;
            let boundaries = boundary_extractor(y, grid, &bc);
            RHS_trad(op, k, y, metrics, &boundaries, wb)
        };
        integrate::integrate::<integrate::Rk4, _, _, _, _>(
            rhs_trad,
            &self.sys.0,
            &mut self.sys.1,
            &mut 0.0,
            dt,
            &mut self.k,
            &self.grid,
            &mut self.wb.0,
        );
        std::mem::swap(&mut self.sys.0, &mut self.sys.1);
    }

    pub fn vortex(&mut self, t: Float, vortex_parameters: VortexParameters) {
        self.sys
            .0
            .vortex(self.grid.0.x(), self.grid.0.y(), t, vortex_parameters);
    }

    #[allow(clippy::many_single_char_names)]
    pub fn init_with_vortex(&mut self, x0: Float, y0: Float) {
        // Should parametrise such that we have radius, drop in pressure at center, etc
        let vortex_parameters = VortexParameters {
            x0,
            y0,
            rstar: 1.0,
            eps: 3.0,
            mach: 0.5,
        };

        self.sys
            .0
            .vortex(self.grid.0.x(), self.grid.0.y(), 0.0, vortex_parameters)
    }

    pub fn field(&self) -> &Field {
        &self.sys.0
    }

    pub fn x(&self) -> ArrayView2<Float> {
        self.grid.0.x.view()
    }
    pub fn y(&self) -> ArrayView2<Float> {
        self.grid.0.y.view()
    }
}

impl<UO: UpwindOperator2d> System<UO> {
    pub fn advance_upwind(&mut self, dt: Float) {
        let bc = BoundaryCharacteristics {
            north: BoundaryCharacteristic::This,
            south: BoundaryCharacteristic::This,
            east: BoundaryCharacteristic::This,
            west: BoundaryCharacteristic::This,
        };
        let op = &self.op;
        let rhs_upwind = |k: &mut Field, y: &Field, _time: Float, gm: &(_, _), wb: &mut _| {
            let (grid, metrics) = gm;
            let boundaries = boundary_extractor(y, grid, &bc);
            RHS_upwind(op, k, y, metrics, &boundaries, wb)
        };
        integrate::integrate::<integrate::Rk4, _, _, _, _>(
            rhs_upwind,
            &self.sys.0,
            &mut self.sys.1,
            &mut 0.0,
            dt,
            &mut self.k,
            &self.grid,
            &mut self.wb.0,
        );
        std::mem::swap(&mut self.sys.0, &mut self.sys.1);
    }
}

#[derive(Clone, Debug)]
/// A 4 x ny x nx array
pub struct Field(pub(crate) Array3<Float>);

impl std::ops::Deref for Field {
    type Target = Array3<Float>;
    fn deref(&self) -> &Self::Target {
        &self.0
    }
}

impl std::ops::DerefMut for Field {
    fn deref_mut(&mut self) -> &mut Self::Target {
        &mut self.0
    }
}

impl Field {
    pub fn new(ny: usize, nx: usize) -> Self {
        let field = Array3::zeros((4, ny, nx));

        Self(field)
    }

    pub fn nx(&self) -> usize {
        self.0.shape()[2]
    }
    pub fn ny(&self) -> usize {
        self.0.shape()[1]
    }

    pub fn rho(&self) -> ArrayView2<Float> {
        self.slice(s![0, .., ..])
    }
    pub fn rhou(&self) -> ArrayView2<Float> {
        self.slice(s![1, .., ..])
    }
    pub fn rhov(&self) -> ArrayView2<Float> {
        self.slice(s![2, .., ..])
    }
    pub fn e(&self) -> ArrayView2<Float> {
        self.slice(s![3, .., ..])
    }

    pub fn rho_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![0, .., ..])
    }
    pub fn rhou_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![1, .., ..])
    }
    pub fn rhov_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![2, .., ..])
    }
    pub fn e_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![3, .., ..])
    }

    #[allow(unused)]
    pub fn components(
        &self,
    ) -> (
        ArrayView2<Float>,
        ArrayView2<Float>,
        ArrayView2<Float>,
        ArrayView2<Float>,
    ) {
        (self.rho(), self.rhou(), self.rhov(), self.e())
    }
    #[allow(unused)]
    pub fn components_mut(
        &mut self,
    ) -> (
        ArrayViewMut2<Float>,
        ArrayViewMut2<Float>,
        ArrayViewMut2<Float>,
        ArrayViewMut2<Float>,
    ) {
        let mut iter = self.0.outer_iter_mut();

        let rho = iter.next().unwrap();
        let rhou = iter.next().unwrap();
        let rhov = iter.next().unwrap();
        let e = iter.next().unwrap();
        assert_eq!(iter.next(), None);

        (rho, rhou, rhov, e)
    }

    pub fn north(&self) -> ArrayView2<Float> {
        self.slice(s![.., self.ny() - 1, ..])
    }
    pub fn south(&self) -> ArrayView2<Float> {
        self.slice(s![.., 0, ..])
    }
    pub fn east(&self) -> ArrayView2<Float> {
        self.slice(s![.., .., self.nx() - 1])
    }
    pub fn west(&self) -> ArrayView2<Float> {
        self.slice(s![.., .., 0])
    }
    fn north_mut(&mut self) -> ArrayViewMut2<Float> {
        let ny = self.ny();
        self.slice_mut(s![.., ny - 1, ..])
    }
    fn south_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![.., 0, ..])
    }
    fn east_mut(&mut self) -> ArrayViewMut2<Float> {
        let nx = self.nx();
        self.slice_mut(s![.., .., nx - 1])
    }
    fn west_mut(&mut self) -> ArrayViewMut2<Float> {
        self.slice_mut(s![.., .., 0])
    }

    pub fn vortex(
        &mut self,
        x: ArrayView2<Float>,
        y: ArrayView2<Float>,
        time: Float,
        vortex_param: VortexParameters,
    ) {
        assert_eq!(x.shape(), y.shape());
        assert_eq!(x.shape()[1], self.nx());
        assert_eq!(x.shape()[0], self.ny());

        let (rho, rhou, rhov, e) = self.components_mut();
        let n = rho.len();

        vortex(
            rho.into_shape((n,)).unwrap(),
            rhou.into_shape((n,)).unwrap(),
            rhov.into_shape((n,)).unwrap(),
            e.into_shape((n,)).unwrap(),
            x.into_shape((n,)).unwrap(),
            y.into_shape((n,)).unwrap(),
            time,
            vortex_param,
        )
    }
}

impl Field {
    /// sqrt((self-other)^T*H*(self-other))
    pub fn h2_err<SBP: SbpOperator2d>(&self, other: &Self, op: &SBP) -> Float {
        assert_eq!(self.nx(), other.nx());
        assert_eq!(self.ny(), other.ny());

        // Resulting structure should be
        // serialized(F0 - F1)^T (Hx kron Hy) serialized(F0 - F1)
        //
        // We accomplish this by serializing along x as fastest dimension
        // Since h is diagonal, it can be iterated with the following iterators

        // This chains the h block into the form [h, 1, 1, 1, rev(h)],
        // and multiplies with a factor
        let itermaker = move |h: &'static [Float], n: usize, factor: Float| {
            h.iter()
                .copied()
                .chain(std::iter::repeat(1.0).take(n - 2 * h.len()))
                .chain(h.iter().copied().rev())
                .map(move |x| x * factor)
        };

        let hxiterator = itermaker(
            op.hxi(),
            self.nx(),
            if op.is_h2xi() {
                1.0 / (self.nx() - 2) as Float
            } else {
                1.0 / (self.nx() - 1) as Float
            },
        );
        // Repeating to get the form
        // [[hx0, hx1, ..., hxn], [hx0, hx1, ..., hxn], ..., [hx0, hx1, ..., hxn]]
        let hxiterator = hxiterator.cycle().take(self.nx() * self.ny());

        let hyiterator = itermaker(
            op.heta(),
            self.ny(),
            1.0 / if op.is_h2eta() {
                (self.ny() - 2) as Float
            } else {
                (self.ny() - 1) as Float
            },
        );
        // Repeating to get the form
        // [[hy0, hy0, ..., hy0], [hy1, hy1, ..., hy1], ..., [hym, hym, ..., hym]]
        let hyiterator = hyiterator.flat_map(|x| std::iter::repeat(x).take(self.nx()));

        let diagiterator = hxiterator.zip(hyiterator).cycle();

        diagiterator
            .zip(self.0.iter())
            .zip(other.0.iter())
            .map(|(((hx, hy), r0), r1)| (*r0 - *r1).powi(2) * hx * hy)
            .sum::<Float>()
            .sqrt()
    }
}

#[test]
fn h2_diff() {
    let mut field0 = Field::new(20, 21);
    for f in field0.0.iter_mut() {
        *f = 1.0
    }
    let field1 = Field::new(20, 21);

    use super::operators::{Upwind4, Upwind9, SBP4, SBP8};

    assert!((field0.h2_err(&field1, &Upwind4).powi(2) - 4.0).abs() < 1e-3);
    assert!((field0.h2_err(&field1, &Upwind9).powi(2) - 4.0).abs() < 1e-3);
    assert!((field0.h2_err(&field1, &SBP4).powi(2) - 4.0).abs() < 1e-3);
    assert!((field0.h2_err(&field1, &SBP8).powi(2) - 4.0).abs() < 1e-3);
}

#[derive(Copy, Clone, Debug)]
pub struct VortexParameters {
    pub x0: Float,
    pub y0: Float,
    pub rstar: Float,
    pub eps: Float,
    pub mach: Float,
}

#[allow(clippy::too_many_arguments)]
pub fn vortex(
    rho: ArrayViewMut1<Float>,
    rhou: ArrayViewMut1<Float>,
    rhov: ArrayViewMut1<Float>,
    e: ArrayViewMut1<Float>,
    x: ArrayView1<Float>,
    y: ArrayView1<Float>,
    time: Float,
    vortex_param: VortexParameters,
) {
    assert_eq!(rho.len(), rhou.len());
    assert_eq!(rho.len(), rhov.len());
    assert_eq!(rho.len(), e.len());
    assert_eq!(rho.len(), x.len());
    assert_eq!(rho.len(), y.len());
    assert_eq!(x.shape(), y.shape());

    let eps = vortex_param.eps;
    let m = vortex_param.mach;
    let rstar = vortex_param.rstar;
    let p_inf = 1.0 / (GAMMA * m * m);
    azip!((rho in rho,
           rhou in rhou,
           rhov in rhov,
           e in e,
           x in x,
           y in y)
    {
        use crate::consts::PI;

        let dx = (x - vortex_param.x0) - time;
        let dy = y - vortex_param.y0;
        let f = (1.0 - (dx*dx + dy*dy))/(rstar*rstar);

        *rho = Float::powf(1.0 - eps*eps*(GAMMA - 1.0)*m*m/(8.0*PI*PI*p_inf*rstar*rstar)*f.exp(), 1.0/(GAMMA - 1.0));
        assert!(*rho > 0.0);
        let p = Float::powf(*rho, GAMMA)*p_inf;
        assert!(p > 0.0);
        let u = 1.0 - eps*dy/(2.0*PI*p_inf.sqrt()*rstar*rstar)*(f/2.0).exp();
        let v =       eps*dx/(2.0*PI*p_inf.sqrt()*rstar*rstar)*(f/2.0).exp();
        *rhou = *rho*u;
        *rhov = *rho*v;
        *e = p/(GAMMA - 1.0) + *rho*(u*u + v*v)/2.0;
    });
}

fn pressure(gamma: Float, rho: Float, rhou: Float, rhov: Float, e: Float) -> Float {
    (gamma - 1.0) * (e - (rhou * rhou + rhov * rhov) / (2.0 * rho))
}

#[allow(non_snake_case)]
pub fn RHS_trad<SBP: SbpOperator2d>(
    op: &SBP,
    k: &mut Field,
    y: &Field,
    metrics: &Metrics,
    boundaries: &BoundaryTerms,
    tmp: &mut (Field, Field, Field, Field, Field, Field),
) {
    let ehat = &mut tmp.0;
    let fhat = &mut tmp.1;
    fluxes((ehat, fhat), y, metrics);
    let dE = &mut tmp.2;
    let dF = &mut tmp.3;

    op.diffxi(ehat.rho(), dE.rho_mut());
    op.diffxi(ehat.rhou(), dE.rhou_mut());
    op.diffxi(ehat.rhov(), dE.rhov_mut());
    op.diffxi(ehat.e(), dE.e_mut());

    op.diffeta(fhat.rho(), dF.rho_mut());
    op.diffeta(fhat.rhou(), dF.rhou_mut());
    op.diffeta(fhat.rhov(), dF.rhov_mut());
    op.diffeta(fhat.e(), dF.e_mut());

    azip!((out in &mut k.0,
                    eflux in &dE.0,
                    fflux in &dF.0,
                    detj in &metrics.detj.broadcast((4, y.ny(), y.nx())).unwrap()) {
        *out = (-eflux - fflux)/detj
    });

    SAT_characteristics(op, k, y, metrics, boundaries);
}

#[allow(non_snake_case)]
pub fn RHS_upwind<UO: UpwindOperator2d>(
    op: &UO,
    k: &mut Field,
    y: &Field,
    metrics: &Metrics,
    boundaries: &BoundaryTerms,
    tmp: &mut (Field, Field, Field, Field, Field, Field),
) {
    let ehat = &mut tmp.0;
    let fhat = &mut tmp.1;
    fluxes((ehat, fhat), y, metrics);
    let dE = &mut tmp.2;
    let dF = &mut tmp.3;

    op.diffxi(ehat.rho(), dE.rho_mut());
    op.diffxi(ehat.rhou(), dE.rhou_mut());
    op.diffxi(ehat.rhov(), dE.rhov_mut());
    op.diffxi(ehat.e(), dE.e_mut());

    op.diffeta(fhat.rho(), dF.rho_mut());
    op.diffeta(fhat.rhou(), dF.rhou_mut());
    op.diffeta(fhat.rhov(), dF.rhov_mut());
    op.diffeta(fhat.e(), dF.e_mut());

    let ad_xi = &mut tmp.4;
    let ad_eta = &mut tmp.5;
    upwind_dissipation(op, (ad_xi, ad_eta), y, metrics, (&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 &metrics.detj.broadcast((4, y.ny(), y.nx())).unwrap()) {
        *out = (-eflux - fflux + ad_xi + ad_eta)/detj
    });

    SAT_characteristics(op, k, y, metrics, boundaries);
}

#[allow(clippy::many_single_char_names)]
fn upwind_dissipation<UO: UpwindOperator2d>(
    op: &UO,
    k: (&mut Field, &mut Field),
    y: &Field,
    metrics: &Metrics,
    tmp: (&mut Field, &mut Field),
) {
    let n = y.nx() * y.ny();
    let yview = y.view().into_shape((4, n)).unwrap();
    let mut tmp0 = tmp.0.view_mut().into_shape((4, n)).unwrap();
    let mut tmp1 = tmp.1.view_mut().into_shape((4, n)).unwrap();

    for (
        ((((((y, mut tmp0), mut tmp1), detj), detj_dxi_dx), detj_dxi_dy), detj_deta_dx),
        detj_deta_dy,
    ) in yview
        .axis_iter(ndarray::Axis(1))
        .zip(tmp0.axis_iter_mut(ndarray::Axis(1)))
        .zip(tmp1.axis_iter_mut(ndarray::Axis(1)))
        .zip(metrics.detj.iter())
        .zip(metrics.detj_dxi_dx.iter())
        .zip(metrics.detj_dxi_dy.iter())
        .zip(metrics.detj_deta_dx.iter())
        .zip(metrics.detj_deta_dy.iter())
    {
        let rho = y[0];
        assert!(rho > 0.0);
        let rhou = y[1];
        let rhov = y[2];
        let e = y[3];

        let u = rhou / rho;
        let v = rhov / rho;

        let uhat = detj_dxi_dx / detj * u + detj_dxi_dy / detj * v;
        let vhat = detj_deta_dx / detj * u + detj_deta_dy / detj * 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;

        tmp0[0] = alpha_u * rho * detj;
        tmp1[0] = alpha_v * rho * detj;

        tmp0[1] = alpha_u * rhou * detj;
        tmp1[1] = alpha_v * rhou * detj;

        tmp0[2] = alpha_u * rhov * detj;
        tmp1[2] = alpha_v * rhov * detj;

        tmp0[3] = alpha_u * e * detj;
        tmp1[3] = alpha_v * e * detj;
    }

    op.dissxi(tmp.0.rho(), k.0.rho_mut());
    op.dissxi(tmp.0.rhou(), k.0.rhou_mut());
    op.dissxi(tmp.0.rhov(), k.0.rhov_mut());
    op.dissxi(tmp.0.e(), k.0.e_mut());

    op.disseta(tmp.1.rho(), k.1.rho_mut());
    op.disseta(tmp.1.rhou(), k.1.rhou_mut());
    op.disseta(tmp.1.rhov(), k.1.rhov_mut());
    op.disseta(tmp.1.e(), k.1.e_mut());
}

fn fluxes(k: (&mut Field, &mut Field), y: &Field, metrics: &Metrics) {
    let j_dxi_dx = metrics.detj_dxi_dx.view();
    let j_dxi_dy = metrics.detj_dxi_dy.view();
    let j_deta_dx = metrics.detj_deta_dx.view();
    let j_deta_dy = metrics.detj_deta_dy.view();

    let rho = y.rho();
    let rhou = y.rhou();
    let rhov = y.rhov();
    let e = y.e();

    for j in 0..y.ny() {
        for i in 0..y.nx() {
            let rho = rho[(j, i)];
            assert!(rho > 0.0);
            let rhou = rhou[(j, i)];
            let rhov = rhov[(j, i)];
            let e = e[(j, i)];
            let p = pressure(GAMMA, rho, rhou, rhov, e);

            assert!(p > 0.0);

            let ef = [
                rhou,
                rhou * rhou / rho + p,
                rhou * rhov / rho,
                rhou * (e + p) / rho,
            ];
            let ff = [
                rhov,
                rhou * rhov / rho,
                rhov * rhov / rho + p,
                rhov * (e + p) / rho,
            ];

            for comp in 0..4 {
                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];

                k.0[(comp, j, i)] = eflux;
                k.1[(comp, j, i)] = fflux;
            }
        }
    }
}

#[derive(Clone, Debug)]
pub struct BoundaryTerms<'a> {
    pub north: ArrayView2<'a, Float>,
    pub south: ArrayView2<'a, Float>,
    pub east: ArrayView2<'a, Float>,
    pub west: ArrayView2<'a, Float>,
}

#[derive(Clone, Debug)]
pub enum BoundaryCharacteristic {
    This,
    Grid(usize),
    Vortex(VortexParameters),
    // Vortices(Vec<VortexParameters>),
    Interpolate(usize),
}

pub struct Direction<T> {
    pub north: T,
    pub south: T,
    pub west: T,
    pub east: T,
}

pub type BoundaryCharacteristics = Direction<BoundaryCharacteristic>;

fn boundary_extractor<'a>(
    field: &'a Field,
    _grid: &Grid,
    bc: &BoundaryCharacteristics,
) -> BoundaryTerms<'a> {
    BoundaryTerms {
        north: match bc.north {
            BoundaryCharacteristic::This => field.south(),
            BoundaryCharacteristic::Vortex(_params) => todo!(),
            BoundaryCharacteristic::Grid(_) | BoundaryCharacteristic::Interpolate(_) => {
                panic!("Only working on self grid")
            }
        },
        south: match bc.south {
            BoundaryCharacteristic::This => field.north(),
            BoundaryCharacteristic::Vortex(_params) => todo!(),
            BoundaryCharacteristic::Grid(_) | BoundaryCharacteristic::Interpolate(_) => {
                panic!("Only working on self grid")
            }
        },
        west: match bc.west {
            BoundaryCharacteristic::This => field.east(),
            BoundaryCharacteristic::Vortex(_params) => todo!(),
            BoundaryCharacteristic::Grid(_) | BoundaryCharacteristic::Interpolate(_) => {
                panic!("Only working on self grid")
            }
        },
        east: match bc.east {
            BoundaryCharacteristic::This => field.west(),
            BoundaryCharacteristic::Vortex(_params) => todo!(),
            BoundaryCharacteristic::Grid(_) | BoundaryCharacteristic::Interpolate(_) => {
                panic!("Only working on self grid")
            }
        },
    }
}

pub type InterpolationOperators = Direction<Option<Box<dyn InterpolationOperator>>>;

pub fn extract_boundaries<'a>(
    fields: &'a [Field],
    bt: &[BoundaryCharacteristics],
    eb: &'a mut [BoundaryStorage],
    grids: &[Grid],
    time: Float,
    interpolation_operators: Option<&[InterpolationOperators]>,
) -> Vec<BoundaryTerms<'a>> {
    bt.iter()
        .zip(eb)
        .zip(grids)
        .zip(fields)
        .enumerate()
        .map(|(ig, (((bt, eb), grid), field))| BoundaryTerms {
            north: match bt.north {
                BoundaryCharacteristic::This => field.south(),
                BoundaryCharacteristic::Grid(g) => fields[g].south(),
                BoundaryCharacteristic::Vortex(v) => {
                    let field = eb.n.as_mut().unwrap();
                    vortexify(field.view_mut(), grid.north(), v, time);
                    field.view()
                }
                BoundaryCharacteristic::Interpolate(g) => {
                    let to = eb.n.as_mut().unwrap();
                    let fine2coarse = field.nx() < fields[g].nx();

                    let operator = interpolation_operators.as_ref().unwrap()[ig]
                        .north
                        .as_ref()
                        .unwrap();

                    for (mut to, from) in to.outer_iter_mut().zip(fields[g].south().outer_iter()) {
                        if fine2coarse {
                            operator.fine2coarse(from.view(), to.view_mut());
                        } else {
                            operator.coarse2fine(from.view(), to.view_mut());
                        }
                    }
                    to.view()
                }
            },
            south: match bt.south {
                BoundaryCharacteristic::This => field.north(),
                BoundaryCharacteristic::Grid(g) => fields[g].north(),
                BoundaryCharacteristic::Vortex(v) => {
                    let field = eb.s.as_mut().unwrap();
                    vortexify(field.view_mut(), grid.south(), v, time);
                    field.view()
                }
                BoundaryCharacteristic::Interpolate(g) => {
                    let to = eb.s.as_mut().unwrap();
                    let fine2coarse = field.nx() < fields[g].nx();
                    let operator = interpolation_operators.as_ref().unwrap()[ig]
                        .south
                        .as_ref()
                        .unwrap();

                    for (mut to, from) in to.outer_iter_mut().zip(fields[g].north().outer_iter()) {
                        if fine2coarse {
                            operator.fine2coarse(from.view(), to.view_mut());
                        } else {
                            operator.coarse2fine(from.view(), to.view_mut());
                        }
                    }
                    to.view()
                }
            },
            west: match bt.west {
                BoundaryCharacteristic::This => field.east(),
                BoundaryCharacteristic::Grid(g) => fields[g].east(),
                BoundaryCharacteristic::Vortex(v) => {
                    let field = eb.w.as_mut().unwrap();
                    vortexify(field.view_mut(), grid.west(), v, time);
                    field.view()
                }
                BoundaryCharacteristic::Interpolate(g) => {
                    let to = eb.w.as_mut().unwrap();
                    let fine2coarse = field.ny() < fields[g].ny();
                    let operator = interpolation_operators.as_ref().unwrap()[ig]
                        .west
                        .as_ref()
                        .unwrap();

                    for (mut to, from) in to.outer_iter_mut().zip(fields[g].east().outer_iter()) {
                        if fine2coarse {
                            operator.fine2coarse(from.view(), to.view_mut());
                        } else {
                            operator.coarse2fine(from.view(), to.view_mut());
                        }
                    }
                    to.view()
                }
            },
            east: match bt.east {
                BoundaryCharacteristic::This => field.west(),
                BoundaryCharacteristic::Grid(g) => fields[g].west(),
                BoundaryCharacteristic::Vortex(v) => {
                    let field = eb.e.as_mut().unwrap();
                    vortexify(field.view_mut(), grid.east(), v, time);
                    field.view()
                }
                BoundaryCharacteristic::Interpolate(g) => {
                    let to = eb.e.as_mut().unwrap();
                    let fine2coarse = field.ny() < fields[g].ny();
                    let operator = interpolation_operators.as_ref().unwrap()[ig]
                        .east
                        .as_ref()
                        .unwrap();

                    for (mut to, from) in to.outer_iter_mut().zip(fields[g].west().outer_iter()) {
                        if fine2coarse {
                            operator.fine2coarse(from.view(), to.view_mut());
                        } else {
                            operator.coarse2fine(from.view(), to.view_mut());
                        }
                    }
                    to.view()
                }
            },
        })
        .collect()
}

#[derive(Debug, Clone)]
/// Used for storing boundary elements
pub struct BoundaryStorage {
    pub n: Option<ndarray::Array2<Float>>,
    pub s: Option<ndarray::Array2<Float>>,
    pub e: Option<ndarray::Array2<Float>>,
    pub w: Option<ndarray::Array2<Float>>,
}

impl BoundaryStorage {
    pub fn new(bt: &BoundaryCharacteristics, grid: &Grid) -> Self {
        Self {
            n: match bt.north {
                BoundaryCharacteristic::Vortex(_) | BoundaryCharacteristic::Interpolate(_) => {
                    Some(ndarray::Array2::zeros((4, grid.nx())))
                }
                _ => None,
            },
            s: match bt.south {
                BoundaryCharacteristic::Vortex(_) | BoundaryCharacteristic::Interpolate(_) => {
                    Some(ndarray::Array2::zeros((4, grid.nx())))
                }
                _ => None,
            },
            e: match bt.east {
                BoundaryCharacteristic::Vortex(_) | BoundaryCharacteristic::Interpolate(_) => {
                    Some(ndarray::Array2::zeros((4, grid.ny())))
                }
                _ => None,
            },
            w: match bt.west {
                BoundaryCharacteristic::Vortex(_) | BoundaryCharacteristic::Interpolate(_) => {
                    Some(ndarray::Array2::zeros((4, grid.ny())))
                }
                _ => None,
            },
        }
    }
}

fn vortexify(
    mut field: ndarray::ArrayViewMut2<Float>,
    yx: (ndarray::ArrayView1<Float>, ndarray::ArrayView1<Float>),
    vparams: VortexParameters,
    time: Float,
) {
    let mut fiter = field.outer_iter_mut();
    let (rho, rhou, rhov, e) = (
        fiter.next().unwrap(),
        fiter.next().unwrap(),
        fiter.next().unwrap(),
        fiter.next().unwrap(),
    );
    let (y, x) = yx;
    vortex(rho, rhou, rhov, e, x, y, time, vparams);
}

#[allow(non_snake_case)]
/// Boundary conditions (SAT)
fn SAT_characteristics<SBP: SbpOperator2d>(
    op: &SBP,
    k: &mut Field,
    y: &Field,
    metrics: &Metrics,
    boundaries: &BoundaryTerms,
) {
    // North boundary
    {
        let hi = if op.is_h2eta() {
            (k.ny() - 2) as Float / op.heta()[0]
        } else {
            (k.ny() - 1) as Float / op.heta()[0]
        };
        let sign = -1.0;
        let tau = 1.0;
        let slice = s![y.ny() - 1, ..];
        SAT_characteristic(
            k.north_mut(),
            y.north(),
            boundaries.north,
            hi,
            sign,
            tau,
            metrics.detj.slice(slice),
            metrics.detj_deta_dx.slice(slice),
            metrics.detj_deta_dy.slice(slice),
        );
    }
    // South boundary
    {
        let hi = if op.is_h2eta() {
            (k.ny() - 2) as Float / op.heta()[0]
        } else {
            (k.ny() - 1) as Float / op.heta()[0]
        };
        let sign = 1.0;
        let tau = -1.0;
        let slice = s![0, ..];
        SAT_characteristic(
            k.south_mut(),
            y.south(),
            boundaries.south,
            hi,
            sign,
            tau,
            metrics.detj.slice(slice),
            metrics.detj_deta_dx.slice(slice),
            metrics.detj_deta_dy.slice(slice),
        );
    }
    // West Boundary
    {
        let hi = if op.is_h2xi() {
            (k.nx() - 2) as Float / op.hxi()[0]
        } else {
            (k.nx() - 1) as Float / op.hxi()[0]
        };
        let sign = 1.0;
        let tau = -1.0;
        let slice = s![.., 0];
        SAT_characteristic(
            k.west_mut(),
            y.west(),
            boundaries.west,
            hi,
            sign,
            tau,
            metrics.detj.slice(slice),
            metrics.detj_dxi_dx.slice(slice),
            metrics.detj_dxi_dy.slice(slice),
        );
    }
    // East Boundary
    {
        let hi = if op.is_h2xi() {
            (k.nx() - 2) as Float / op.hxi()[0]
        } else {
            (k.nx() - 1) as Float / op.hxi()[0]
        };
        let sign = -1.0;
        let tau = 1.0;
        let slice = s![.., y.nx() - 1];
        SAT_characteristic(
            k.east_mut(),
            y.east(),
            boundaries.east,
            hi,
            sign,
            tau,
            metrics.detj.slice(slice),
            metrics.detj_dxi_dx.slice(slice),
            metrics.detj_dxi_dy.slice(slice),
        );
    }
}

#[allow(non_snake_case)]
#[allow(clippy::many_single_char_names)]
#[allow(clippy::too_many_arguments)]
/// Boundary conditions (SAT)
fn SAT_characteristic(
    mut k: ArrayViewMut2<Float>,
    y: ArrayView2<Float>,
    z: ArrayView2<Float>, // Size 4 x n (all components in line)
    hi: Float,
    sign: Float,
    tau: Float,
    detj: ArrayView1<Float>,
    detj_d_dx: ArrayView1<Float>,
    detj_d_dy: ArrayView1<Float>,
) {
    assert_eq!(detj.shape(), detj_d_dx.shape());
    assert_eq!(detj.shape(), detj_d_dy.shape());
    assert_eq!(y.shape(), z.shape());
    assert_eq!(y.shape()[0], 4);
    assert_eq!(y.shape()[1], detj.shape()[0]);

    for (((((mut k, y), z), detj), detj_d_dx), detj_d_dy) in k
        .axis_iter_mut(ndarray::Axis(1))
        .zip(y.axis_iter(ndarray::Axis(1)))
        .zip(z.axis_iter(ndarray::Axis(1)))
        .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 / detj;
        let ky_ = detj_d_dy / detj;

        let (kx, ky) = {
            let r = Float::hypot(kx_, ky_);
            (kx_ / r, ky_ / r)
        };

        let u = rhou / rho;
        let v = rhov / rho;

        let theta = kx * u + ky * v;

        let p = pressure(GAMMA, rho, rhou, rhov, e);
        let c = (GAMMA * p / rho).sqrt();
        let phi2 = (GAMMA - 1.0) * (u * u + v * v) / 2.0;

        let phi2_c2 = (phi2 + c * c) / (GAMMA - 1.0);

        #[rustfmt::skip]
        let T = [
            [                 1.0,             0.0,                 1.0,                 1.0],
            [                   u,              ky,          u + kx * c,          u - kx * c],
            [                   v,             -kx,          v + ky * c,          v - ky * c],
            [phi2 / (GAMMA - 1.0), ky * u - kx * v, phi2_c2 + c * theta, phi2_c2 - c * theta],
        ];
        let U = kx_ * u + ky_ * v;
        let L = [
            U,
            U,
            U + c * Float::hypot(kx_, ky_),
            U - c * Float::hypot(kx_, ky_),
        ];
        let beta = 1.0 / (2.0 * c * c);
        #[rustfmt::skip]
        let TI = [
            [     1.0 - phi2 / (c * c),          (GAMMA - 1.0) * u / (c * c),          (GAMMA - 1.0) * v / (c * c), -(GAMMA - 1.0) / (c * c)],
            [       -(ky * u - kx * v),                                   ky,                                  -kx,                      0.0],
            [beta * (phi2 - c * theta),  beta * (kx * c - (GAMMA - 1.0) * u),  beta * (ky * c - (GAMMA - 1.0) * v),     beta * (GAMMA - 1.0)],
            [beta * (phi2 + c * theta), -beta * (kx * c + (GAMMA - 1.0) * u), -beta * (ky * c + (GAMMA - 1.0) * v),     beta * (GAMMA - 1.0)],
        ];

        let res = [rho - z[0], rhou - z[1], rhov - z[2], e - z[3]];
        let mut TIres = [0.0; 4];
        #[allow(clippy::needless_range_loop)]
        for row in 0..4 {
            for col in 0..4 {
                TIres[row] += TI[row][col] * res[col];
            }
        }

        // L + sign(abs(L)) * TIres
        let mut LTIres = [0.0; 4];
        for row in 0..4 {
            LTIres[row] = (L[row] + sign * L[row].abs()) * TIres[row];
        }

        // T*LTIres
        let mut TLTIres = [0.0; 4];
        #[allow(clippy::needless_range_loop)]
        for row in 0..4 {
            for col in 0..4 {
                TLTIres[row] += T[row][col] * LTIres[col];
            }
        }

        for comp in 0..4 {
            k[comp] += hi * tau * TLTIres[comp];
        }
    }
}

#[derive(Debug)]
pub struct WorkBuffers(pub (Field, Field, Field, Field, Field, Field));

impl WorkBuffers {
    pub fn new(nx: usize, ny: usize) -> Self {
        let arr3 = Field::new(nx, ny);
        Self((
            arr3.clone(),
            arr3.clone(),
            arr3.clone(),
            arr3.clone(),
            arr3.clone(),
            arr3,
        ))
    }
}