297 lines
9.3 KiB
Rust
297 lines
9.3 KiB
Rust
use super::Float;
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use sbp::operators::{SbpOperator2d, UpwindOperator2d};
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use sbp::utils::kronecker_product;
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fn eye(n: usize) -> sprs::CsMat<Float> {
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sprs::CsMat::eye(n)
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}
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/// Implicit version of the system
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/// C u_{t+1} = u_t
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pub fn implicit_matrix(rhs: sprs::CsMatView<Float>, dt: Float) -> sprs::CsMat<Float> {
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let n = rhs.rows();
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let f = rhs.map(|x| x * dt);
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&eye(n) - &f
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}
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fn diagonal(values: &[Float]) -> sprs::CsMat<Float> {
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let values = values.to_vec();
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let indptr = (0..values.len() + 1).collect();
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let indices = (0..values.len()).collect();
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sprs::CsMat::new((values.len(), values.len()), indptr, indices, values)
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}
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pub struct Implicit {
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pub(crate) rhs: sprs::CsMat<Float>,
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/// Diagonal matrix
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pub(crate) lhs: sprs::CsMat<Float>,
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}
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/// Assumes self boundaries
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pub fn rhs_matrix(op: &dyn SbpOperator2d, grid: &super::Grid) -> Implicit {
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let metrics = grid.metrics(op).unwrap();
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let nx = grid.nx();
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let ny = grid.ny();
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let fluxes = {
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let d1_xi = op.op_eta().diff_matrix(nx);
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let d1_eta = op.op_xi().diff_matrix(ny);
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let d1_xi = kronecker_product(eye(ny).view(), d1_xi.view());
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let d1_eta = kronecker_product(d1_eta.view(), eye(nx).view());
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let mut a_flux = sprs::CsMat::zero((3, 3));
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a_flux.insert(1, 2, -1.0);
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a_flux.insert(2, 1, -1.0);
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let mut b_flux = sprs::CsMat::zero((3, 3));
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b_flux.insert(0, 1, 1.0);
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b_flux.insert(1, 0, 1.0);
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let detj_dxi_dx = diagonal(metrics.detj_dxi_dx().as_slice().unwrap());
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let detj_dxi_dx = &d1_xi * &detj_dxi_dx;
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// Can multiply with the constant matrix after differentiation
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let f_flux_dxi = kronecker_product(a_flux.view(), detj_dxi_dx.view());
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let detj_dxi_dy = diagonal(metrics.detj_dxi_dy().as_slice().unwrap());
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let detj_dxi_dy = &d1_xi * &detj_dxi_dy;
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let f_flux_deta = kronecker_product(b_flux.view(), detj_dxi_dy.view());
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let detj_deta_dx = diagonal(metrics.detj_deta_dx().as_slice().unwrap());
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let detj_deta_dx = &d1_eta * &detj_deta_dx;
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let g_flux_dxi = kronecker_product(a_flux.view(), detj_deta_dx.view());
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let detj_deta_dy = diagonal(metrics.detj_deta_dy().as_slice().unwrap());
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let detj_deta_dy = &d1_eta * &detj_deta_dy;
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let g_flux_deta = kronecker_product(b_flux.view(), detj_deta_dy.view());
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let f_flux = &f_flux_dxi + &f_flux_deta;
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let g_flux = &g_flux_dxi + &g_flux_deta;
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&f_flux + &g_flux
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};
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fn flux_matrix(
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kx: ndarray::ArrayView2<Float>,
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ky: ndarray::ArrayView2<Float>,
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positive: bool,
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) -> sprs::CsMat<Float> {
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let mut r = &(&kx * &kx) + &(&ky * &ky);
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r.map_inplace(|v| *v = v.sqrt());
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let a00 = if positive {
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&ky * &ky / (2.0 * &r)
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} else {
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-&ky * ky / (2.0 * &r)
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};
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let a00 = diagonal(a00.as_slice().unwrap());
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let a01 = &ky / 2.0;
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let a01 = diagonal(a01.as_slice().unwrap());
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let a02 = &kx * &ky / (2.0 * &r);
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let a02 = diagonal(a02.as_slice().unwrap());
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let a10 = &a01;
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let a11 = if positive { &r / 2.0 } else { -&r / 2.0 };
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let a11 = diagonal(a11.as_slice().unwrap());
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let a12 = -&kx / 2.0;
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let a12 = diagonal(a12.as_slice().unwrap());
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let a20 = &a02;
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let a21 = &a12;
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let a22 = if positive {
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&kx * &kx / (2.0 * &r)
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} else {
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-&kx * kx / (2.0 * &r)
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};
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let a22 = diagonal(a22.as_slice().unwrap());
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sprs::bmat(&[
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[Some(a00.view()), Some(a01.view()), Some(a02.view())],
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[Some(a10.view()), Some(a11.view()), Some(a12.view())],
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[Some(a20.view()), Some(a21.view()), Some(a22.view())],
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])
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}
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let e0 = |n| {
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let mut e0 = sprs::CsMat::zero((n, 1));
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e0.insert(0, 0, 1.0);
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e0
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};
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let en = |n| {
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let mut en = sprs::CsMat::zero((n, 1));
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en.insert(n - 1, 0, 1.0);
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en
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};
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let sat_west = {
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// West boundary
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let e0 = e0(nx);
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let en = en(nx);
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// Periodic => (e_0 - e_n)q => 0
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let p = &e0 - &en;
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// Forming the matrix of size (nx,nx)
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let mat = &e0 * &p.transpose_view();
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// Must be scaled by the h norm
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let hi = op.op_xi().h_matrix(nx).map(|h| 1.0 / h);
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let mat = &hi * &mat;
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// Upscaling to (nx * ny, nx * ny)
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let mat = kronecker_product(eye(ny).view(), mat.view());
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let aminus = flux_matrix(metrics.detj_dxi_dx(), metrics.detj_dxi_dy(), false);
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let mut sat = &aminus * &kronecker_product(eye(3).view(), mat.view());
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let tau = 1.0;
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// Scaling by tau
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sat.map_inplace(|x| tau * x);
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sat
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};
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let sat_east = {
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// East boundary
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let e0 = e0(nx);
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let en = en(nx);
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// Periodic => (e_0 - e_n) => 0
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let p = &en - &e0;
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// Forming the matrix of size (nx,nx)
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let mat = &en * &p.transpose_view();
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// Must be scaled by the h norm
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let hi = op.op_xi().h_matrix(nx).map(|h| 1.0 / h);
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let mat = &hi * &mat;
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// Upscaling to (nx * ny, nx * ny)
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let mat = kronecker_product(eye(ny).view(), mat.view());
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let aplus = flux_matrix(metrics.detj_dxi_dx(), metrics.detj_dxi_dy(), true);
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let mut sat = &aplus * &kronecker_product(eye(3).view(), mat.view());
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let tau = -1.0;
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// Scaling by tau
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sat.map_inplace(|x| tau * x);
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sat
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};
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let sat_south = {
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// South boundary
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let e0 = e0(ny);
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let en = en(ny);
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// Periodic => (e_0 - e_n) => 0
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let p = &e0 - &en;
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// Forming the matrix of size (ny,ny)
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let mat = &e0 * &p.transpose_view();
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// Must be scaled by the h norm
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let hi = op.op_eta().h_matrix(ny).map(|h| 1.0 / h);
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let mat = &hi * &mat;
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// Upscaling to (nx * ny, nx * ny)
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let mat = kronecker_product(mat.view(), eye(nx).view());
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let bminus = flux_matrix(metrics.detj_deta_dx(), metrics.detj_deta_dy(), false);
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let mut sat = &bminus * &kronecker_product(eye(3).view(), mat.view());
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let tau = 1.0;
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// Scaling by tau
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sat.map_inplace(|x| tau * x);
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sat
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};
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let sat_north = {
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// North boundary
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let e0 = e0(ny);
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let en = en(ny);
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// Periodic => (e_0 - e_n) => 0
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let p = &en - &e0;
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// Forming the matrix of size (ny,ny)
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let mat = &en * &p.transpose_view();
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// Must be scaled by the h norm
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let hi = op.op_eta().h_matrix(ny).map(|h| 1.0 / h);
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let mat = &hi * &mat;
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// Upscaling to (nx * ny, nx * ny)
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let mat = kronecker_product(mat.view(), eye(nx).view());
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let bminus = flux_matrix(metrics.detj_deta_dx(), metrics.detj_deta_dy(), true);
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let mut sat = &bminus * &kronecker_product(eye(3).view(), mat.view());
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let tau = -1.0;
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// Scaling by tau
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sat.map_inplace(|x| tau * x);
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sat
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};
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let rhs = &fluxes + &(&(&sat_west + &sat_east) + &(&sat_north + &sat_south));
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Implicit {
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rhs,
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lhs: kronecker_product(
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sprs::CsMat::eye(3).view(),
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diagonal(metrics.detj().as_slice().unwrap()).view(),
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),
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}
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}
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/// RHS with some additional dissipation from the upwind operator
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pub fn rhs_matrix_with_upwind_dissipation(
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op: &dyn UpwindOperator2d,
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grid: &super::Grid,
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) -> sprs::CsMat<Float> {
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let rhs = rhs_matrix(op.as_sbp(), grid).rhs;
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let metrics = grid.metrics(op.as_sbp()).unwrap();
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let nx = grid.nx();
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let ny = grid.ny();
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let diss = |kx: ndarray::ArrayView2<Float>, ky: ndarray::ArrayView2<Float>| {
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let r = &kx * &kx + &ky * &ky;
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let s00 = &ky * &ky / &r;
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let s00 = diagonal(s00.as_slice().unwrap());
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let s02 = -&kx * ky / &r;
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let s02 = diagonal(s02.as_slice().unwrap());
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let s11 = diagonal(r.as_slice().unwrap());
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let s20 = &s02;
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let s22 = &kx * &kx / &r;
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let s22 = diagonal(s22.as_slice().unwrap());
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sprs::bmat(&[
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[Some(s00.view()), None, Some(s02.view())],
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[None, Some(s11.view()), None],
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[Some(s20.view()), None, Some(s22.view())],
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])
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};
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let diss_x = {
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let diss_x = UpwindOperator2d::op_xi(op).diss_matrix(nx);
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let diss_x = kronecker_product(eye(ny).view(), diss_x.view());
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let met = diss(metrics.detj_dxi_dx(), metrics.detj_dxi_dy());
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&met * &kronecker_product(eye(3).view(), diss_x.view())
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};
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let diss_y = {
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let diss_y = UpwindOperator2d::op_eta(op).diss_matrix(ny);
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let diss_y = kronecker_product(diss_y.view(), eye(nx).view());
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let met = diss(metrics.detj_deta_dx(), metrics.detj_deta_dy());
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&met * &kronecker_product(eye(3).view(), diss_y.view())
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};
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&rhs + &(&diss_x + &diss_y)
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}
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#[test]
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fn creation() {
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let ny = 16;
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let nx = 170;
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let x = ndarray::Array::from_shape_fn((ny, nx), |(_j, i)| i as Float / (nx - 1) as Float);
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let y = ndarray::Array::from_shape_fn((ny, nx), |(j, _i)| j as Float / (ny - 1) as Float);
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let op = &sbp::operators::Upwind4;
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let grid = sbp::grid::Grid::new(x, y).unwrap();
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let _rhs = rhs_matrix(op, &grid);
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// let _lhs = implicit_matrix(rhs.view(), 1e-2);
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let _rhs_upwind = rhs_matrix_with_upwind_dissipation(op, &grid);
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}
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