Add matrix approach to heat equation

heat-equation/src/main.rs

@@ -2,7 +2,7 @@ use ndarray::{Array1, ArrayView1};
 use plotters::prelude::*;
 use sbp::{
     integrate::{integrate, Rk4},
-    operators::{SbpOperator1d2, SBP4},
+    operators::{SbpOperator1d, SbpOperator1d2, SBP4},
     Float,
 };
 
@@ -13,6 +13,9 @@ fn main() {
     dual_dirichlet(v0.view(), 1.0, 1.0);
     // Neumann boundary is introducing energy into the system
     neumann_dirichlet(v0.view(), -0.2, 1.0);
+
+    // The sparse formulation
+    dual_dirichlet_sparse(v0.view(), 1.0, 1.0);
 }
 
 fn dual_dirichlet(v: ArrayView1<Float>, v0: Float, vn: Float) {
@@ -134,3 +137,90 @@ fn neumann_dirichlet(v: ArrayView1<Float>, v0: Float, vn: Float) {
         std::mem::swap(&mut v1, &mut v2);
     }
 }
+
+fn dual_dirichlet_sparse(v: ArrayView1<Float>, v0: Float, vn: Float) {
+    let drawing_area = BitMapBackend::gif("dual_dirichlet_sparse.gif", (300, 300), 100)
+        .unwrap()
+        .into_drawing_area();
+    let mut chart = ChartBuilder::on(&drawing_area)
+        .x_label_area_size(40)
+        .y_label_area_size(40)
+        .build_cartesian_2d(0.0..1.01, -1.0..2.0)
+        .unwrap();
+
+    let op = SBP4;
+    let nx = v.len();
+    let system = op.diff2_matrix(nx);
+    let e0 = {
+        let mut e0 = sprs::CsMat::zero((nx, 1));
+        e0.insert(0, 0, 1.0);
+        e0
+    };
+    let en = {
+        let mut en = sprs::CsMat::zero((nx, 1));
+        en.insert(nx - 1, 0, 1.0);
+        en
+    };
+
+    let mut hi = op.h_matrix(nx);
+    hi.map_inplace(|v| 1.0 / v);
+
+    let tau = (1.0, -1.0);
+    let sat0 = {
+        let d1 = op.d1_vec(nx, true);
+        let mut mat = &hi * &d1.transpose_view();
+        mat.map_inplace(|v| v * tau.0);
+        (&mat * &e0.transpose_view(), mat)
+    };
+    let satn = {
+        let dn = op.d1_vec(nx, false);
+        let mut mat = &hi * &dn.transpose_view();
+        mat.map_inplace(|v| v * tau.1);
+        (&mat * &en.transpose_view(), mat)
+    };
+
+    let system = &system + &(&sat0.0 + &satn.0);
+    // Stack the two matrices to allow easy book-keeping
+    // of boundary conditions
+    let mut bc = sprs::hstack(&[sat0.1.view(), satn.1.view()]).to_csr();
+    bc.map_inplace(|v| -v);
+    let system = &system;
+    let bc = &bc;
+
+    let dt = 0.2 / nx.pow(2) as Float / 3.0;
+    let x = Array1::from_shape_fn((nx,), |i| i as Float / (nx - 1) as Float);
+
+    let mut k = [v.to_owned(), v.to_owned(), v.to_owned(), v.to_owned()];
+    let rhs = move |fut: &mut Array1<Float>, prev: &Array1<Float>, _t: Float| {
+        fut.fill(0.0);
+        let prev = prev.as_slice().unwrap();
+        let fut = fut.as_slice_mut().unwrap();
+
+        sprs::prod::mul_acc_mat_vec_csr(system.view(), prev, fut);
+
+        sprs::prod::mul_acc_mat_vec_csr(bc.view(), &[v0, vn][..], fut);
+    };
+
+    let mut v1 = v.to_owned();
+    let mut v2 = v.to_owned();
+    for i in 0..90 {
+        if i % 3 == 0 {
+            drawing_area.fill(&WHITE).unwrap();
+            chart
+                .configure_mesh()
+                .x_desc("x")
+                .y_desc("y")
+                .draw()
+                .unwrap();
+            chart
+                .draw_series(LineSeries::new(
+                    x.iter().zip(v1.iter()).map(|(&x, &y)| (x, y)),
+                    &BLACK,
+                ))
+                .unwrap();
+            drawing_area.present().unwrap();
+        }
+        integrate::<Rk4, _, _, _>(rhs, &v1, &mut v2, &mut 0.0, dt, &mut k);
+        std::mem::swap(&mut v1, &mut v2);
+    }
+}