diff --git a/euler/src/lib.rs b/euler/src/lib.rs
index 9de5928..13e605b 100644
--- a/euler/src/lib.rs
+++ b/euler/src/lib.rs
@@ -610,18 +610,15 @@ fn upwind_dissipation(
     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())
+    for ((((((y, mut tmp0), mut tmp1), 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_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);
@@ -632,27 +629,27 @@ fn upwind_dissipation(
         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 uhat = detj_dxi_dx * u + detj_dxi_dy * v;
+        let vhat = detj_deta_dx * u + detj_deta_dy * 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;
+        let alpha_u = uhat.abs() + c * Float::hypot(*detj_dxi_dx, *detj_dxi_dy);
+        let alpha_v = vhat.abs() + c * Float::hypot(*detj_deta_dx, *detj_deta_dy);
 
-        tmp0[0] = alpha_u * rho * detj;
-        tmp1[0] = alpha_v * rho * detj;
+        tmp0[0] = alpha_u * rho;
+        tmp1[0] = alpha_v * rho;
 
-        tmp0[1] = alpha_u * rhou * detj;
-        tmp1[1] = alpha_v * rhou * detj;
+        tmp0[1] = alpha_u * rhou;
+        tmp1[1] = alpha_v * rhou;
 
-        tmp0[2] = alpha_u * rhov * detj;
-        tmp1[2] = alpha_v * rhov * detj;
+        tmp0[2] = alpha_u * rhov;
+        tmp1[2] = alpha_v * rhov;
 
-        tmp0[3] = alpha_u * e * detj;
-        tmp1[3] = alpha_v * e * detj;
+        tmp0[3] = alpha_u * e;
+        tmp1[3] = alpha_v * e;
     }
 
     op.dissxi(tmp.0.rho(), k.0.rho_mut());
@@ -1099,15 +1096,16 @@ fn SAT_characteristic(
         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 alpha = rho / (sbp::consts::SQRT_2 * c);
 
         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],
+            [                 1.0,                   0.0,                       alpha,                       alpha],
+            [                   u,                    ky,          alpha*(u + kx * c),          alpha*(u - kx * c)],
+            [                   v,                   -kx,          alpha*(v + ky * c),          alpha*(v - ky * c)],
+            [phi2 / (GAMMA - 1.0), rho*(ky * u - kx * v), alpha*(phi2_c2 + c * theta), alpha*(phi2_c2 - c * theta)],
         ];
         let U = kx_ * u + ky_ * v;
         let L = [
@@ -1120,7 +1118,7 @@ fn SAT_characteristic(
         #[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],
+            [   -(ky * u - kx * v)/rho,                               ky/rho,                              -kx/rho,                      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)],
         ];