SWE: more comments and update the docs

Author: laylagi

examples/fluids/index.rst

@@ -377,7 +377,9 @@ where the time derivative :math:`\bm{\dot q}` is defined by
   \bm{\dot q}(\bm q) = \sigma \bm q + \bm z
 
 in terms of :math:`\bm z` from prior state and :math:`\sigma > 0`,
-both of which depend on the specific time integration scheme. We split the implicit and explicit terms as follows:
+both of which depend on the specific time integration scheme. We split the time 
+integration scheme into implicit and explicit parts where the implicit terms is 
+given in what follows.
 
 .. math::
    :label: eq-swe-implicit-part
@@ -399,8 +401,8 @@ While for the explicit part we specify
    \bm G(t, \bm q) := 
    \left\{
          \begin{array}{l}
-             G_1 (u_{\lambda}, u_{\theta}, h) = - u_{\lambda} \frac{\partial u_{\lambda}}{\partial \alpha} - u_{\theta} \frac{\partial u_{\lambda}}{\partial \beta} - f u_{\theta}\\
-             G_2 (u_{\lambda}, u_{\theta}, h) =  - u_{\lambda} \frac{\partial u_{\theta}}{\partial \alpha} - u_{\theta} \frac{\partial u_{\theta}}{\partial \beta} + f u_{\lambda}\\
+             G_1 (u_{\lambda}, u_{\theta}, h) = - u_{\lambda} \frac{\partial u_{\lambda}}{\partial \alpha} - u_{\theta} \frac{\partial u_{\theta}}{\partial \alpha} + f u_{\theta}\\
+             G_2 (u_{\lambda}, u_{\theta}, h) =  - u_{\lambda} \frac{\partial u_{\lambda}}{\partial \beta} - u_{\theta} \frac{\partial u_{\theta}}{\partial \beta} - f u_{\lambda}\\
              G_3 (u_{\lambda}, u_{\theta}, h) = 0 .
          \end{array}
    \right.

examples/fluids/shallow-water/qfunctions/geostrophic.h

@@ -153,19 +153,21 @@ CEED_QFUNCTION(SWExplicit)(void *ctx, CeedInt Q, const CeedScalar *const *in,
     // The Physics
     // Explicit spatial terms of G_1(t,q):
     // Explicit terms multiplying v
-    // - (omega + f) * khat curl u - grad(|u|^2/2) // TODO: needs fix with weak form
-    // --  grad(|u|^2/2) = u^T grad u
-    // --  khat curl u = [-u_1, u_0]
+    // - (omega + f) * khat curl u - grad(|u|^2/2)
+    // --  omega khat curl u = 0 (u_3 = 0)
+    // --  f khat curl u = f [-u_1, u_0]
+    // --  grad(|u|^2/2) = (grad u)^T u
     v[0][i] = - wdetJ*(u[0]*du[0][0] + u[1]*du[1][0] - f*u[1]);
     // No explicit terms multiplying dv
     dv[0][0][i] = 0;
     dv[1][0][i] = 0;
 
     // Explicit spatial terms of G_2(t,q):
     // Explicit terms multiplying v
-    // - (omega + f) * khat curl u - grad(|u|^2/2) // TODO: needs fix with weak form
-    // --  grad(|u|^2/2) = u^T grad u
-    // --  khat curl u = [-u_1, u_0]
+    // - (omega + f) * khat curl u - grad(|u|^2/2)
+    // --  omega khat curl u = 0 (u_3 = 0)
+    // --  f khat curl u = f [-u_1, u_0]
+    // --  grad(|u|^2/2) = (grad u)^T u
     v[1][i] = - wdetJ*(u[0]*du[0][1] + u[1]*du[1][1] + f*u[0]);
     // No explicit terms multiplying dv
     dv[0][1][i] = 0;