mesh: implement differentiate for FVElement

src/mesh.jl

@@ -411,6 +411,27 @@ function differentiate!(du, u, mesh::Mesh1d)
     duk .*= invdetJk
   end
 end
+function differentiate!(du, u, mesh::Mesh1d{FVElement})
+  @unpack x, invdetJ = get_static_variables(mesh.cache)
+  K, = mesh.tree.dims
+  for k in 2:K-1
+    du[k] = (u[k+1]-u[k-1])/(x[k+1]-x[k-1])
+  end
+  x1, x2, x3 = x[1], x[2], x[3]
+  M = [ 1 x1 x1^2;
+        1 x2 x2^2;
+        1 x3 x3^2 ]
+  uu = u[1:3]
+  b = M \ uu
+  du[1] = b[2] + 2*b[3]*x1
+  x1, x2, x3 = x[end], x[end-1], x[end-2]
+  M = [ 1 x1 x1^2;
+        1 x2 x2^2;
+        1 x3 x3^2 ]
+  uu = u[end:-1:end-2]
+  b = M \ uu
+  du[end] = b[2] + 2*b[3]*x1
+end
 function differentiate(u, mesh)
   du = similar(u)
   differentiate!(du, u, mesh)