Correct Finite Volume Integration of a Tensor Field on a Manifold - Example
by Christopher Moore
Let's look at the (0,0)-degree scalar case. Remember, no state indexes means no parallel propagators.
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
-
\Sigma_{k=1}^n
\left(
\int_{x^1_L}^{x^1_R}
\overset{-\{x^k\}}{...}
\int_{x^n_L}^{x^n_R}
\frac{1}{\mathcal{V}(x_C)}
\left(
J(x^k_R) {e_k}^\bar{k}(x^k_R) F^k(x^k_R)
- J(x^k_L) {e_k}^\bar{k}(x^k_L) F^k(x^k_L)
\right)
dx^n
\overset{-\{x^k\}}{...}
dx^1
\right)
+ S(x_C)
\right)$
Notice the ${e_k}^{\bar{k}} F^k$ term. Remember the ordinary letters are an arbitrary basis while the barred letters are a coordinate basis.
This tells us we should be calculating our flux with respect to a coordinate basis.
If you want to calculate the flux using your non-coordinate basis then you will have to re-transform it back to the coordinate basis.
For example, let's say you're using a polar grid. If you want to use coordinate basis then you'll have to scale and unscale your $\phi$ component by r, but you can calculate your flux without any extra manipulation.
If you want to use an orthonormal basis then you no longer need to worry about coordinate scaling, but you have to be sure to rescale the $\phi$ flux by $\frac{1}{r}$ so that it is a component of the coordinate basis.
If you want to calculate your flux in a global Cartesian basis then your ${e_k}^{\bar{k}}$ becomes the inverse of the polar coordinate basis coefficients in terms of the global Cartesian basis (a rotation and a scale of $\phi$ by $r$),
and you will have to undo that - by calculating $F^\bar{k} = {e_k}^{\bar{k}} F^k$ which is a linear combination of the $F^x$ and $F^y$ terms.
You will need to calculate both the $F^x$ and $F^y$ flux at $r_L$ and $r_R$ and then linearly transform these into the $F^\bar{r}$ flux.
So finite volume integration can be implemented with volume weighting across your cell sides (and transforming between your coordinate and non-coordinate basis).
Just like in 1987 Thompson "Lecture Series in Computational Fluid Dynamics" ... somewhere? Section 4, equation 4.127 maybe?
Let's look at specific grids for the scalar case. How about Cartesian for starters. $J(x) = 1$ everywhere so $\mathcal{V}(x) = \Pi_{k=1}^n (x^k_R - x^k_L)$.
Our basis is a coordinate basis so ${e_k}^{\bar{k}} = \delta_k^{\bar{k}}$.
Let $\Delta f = f_R - f_L$.
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
-
\Sigma_{k=1}^n
\left(
\frac{\Pi_{j=1\\j \ne k}^n (\Delta x^j)}{\Pi_{j=1}^n (\Delta x^j)}
\left(
F^k(x^k_R)
- F^k(x^k_L)
\right)
\right)
+ S(x_C)
\right)$
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
-
\Sigma_{k=1}^n
\left(
\frac{1}{(\Delta x^k)}
\left(
F^k(x^k_R)
- F^k(x^k_L)
\right)
\right)
+ S(x_C)
\right)$
This looks like the typical finite volume equation.
How about cylinder coordinate basis?
$e_r = \partial_r, e_\phi, e_z = \partial_\phi, g_{rr} = g_{zz} = 1, g_{\phi\phi} = r^2, J = r,
\mathcal{V} = \frac{1}{2} ( (r_R)^2 - (r_L)^2 ) (\phi_R - \phi_L) (z_R - z_L) = \frac{1}{2} \Delta (r^2) \Delta \phi \Delta z$.
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
- \left(
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\frac{1}{\mathcal{V}(x_C)}
\left(
J(r_R, \phi, z) F^r(r_R, \phi, z)
- J(r_L, \phi, z) F^r(r_L, \phi, z)
\right)
dz d\phi
\right)
- \left(
\int_{r_L}^{r_R}
\int_{z_L}^{z_R}
\frac{1}{\mathcal{V}(x_C)}
\left(
J(r, \phi_R, z) F^\phi(r, \phi_R, z)
- J(r, \phi_L, z) F^\phi(r, \phi_L, z)
\right)
dr dz
\right)
- \left(
\int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\frac{1}{\mathcal{V}(x_C)}
\left(
J(r, \phi, z_R) F^z(r, \phi, z_R)
- J(r, \phi, z_L) F^z(r, \phi, z_L)
\right)
d\phi dr
\right)
+ S(x_C)
\right)$
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
- \left(
\Delta \phi \Delta z
\frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
r_R F^r(r_R)
- r_L F^r(r_L)
\right)
\right)
- \left(
\frac{1}{2} \Delta (r^2) \Delta z
\frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
F^\phi(\phi_R)
- F^\phi(\phi_L)
\right)
\right)
- \left(
\Delta \phi \frac{1}{2} \Delta (r^2)
\frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
F^z(z_R)
- F^z(z_L)
\right)
\right)
+ S(x_C)
\right)$
$
u(x_C, t_R)
= u(x_C, t_L)
+ \Delta t \left(
- \frac{r_R F^r(r_R) - r_L F^r(r_L)}{\frac{1}{2} \Delta (r^2)}
- \frac{F^\phi(\phi_R) - F^\phi(\phi_L)}{\Delta \phi}
- \frac{F^z(z_R) - F^z(z_L)}{\Delta z}
+ S(x_C)
\right)$
Alright, how about the vector case?
Now we have to use our connections: ${\Gamma^r}_{\phi\phi} = -r, {\Gamma^\phi}_{\phi r} = {\Gamma^\phi}_{r \phi} = \frac{1}{r}$
...and our propagators: $P(x_L, x_R) = S(1, \frac{1}{r_R}) \cdot R_z(\phi_L - \phi_R) \cdot S(1, r_L)$
so $P_r(r_L, r_R) = S(1, \frac{r_L}{r_R})$
and $P_\phi(\phi_L, \phi_R) = S(1, \frac{1}{r}) \cdot R_z(\phi_L - \phi_R) \cdot S(1, r)$
and $P_z(z_L, z_R) = I$
$
u^l(x_C, t_R)
= u^l(x_C, t_L)
+ \Delta t \left(
\\
- \frac{1}{\mathcal{V}(x_C)}
\Sigma_{k=1}^n
\left(
\int_{x^1_L}^{x^1_R}
\overset{-\{x^k\}}{...}
\int_{x^n_L}^{x^n_R}
\left(
{P^l}_i(x^k_R,x_C)
J(x^k_R) F^{ki}(x^k_R)
-
{P^l}_i(x^k_L,x_C)
J(x^k_L) F^{ki}(x^k_L)
\right)
dx^n
\overset{-\{x^k\}}{...}
dx^1
\\
-
\int_{x^1_L}^{x^1_R}
...
\int_{x^n_L}^{x^n_R}
\partial_\bar{k}
\left(
{P^l}_i(x,x_C)
\right)
J(x) F^{ki}(x)
dx^n ... dx^1
\\
+
\int_{x^1_L}^{x^1_R}
...
\int_{x^n_L}^{x^n_R}
\left(
{P^l}_i(x,x_C)
J(x) \left(
F^{km}(x) {\Gamma^i}_{km}(x)
\right)
\right)
dx^n ... dx^1
\right)
\\
+ S^l(x_C)
\right)$
$
u^l(x_C, t_R)
= u^l(x_C, t_L)
+ \Delta t \left(
\\
- \frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
{P^l}_i(r_R,x_C)
r_R F^{ r i}(r_R)
-
{P^l}_i(r_L,x_C)
r_L F^{ r i}(r_L)
\right)
d\phi dz
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\partial_\bar{ r }
\left(
{P^l}_i(x,x_C)
\right)
r F^{ r i}(x)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
{P^l}_i(x,x_C)
r \left(
F^{ r m}(x) {\Gamma^i}_{ r m}(x)
\right)
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{z_L}^{z_R}
\left(
{P^l}_i(x^\phi _R,x_C)
r F^{\phi i}(x^\phi _R)
-
{P^l}_i(x^\phi _L,x_C)
r F^{\phi i}(x^\phi _L)
\right)
dz dr
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\partial_\bar{\phi }
\left(
{P^l}_i(x,x_C)
\right)
r F^{\phi i}(x)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
{P^l}_i(x,x_C)
r \left(
F^{\phi m}(x) {\Gamma^i}_{\phi m}(x)
\right)
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left(
{P^l}_i(x^ z _R,x_C)
r F^{ z i}(x^ z _R)
-
{P^l}_i(x^ z _L,x_C)
r F^{ z i}(x^ z _L)
\right)
d\phi dr
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\partial_\bar{ z }
\left(
{P^l}_i(x,x_C)
\right)
r F^{ z i}(x)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
{P^l}_i(x,x_C)
r \left(
F^{ z m}(x) {\Gamma^i}_{ z m}(x)
\right)
\right)
dz d\phi dr
\right)
\\
+ S^l(x_C)
\right)$
Alright, now time to switch to matrix notation. Notice ${P^i}_j = \downarrow_i \overset{\rightarrow j}{[{P^i}_j]}$ and $F^{ij} = \downarrow_i \overset{\rightarrow j}{[F^{ij}]}$.
Notice I had to break convention with my matrix representation of F, since it has two contravariant indexes, so one of them will have to be a row.
Notice however that most the uses of F are with one index fixed, so it would be used in context of a column vector regardless.
$
u^l(x_C, t_R)
= u^l(x_C, t_L)
+ \Delta t \left(
\\
- \frac{1}{\mathcal{V}(x_C)}
\Sigma_{k=1}^n
\left(
\int_{x^1_L}^{x^1_R}
\overset{-\{x^k\}}{...}
\int_{x^n_L}^{x^n_R}
\left(
{P^l}_i(x^k_R,x_C)
J(x^k_R) F^{ki}(x^k_R)
-
{P^l}_i(x^k_L,x_C)
J(x^k_L) F^{ki}(x^k_L)
\right)
dx^n
\overset{-\{x^k\}}{...}
dx^1
\\
-
\int_{x^1_L}^{x^1_R}
...
\int_{x^n_L}^{x^n_R}
\partial_\bar{k}
\left(
{P^l}_i(x,x_C)
\right)
J(x) F^{ki}(x)
dx^n ... dx^1
\\
+
\int_{x^1_L}^{x^1_R}
...
\int_{x^n_L}^{x^n_R}
\left(
{P^l}_i(x,x_C)
J(x) \left(
F^{km}(x) {\Gamma^i}_{km}(x)
\right)
\right)
dx^n ... dx^1
\right)
\\
+ S^l(x_C)
\right)$
$
\left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_R)}
= \left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_L)}
+ \Delta t \left(
\\
- \frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r_R)
r_R
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_R)}
-
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r_L)
r_L
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_L)}
\right)
d\phi dz
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\partial_\bar{ r }
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
\right)
r
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(x)}
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
r
\left[\begin{matrix}
0 \\
F^{r\phi} {\Gamma^\phi}_{r\phi} \\
0
\end{matrix}\right]_{(x)}
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{z_L}^{z_R}
\left(
S(1, \frac{1}{r_C})
R_z(\phi_R - \phi_C)
S(1, r)
r
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_R)}
-
S(1, \frac{1}{r_C})
R_z(\phi_L - \phi_C)
S(1, r)
r
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_L)}
\right)
dz dr
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\partial_\bar{\phi }
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
\right)
r
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(x)}
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
r
\left[\begin{matrix}
F^{\phi\phi} {\Gamma^r}_{\phi\phi} \\
F^{\phi r} {\Gamma^\phi}_{\phi r} \\
0
\end{matrix}\right]_{(x)}
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left(
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
r
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_R)}
-
S(1, \frac{1}{r_C})
R_z(\phi - \phi_C)
S(1, r)
r
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_L)}
\right)
d\phi dr
\right)
\\
+ \left[\begin{matrix}
S^r \\ S^\phi \\ S^z
\end{matrix}\right]_{(x_C)}
\right)$
Here I'm going to assume the fluxes integrated across the volume are constant and equal to the volume center, and therefore do not need to be propagated to the center of the volume.
Likewise the fluxes integrated across the surface are constant and equal to the surface center.
$
\left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_R)}
= \left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_L)}
+ \Delta t \left(
\\
- \frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
r_R
\left[\begin{matrix}
cos(\phi - \phi_C) & -r_R sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r_R}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_R)}
-
r_L
\left[\begin{matrix}
cos(\phi - \phi_C) & -r_L sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r_L}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_L)}
\right)
d\phi dz
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
r
\partial_\bar{ r }
\left(
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\right)
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(x)}
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
r
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
0 \\
F^{r\phi} {\Gamma^\phi}_{r\phi} \\
0
\end{matrix}\right]_{(x)}
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{z_L}^{z_R}
\left(
r
\left[\begin{matrix}
cos(\phi_R - \phi_C) & -r sin(\phi_R - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi_R - \phi_C) & \frac{r}{r_C} cos(\phi_R - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_R)}
-
r
\left[\begin{matrix}
cos(\phi_L - \phi_C) & -r sin(\phi_L - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi_L - \phi_C) & \frac{r}{r_C} cos(\phi_L - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_L)}
\right)
dz dr
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
r
\partial_\bar{\phi }
\left(
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\right)
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(x)}
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\int_{z_L}^{z_R}
\left(
r
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi\phi} {\Gamma^r}_{\phi\phi} \\
F^{\phi r} {\Gamma^\phi}_{\phi r} \\
0
\end{matrix}\right]_{(x)}
\right)
dz d\phi dr
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left(
r
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_R)}
-
r
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_L)}
\right)
d\phi dr
\right)
\\
+ \left[\begin{matrix}
S^r \\ S^\phi \\ S^z
\end{matrix}\right]_{(x_C)}
\right)$
$
\left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_R)}
= \left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_L)}
+ \Delta t \left(
\\
- \frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
\int_{\phi_L}^{\phi_R}
\left(
r_R
\left[\begin{matrix}
cos(\phi - \phi_C) & -r_R sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r_R}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_R)}
-
r_L
\left[\begin{matrix}
cos(\phi - \phi_C) & -r_L sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r_L}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_L)}
\right)
d\phi \Delta z
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left[\begin{matrix}
0 & -r sin(\phi - \phi_C) & 0 \\
0 & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 0
\end{matrix}\right]
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(x)}
d\phi dr \Delta z
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
0 \\
F^{r\phi} \\
0
\end{matrix}\right]_{(x)}
d\phi dr \Delta z
\\
+ \int_{r_L}^{r_R}
\left(
\left[\begin{matrix}
r cos(\phi_R - \phi_C) & -r^2 sin(\phi_R - \phi_C) & 0 \\
r \frac{1}{r_C} sin(\phi_R - \phi_C) & r^2 \frac{1}{r_C} cos(\phi_R - \phi_C) & 0 \\
0 & 0 & r
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_R)}
-
\left[\begin{matrix}
r cos(\phi_L - \phi_C) & -r^2 sin(\phi_L - \phi_C) & 0 \\
r \frac{1}{r_C} sin(\phi_L - \phi_C) & r^2 \frac{1}{r_C} cos(\phi_L - \phi_C) & 0 \\
0 & 0 & r
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_L)}
\right)
dr \Delta z
\\
- \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left[\begin{matrix}
-r sin(\phi - \phi_C) & -r^2 cos(\phi - \phi_C) & 0 \\
r \frac{1}{r_C} cos(\phi - \phi_C) & -r^2 \frac{1}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & 0
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(x)}
d\phi dr \Delta z
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left[\begin{matrix}
cos(\phi - \phi_C) & -r sin(\phi - \phi_C) & 0 \\
\frac{1}{r_C} sin(\phi - \phi_C) & \frac{r}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
-r^2 F^{\phi\phi} \\
F^{\phi r} \\
0
\end{matrix}\right]_{(x)}
d\phi dr \Delta z
\\
+ \int_{r_L}^{r_R}
\int_{\phi_L}^{\phi_R}
\left(
\left[\begin{matrix}
r cos(\phi - \phi_C) & -r^2 sin(\phi - \phi_C) & 0 \\
r \frac{1}{r_C} sin(\phi - \phi_C) & r^2 \frac{1}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_R)}
-
\left[\begin{matrix}
r cos(\phi - \phi_C) & -r^2 sin(\phi - \phi_C) & 0 \\
r \frac{1}{r_C} sin(\phi - \phi_C) & r^2 \frac{1}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 1
\end{matrix}\right]
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_L)}
\right)
d\phi dr
\right)
\\
+ \left[\begin{matrix}
S^r \\ S^\phi \\ S^z
\end{matrix}\right]_{(x_C)}
\right)$
$
\left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_R)}
= \left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_L)}
+ \Delta t \left(
\\
- \frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi \Delta z}
\left(
\left(
r_R
\left[\begin{matrix}
sin(\phi - \phi_C) & r_R cos(\phi - \phi_C) & 0 \\
-\frac{1}{r_C} cos(\phi - \phi_C) & \frac{r_R}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & \phi
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_R)}
-
r_L
\left[\begin{matrix}
sin(\phi - \phi_C) & r_L cos(\phi - \phi_C) & 0 \\
-\frac{1}{r_C} cos(\phi - \phi_C) & \frac{r_L}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & \phi
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
F^{rr} \\ F^{r\phi} \\ F^{rz}
\end{matrix}\right]_{(r_L)}
\right)
\Delta z
\\
- \frac{1}{2} \Delta (r^2)
\left[\begin{matrix}
cos(\phi - \phi_C) F^{r\phi}(x) \\
\frac{1}{r_C} sin(\phi - \phi_C) F^{r\phi}(x) \\
0
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\Delta z
\\
+ \left[\begin{matrix}
\Delta r sin(\phi - \phi_C) & \frac{1}{2} \Delta (r^2) cos(\phi - \phi_C) & 0 \\
-\Delta r \frac{1}{r_C} cos(\phi - \phi_C) & \frac{1}{2} \Delta (r^2) \frac{1}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & \Delta r \Delta \phi
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
0 \\
F^{r\phi} \\
0
\end{matrix}\right]_{(x)}
\Delta z
\\
+ \left(
\left[\begin{matrix}
\frac{1}{2} \Delta (r^2) cos(\phi_R - \phi_C) & -\frac{1}{3} \Delta (r^3) sin(\phi_R - \phi_C) & 0 \\
\frac{1}{2} \Delta (r^2) \frac{1}{r_C} sin(\phi_R - \phi_C) & \frac{1}{3} \Delta (r^3) \frac{1}{r_C} cos(\phi_R - \phi_C) & 0 \\
0 & 0 & \frac{1}{2} \Delta (r^2)
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_R)}
-
\left[\begin{matrix}
\frac{1}{2} \Delta (r^2) cos(\phi_L - \phi_C) & -\frac{1}{3} \Delta (r^3) sin(\phi_L - \phi_C) & 0 \\
\frac{1}{2} \Delta (r^2) \frac{1}{r_C} sin(\phi_L - \phi_C) & \frac{1}{3} \Delta (r^3) \frac{1}{r_C} cos(\phi_L - \phi_C) & 0 \\
0 & 0 & \frac{1}{2} \Delta (r^2)
\end{matrix}\right]
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(\phi_L)}
\right)
\Delta z
\\
- \left[\begin{matrix}
\frac{1}{2} \Delta (r^2) cos(\phi - \phi_C) & -\frac{1}{3} \Delta (r^3) sin(\phi - \phi_C) & 0 \\
\frac{1}{2} \Delta (r^2) \frac{1}{r_C} sin(\phi - \phi_C) & \frac{1}{3} \Delta (r^3) \frac{1}{r_C} cos(\phi - \phi_C) & 0 \\
0 & 0 & 0
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
F^{\phi r} \\ F^{\phi\phi} \\ F^{\phi z}
\end{matrix}\right]_{(x)}
\Delta z
\\
+ \left[\begin{matrix}
\frac{1}{3} \Delta (r^3) sin(\phi - \phi_C) F^{\phi\phi}(x) + \frac{1}{2} \Delta (r^2) cos(\phi - \phi_C) F^{\phi r}(x) \\
-\frac{1}{3} \Delta (r^3) \frac{1}{r_C} cos(\phi - \phi_C) F^{\phi\phi}(x) + \frac{1}{2} \Delta (r^2) \frac{1}{r_C} sin(\phi - \phi_C) F^{\phi r}(x) \\
0
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\Delta z
\\
+ \left(
\left[\begin{matrix}
\frac{1}{2} \Delta (r^2) sin(\phi - \phi_C) & \frac{1}{3} \Delta (r^3) cos(\phi - \phi_C) & 0 \\
-\frac{1}{2} \Delta (r^2) \frac{1}{r_C} cos(\phi - \phi_C) & \frac{1}{3} \Delta (r^3) \frac{1}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & \Delta r \Delta \phi
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_R)}
-
\left[\begin{matrix}
\frac{1}{2} \Delta (r^2) sin(\phi - \phi_C) & \frac{1}{3} \Delta (r^3) cos(\phi - \phi_C) & 0 \\
-\frac{1}{2} \Delta (r^2) \frac{1}{r_C} cos(\phi - \phi_C) & \frac{1}{3} \Delta (r^3) \frac{1}{r_C} sin(\phi - \phi_C) & 0 \\
0 & 0 & \Delta r \Delta \phi
\end{matrix}\right]|_{\phi_L}^{\phi_R}
\left[\begin{matrix}
F^{zr} \\ F^{z\phi} \\ F^{zz}
\end{matrix}\right]_{(z_L)}
\right)
\right)
\\
+ \left[\begin{matrix}
S^r \\ S^\phi \\ S^z
\end{matrix}\right]_{(x_C)}
\right)$
$
\left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_R)}
= \left[\begin{matrix}
u^r \\ u^\phi \\ u^z
\end{matrix}\right]_{(x_C, t_L)}
+ \Delta t \left(
\\
-
\frac{1}{\frac{1}{2} \Delta (r^2) \Delta \phi}
\left[\begin{matrix}
(sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) (r_R F^{rr}(r_R) - r_L F^{rr}(r_L))
+ (cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) ((r_R)^2 F^{r\phi}(r_R) - (r_L)^2 F^{r\phi}(r_L)) \\
-\frac{1}{r_C} (cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) (r_R F^{rr}(r_R) - r_L F^{rr}(r_L))
+ \frac{1}{r_C} (sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) ((r_R)^2 F^{r\phi}(r_R) - (r_L)^2 F^{r\phi}(r_L)) \\
\Delta \phi (r_R F^{rz}(r_R) - r_L F^{rz}(r_L))
\end{matrix}\right]
\\
+
\frac{1}{\Delta \phi}
\left[\begin{matrix}
cos(\phi_R - \phi_C) F^{r\phi}(\phi_R) - cos(\phi_L - \phi_C) F^{r\phi}(\phi_L) \\
\frac{1}{r_C} (sin(\phi_R - \phi_C) F^{r\phi}(\phi_R) - sin(\phi_L - \phi_C) F^{r\phi}(\phi_L)) \\
0
\end{matrix}\right]
\\
-
\frac{1}{\Delta \phi}
\left[\begin{matrix}
(cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) F^{r\phi}(x_C) \\
\frac{1}{r_C} (sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) F^{r\phi}(x_C) \\
0
\end{matrix}\right]
\\
-
\frac{1}{\Delta \phi}
\left[\begin{matrix}
cos(\phi_R - \phi_C) F^{\phi r}(\phi_R) - \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} sin(\phi_R - \phi_C) F^{\phi\phi}(\phi_R) \\
\frac{1}{r_C} sin(\phi_R - \phi_C) F^{\phi r}(\phi_R) + \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} \frac{1}{r_C} cos(\phi_R - \phi_C) F^{\phi\phi}(\phi_R) \\
F^{\phi z}(\phi_R)
\end{matrix}\right]
\\
+
\frac{1}{\Delta \phi}
\left[\begin{matrix}
cos(\phi_L - \phi_C) F^{\phi r}(\phi_L) - \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} sin(\phi_L - \phi_C) F^{\phi\phi}(\phi_L) \\
\frac{1}{r_C} sin(\phi_L - \phi_C) F^{\phi r}(\phi_L) + \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} \frac{1}{r_C} cos(\phi_L - \phi_C) F^{\phi\phi}(\phi_L) \\
F^{\phi z}(\phi_L)
\end{matrix}\right]
\\
-
\frac{4}{3} \frac{\Delta (r^3)}{\Delta (r^2) \Delta \phi}
\left[\begin{matrix}
(sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) F^{\phi\phi}(x_C) \\
-\frac{1}{r_C} (cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) F^{\phi\phi}(x_C) \\
0
\end{matrix}\right]
\\
-
\frac{1}{\Delta \phi \Delta z}
\left[\begin{matrix}
(sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) (F^{zr}(z_R) - F^{zr}(z_L)) + \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} (cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) (F^{z\phi}(z_R) - F^{z\phi}(z_L)) \\
-\frac{1}{r_C} (cos(\phi_R - \phi_C) - cos(\phi_L - \phi_C)) (F^{zr}(z_R) - F^{zr}(z_L)) + \frac{2}{3} \frac{\Delta (r^3)}{\Delta (r^2)} \frac{1}{r_C} (sin(\phi_R - \phi_C) - sin(\phi_L - \phi_C)) (F^{z\phi}(z_R) - F^{z\phi}(z_L)) \\
2 \frac{\Delta r}{\Delta (r^2)} \Delta \phi (F^{zz}(z_R) - F^{zz}(z_L))
\end{matrix}\right]
\\
+ \left[\begin{matrix}
S^r \\ S^\phi \\ S^z
\end{matrix}\right]_{(x_C)}
\right)$