## Conservation Laws
**Derivative form:** $\boxed{\frac{ \partial \phi }{ \partial x }+\nabla\cdot \vec{F}=s}$
$\phi =$ conserved quantity
$\vec{F} =$ flux of conserved quantity: $\phi$ is transported through boundaries
$S =$ sink and source terms: $\phi$ is coming from outside
**Integral form:** $\frac{d}{dt}\int_{V} \phi \, dV + \int _{ CS } \vec{F} \cdot \vec{n} \, dA = \int _{V}s \, dV$
**Discretized integral form:** $\boxed{V \frac{d\bar{\phi}}{dt}+\int _{S}\vec{F}_{\phi}\cdot \vec{n} \, dA = S_{\phi}}$
$\bar{\phi} = \frac{1}{V} \int _{V} \rho \phi \, dV$
$S_{\phi} = \int _{V}s_{\phi} \, dV$
| Conservation Law | Variables | Integral Form |
| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------- |
| **Mass** <br> $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$ | ${} \phi = \rho {}$ <br> $\vec{F}_\phi = \rho \vec{u}$ <br> $s_\phi = 0$ | $V \frac{d\bar{\phi}}{dt} + \int_S \rho \vec{u} \cdot \vec{n} \, dA = 0$ |
| **Momentum** <br> $\frac{\partial u}{\partial t} + \frac{\partial (u^2)}{\partial x} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \frac{\partial^2 u}{\partial x^2}$ | $\phi = u$ <br> $\vec{F}_\phi = u^2 - \nu \frac{\partial u}{\partial x}$ <br> $s_\phi = -\frac{1}{\rho} \frac{\partial p}{\partial x}$ | ${} V \frac{d\bar{u}}{dt} + \int_S \left( u^2 - \nu \frac{\partial u}{\partial x} \right) n_x \, dA = S_{u} {}$ |
| **Energy** <br> $\frac{\partial \theta}{\partial t} = \frac{\partial}{\partial x} \left( \alpha \frac{\partial \theta}{\partial x} \right)$ | $\phi = \theta$ <br> $\vec{F}_\phi = -\alpha \frac{\partial \theta}{\partial x}$ <br> $s_\phi = 0$ | $V \frac{d\bar{\theta}}{dt} + \int_S \left( -\alpha \frac{\partial \theta}{\partial x} \right) n_x \, dA = 0$ |
### Convective Fluxes
- Aligned with flow
$\vec{F} = \rho \vec{u}\phi$
### Diffusive Fluxes
- Molecular mixing
- acts against gradient
$\vec{F} = -D \nabla \phi$
## Surface Integrals
You don't know $\phi$ everywhere on the surface, only at nodal values
- integral expressed in terms of symbolic values on the cell face
- the cell face values are interpolated from nodal values
### General Forms
Goal: estimate $F_{e} = \int _{S_{e}}f_{\phi} \, dA$
| | Accuracy | Approximation |
| ----------- | -------- | ---------------------------------------------------- |
| Midpoint | 2 | $F_{e} \approx f_{e}S_{e}$ |
| Trapezoidal | 2 | ${} F_{e} \approx S_{e} \frac{f_{ne}+f_{se}}{2} {}$ |
| Simpson's | 4 | $F_{e} \approx S_{e} \frac{f_{ne}+4f_{e}+f_{se}}{6}$ |
### Interpolation
| Method | | Accuracy | Comments |
| -------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
| UDS (upwind) | $\phi_{e}=\begin{cases}\phi_{P} & \text{if }(\vec{v}.\vec{n})_{e}>0\\\phi_{E} & \text{if }(\vec{v}.\vec{n})_{e}<0\end{cases}$ | 1st Order | - Very stable and robust<br>- Highly diffusive<br> - Best for convection-dominated flows |
| CDS (linear) | $\phi_{e}=\phi_{E}\lambda_{e}+\phi_{P}(1-\lambda_{e})$ <br>where $\lambda_{e}=\frac{x_{e}-x_{p}}{x_{E}-x_{P}}$ | 2nd Order | - Can oscillate when advection dominates <br> - Good at diffusion problems |
| QUICK | $\phi_{e} = \phi_{U}+g_{1}(\phi_{D}-\phi_{U})+g_{2}(\phi_{U}-\phi_{UU})
lt;br>For uniform grids,<br>${} \phi_{e}=\frac{6}{8}\phi_{U}+\frac{3}{8}\phi_{D}-\frac{1}{8}\phi_{UU} {}$… | 3rd Order | - Higher accuracy than CDS or UDS <br> - Some upwind bias for stability <br> - More complex stencil; may overshoot near discontinuities |
| Compact / High-order | Polynomial-based (e.g., Hermite, Padé) | | - Very high accuracy <br> - Sensitive to nonuniform grids and BC's |
## Volume Integrals
$S_{\phi} = \int _{V}s_{\phi}dV \, dx$
$\bar{\phi}=\frac{1}{V}\int _{V}\rho \phi \, dV$
#### Simplest approx:
approximate mean value of the integrand as the value at the center of the control volume
#### Higher Order Approximations:
Interpolate centers, corners, midfaces
Use those 9 points to fit the bi-quadratic shape function $s(x,y)$, where
$s(x,y) = a_{0}+a_{1}x+a_{2}y+a_{3}x^{2}+a_{4}y^{2}+a_{5}xy+a_{6}x^{2}y+a_{7}xy^{2}+a_{8}x^{2}y^{2}$
4th order approx
## FV Scheme
1. Grid generation
2. Discretize integral/conservation equation on FVs
$V\frac{d\bar{\phi}}{dt} + \int _{S}\vec{F}_{\phi}\vec{n} \, dA = S_{\phi}$
3. Solve resultant discrete integral/flux equations
![[Pasted image 20250408121026.png|375]]
### Approach 1
1. Evaluate integrals symbolically
2. Interpolate for unknowns using nodal values
$\phi_{e}$ = value at face center
$\phi_{E}$ = value at nearest node center
| Method | | Accuracy | Comments |
| ------------ | ------------------------------------------------------------------------------------------------------------------------------- | --------- | -------------------------------------------------------------- |
| UDS (upwind) | $\phi_{e}=\begin{cases}\phi_{P} & \text{if }(\vec{v}.\vec{n})_{e}>0\\\phi_{E} & \text{if }(\vec{v}.\vec{n})_{e}<0\end{cases}$ | 1st Order | No oscillations, very diffusive, for convective fluxes |
| CDS (linear) | $\phi_{e}=\phi_{E}\lambda_{e}+\phi_{P}(1-\lambda_{e})$ <br>where $\lambda_{e}=\frac{x_{e}-x_{p}}{x_{E}-x_{P}}$ | 2nd Order | centered differences<br>Can oscillate<br>For convective fluxes |
| QUICK | $\phi_{e} = \phi_{U}+g_{1}(\phi_{D}-\phi_{U})+g_{2}(\phi_{U}-\phi_{UU})$ | 3rd Order | |
### Approach 2
1. Select shape of solution within CV (piecewise approximation
$\xi = x - x_{j}$
$\phi(\xi) = a\xi^{2}+b\xi+c$
2. Impose volume averaging constraints to express coefficients in terms of nodal values
Express $a$, $b$, and $c$ in terms of $\phi_{i}$ such that:
$\frac{1}{V} \int _{V_{p}} \phi_{a_{i}}(x) \, dV = \bar{\phi}_{P}$
3. Integrate the fluxes and average the fluxes on the two sides for continuity
4. Substitute into integral equation
![[Pasted image 20250408115800.png]]
$d
\begin{bmatrix}
0 & 1 & \dots &0 & -1\\
-1 & 0 & 1 & \dots & 0\\
\dots & & & & 1 \\
1 & \dots & -1 & 0
\end{bmatrix}\begin{bmatrix}
\phi_{1} \\
\phi_{2} \\
\dots \\
\phi_{n}
\end{bmatrix} = 0
$
Main difference between finite difference and finite volume? You use an average of points for finite volume and you use exact points for finite difference
Use 4th order accurate surface/volume integrals
Piece-wise quadratic
Express $a$, $b$, and $c$ in terms of $\phi_{i}$
bruh all this is just centered differences…
why finite volume?
directly conservative
higher dimension
*irregular grids!*