# Finite Difference and Finite Volume Discretization for the 2D Diffusion Equation We assume the 2D diffusion equation for scalar $\phi$: \[ \frac{\partial \phi}{\partial t} = D \left(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}\right) \] where $D$ is the diffusion coefficient. Let: - Grid spacings: $\Delta x$, $\Delta y$ - Time step: $\Delta t$ - $\phi_{i,j}^n$ = value at grid point $(i, j)$ and time $n$ We will derive: 1. **Linear algebraic equations for the interior grid point A and boundary grid point B using finite differences** 2. **Linear algebraic equations for the same, but using the finite volume approach** --- ## 1. **Finite Difference Discretization** ### **A. Interior Point (A)** **Time Derivative (Explicit Euler):** \[ \frac{\partial \phi}{\partial t} \approx \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} \] **Spatial Derivatives (Central Difference):** - $\dfrac{\partial^2 \phi}{\partial x^2} \approx \dfrac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2}$ - $\dfrac{\partial^2 \phi}{\partial y^2} \approx \dfrac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2}$ **Full Discretized Equation:** \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = D \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] **Linear Algebraic Form (for $\phi_{i,j}^{n+1}$):** \[ \phi_{i,j}^{n+1} = \phi_{i,j}^n + D \Delta t \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] --- ### **B. Boundary Point (B)** Assume $B$ is on a boundary at $i=$ (e.g., left edge). Suppose Dirichlet BC: $\phi_{,j}^n = \phi_{BC}$ (known value). - The linear equation is simply: \[ \phi_{,j}^{n+1} = \phi_{BC} \] If Neumann BC (e.g., $\frac{\partial \phi}{\partial x}|_{i=} = $): - Approximate: $\frac{\phi_{1,j}^n - \phi_{-1,j}^n}{2\Delta x} = \implies \phi_{-1,j}^n = \phi_{1,j}^n$ - For $i=$ in the finite difference equation, replace $\phi_{-1,j}^n$ with $\phi_{1,j}^n$. --- ## 2. **Finite Volume Discretization (Central Difference on Fluxes)** ### **A. Interior Point (A)** - The control volume around $(i,j)$ has face fluxes at $i\pm \frac{1}{2}$ and $j\pm \frac{1}{2}$. - Discretize as: \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = \frac{F_{i+\frac{1}{2},j}^n - F_{i-\frac{1}{2},j}^n}{\Delta x} + \frac{F_{i,j+\frac{1}{2}}^n - F_{i,j-\frac{1}{2}}^n}{\Delta y} \] where $F$ is the diffusive flux at the face. **Central Difference for Face Fluxes:** \[ F_{i+\frac{1}{2},j}^n = -D \frac{\phi_{i+1,j}^n - \phi_{i,j}^n}{\Delta x} \] \[ F_{i-\frac{1}{2},j}^n = -D \frac{\phi_{i,j}^n - \phi_{i-1,j}^n}{\Delta x} \] **Substitute into the finite volume balance:** \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = \frac{1}{\Delta x} \left[ - D \frac{\phi_{i+1,j}^n - \phi_{i,j}^n}{\Delta x} + D \frac{\phi_{i,j}^n - \phi_{i-1,j}^n}{\Delta x} \right] + \text{(same in $y$)} \] \[ = D \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] > **Result:** The finite volume discretization with central differences for diffusive fluxes gives the **same algebraic form** as the finite difference method for interior points. --- ### **B. Boundary Point (B)** #### **Dirichlet BC:** - Control volume at the boundary is partial, but the face at the boundary is set by the BC. - Equation is just $\phi_{B}^{n+1} = \phi_{BC}$. #### **Neumann BC:** - If $\left. \frac{\partial \phi}{\partial x}\right|_{B} = $ at $i=$, then - The flux at the boundary face is zero: $F_{i-\frac{1}{2},j}=$. - The update equation: \[ \frac{\phi_{,j}^{n+1} - \phi_{,j}^n}{\Delta t} = \frac{F_{1/2,j}^n}{\Delta x} + \text{(y terms)} \] With $F_{1/2,j}^n = -D \frac{\phi_{1,j}^n - \phi_{,j}^n}{\Delta x}$: \[ \frac{\phi_{,j}^{n+1} - \phi_{,j}^n}{\Delta t} = -D \frac{\phi_{1,j}^n - \phi_{,j}^n}{\Delta x^2} + \text{(y terms)} \] --- ## **Summary Table** | Point | Discretization | Algebraic Equation | |-----------|---------------|--------------------| | **A (Interior)** | FD/FV | $\phi_{i,j}^{n+1} = \phi_{i,j}^n + D \Delta t \left[\frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2}\right]$ | | **B (Boundary, Dirichlet)** | FD/FV | $\phi_{B}^{n+1} = \phi_{BC}$ | | **B (Boundary, Neumann)** | FV | $\phi_{B}^{n+1} = \phi_{B}^n + D\Delta t \left[ -\frac{\phi_{1,j}^n - \phi_{,j}^n}{\Delta x^2} + \text{(y terms)}\right]$ | --- ## **Final Answer** **For both finite difference and finite volume (with central differences for fluxes), the linear algebraic equations for interior and boundary points are as above.** - **Interior point equation** is a standard 5-point stencil update. - **Boundary point equation** is determined by the boundary condition (Dirichlet or Neumann). - **Finite volume and finite difference schemes are algebraically equivalent for uniform grids and central differences.** --- **Note:** If you provide "Figure 2" or further details about the grid (e.g., locations of A and B), the equations can be written for specific indices. --- **Example schematic of grid:**  *Alt text: Illustration of a 5-point stencil for a 2D grid, showing a central interior point and its four neighbors*

# Step-by-Step Solution for 2D Diffusion Equation Discretization ## Problem Statement We will derive linear algebraic equations for an interior grid point \( A \) and a boundary grid point \( B \) using: 1. Finite Difference Method (FDM) 2. Finite Volume Method (FVM) Assuming uniform grid spacing \( \Delta x \) and \( \Delta y \) in the x and y directions respectively, and uniform time stepping \( \Delta t \). The governing equation is: \[ \frac{\partial \phi}{\partial t} = D \left(\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}\right) \] Where \( D \) is the diffusion coefficient. ## 1. Finite Difference Discretization ### A. Interior Point (A) **Time Derivative (Explicit Euler):** \[ \frac{\partial \phi}{\partial t} \approx \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} \] **Spatial Derivatives (Central Difference):** - For \( x \): \[ \frac{\partial^2 \phi}{\partial x^2} \approx \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} \] - For \( y \): \[ \frac{\partial^2 \phi}{\partial y^2} \approx \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \] **Full Discretized Equation:** \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = D \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] **Linear Algebraic Form for \( \phi_{i,j}^{n+1} \):** \[ \phi_{i,j}^{n+1} = \phi_{i,j}^n + D \Delta t \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] ### B. Boundary Point (B) Assuming \( B \) is on a boundary (e.g., left edge): - **Dirichlet Boundary Condition**: Assume \( \phi_{BC} \) is a known value. \[ \phi_{B}^{n+1} = \phi_{BC} \] - **Neumann Boundary Condition**: \[ \frac{\partial \phi}{\partial x}\bigg|_{i=B} \approx \frac{\phi_{1,j}^n - \phi_{B,j}^n}{\Delta x} = 0 \implies \phi_{1,j}^n = \phi_{B,j}^n \] ## 2. Finite Volume Discretization ### A. Interior Point (A) The control volume around \( (i,j) \): \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = \frac{F_{i+\frac{1}{2},j}^n - F_{i-\frac{1}{2},j}^n}{\Delta x} + \frac{F_{i,j+\frac{1}{2}}^n - F_{i,j-\frac{1}{2}}^n}{\Delta y} \] **Central Difference for Face Fluxes:** \[ F_{i+\frac{1}{2},j}^n = -D \frac{\phi_{i+1,j}^n - \phi_{i,j}^n}{\Delta x}, \quad F_{i-\frac{1}{2},j}^n = -D \frac{\phi_{i,j}^n - \phi_{i-1,j}^n}{\Delta x} \] **Substituting into the finite volume balance:** \[ \frac{\phi_{i,j}^{n+1} - \phi_{i,j}^n}{\Delta t} = D \left[ \frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2} \right] \] ### B. Boundary Point (B) #### Dirichlet Boundary Condition: - Control volume is incomplete, but the value is set by the boundary condition: \[ \phi_{B}^{n+1} = \phi_{BC} \] #### Neumann Boundary Condition: - For the boundary face: \[ \frac{\partial \phi}{\partial x}\bigg|_{B} = 0 \implies F_{i-\frac{1}{2},j} = 0 \] Thus: \[ \frac{\phi_{B}^{n+1} - \phi_{B}^n}{\Delta t} = -D \frac{\phi_{1,j}^n - \phi_{B,j}^n}{\Delta x^2} \] ## Summary Table | Point | Method | Algebraic Equation | |-----------|-------------------|--------------------| | **A (Interior)** | Finite Difference / Finite Volume | \(\phi_{i,j}^{n+1} = \phi_{i,j}^n + D \Delta t \left[\frac{\phi_{i+1,j}^n - 2\phi_{i,j}^n + \phi_{i-1,j}^n}{\Delta x^2} + \frac{\phi_{i,j+1}^n - 2\phi_{i,j}^n + \phi_{i,j-1}^n}{\Delta y^2}\right]\) | | **B (Boundary, Dirichlet)** | Finite Difference / Finite Volume | \(\phi_{B}^{n+1} = \phi_{BC}\) | | **B (Boundary, Neumann)** | Finite Volume | \(\phi_{B}^{n+1} = \phi_{B}^n - D \Delta t \frac{\phi_{1,j}^n - \phi_{B,j}^n}{\Delta x^2}\) | ## Final Answer The derived equations provide a framework for solving the 2D diffusion equation using both finite difference and finite volume methods for both interior and boundary points. The approaches yield similar algebraic forms under the specified conditions.