2D Heat Equation Solver

Governing PDE

We are solving the 2D heat equation:

$$ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) $$

where:

• $T(x, y, t)$ is the temperature field
• $\alpha$ is the thermal diffusivity
• $x, y$ are spatial coordinates
• $t$ is time

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Discretization

We discretize the domain using a uniform grid:

• Grid spacing: $h_x = \Delta x$, $h_y = \Delta y$
• Time step: $\Delta t$
• Indices: $i$ for $x$, $j$ for $y$, $n$ for time

For uniform grid spacing ($h_x = h_y = h$), the discretized explicit scheme is:

$$ T_{i,j}^{n+1} = T_{i,j}^n + \lambda \Big( T_{i+1,j}^n + T_{i-1,j}^n + T_{i,j+1}^n + T_{i,j-1}^n - 4T_{i,j}^n \Big) $$

where:

$$ \lambda = \frac{\alpha , \Delta t}{h^2} $$

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Stability Condition

For the explicit 2D scheme, stability requires:

$$ \lambda \leq \frac{1}{4} $$

which gives:

$$ \Delta t \leq \frac{h^2}{4 \alpha} $$

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Implementation Parameters

From the given configuration:

• $N_x = 100$, $N_y = 100$
• $\Delta x = \Delta y = 0.1 ; (h = 0.1)$
• $\alpha = 0.1$
• $\Delta t = 0.01$

Compute $\lambda$:

$$ \lambda = \frac{0.1 \times 0.01}{0.1^2} = \frac{0.001}{0.01} = 0.1 $$

This satisfies the stability condition since $0.1 < 0.25$.

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Boundary and Initial Conditions

• Initial condition: hot spot at center $(50, 50)$ with $T = 100$
• Boundary conditions: Dirichlet fixed temperatures
  • North, South, East, West = $0$