2-dimensional Heat Conduction Problem

L23: 2D Heat Conduction using FDM

Lecture 23: Two-Dimensional Heat Conduction Using Finite Difference Method (FDM)

Introduction

Two-dimensional heat conduction problems aim to determine the temperature distribution \( T(X, Y, t) \) over time in a region. The governing PDE is:

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

Where:

  • \( T \): Temperature
  • \( \alpha \): Thermal diffusivity
  • \( X, Y \): Spatial coordinates
  • \( t \): Time

Finite Difference Representation

The domain is discretized into a grid, and partial derivatives are approximated using finite differences. Grid points are labeled \( (i, j) \), with superscript \( n \) for time level.

1. Simple Explicit Method

$$ \frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t} = \alpha \left( \frac{T_{i+1,j}^n - 2T_{i,j}^n + T_{i-1,j}^n}{(\Delta X)^2} + \frac{T_{i,j+1}^n - 2T_{i,j}^n + T_{i,j-1}^n}{(\Delta Y)^2} \right) $$

  • Truncation Error: \( O[\Delta t, (\Delta X)^2, (\Delta Y)^2] \)
  • Stability: Conditionally stable, requires \( r_X + r_Y \le \frac{1}{2} \), where:
    • \( r_X = \frac{\alpha \Delta t}{(\Delta X)^2} \)
    • \( r_Y = \frac{\alpha \Delta t}{(\Delta Y)^2} \)

2. Simple Implicit (Laasonen) Method

$$ \frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t} = \alpha \left( \frac{T_{i+1,j}^{n+1} - 2T_{i,j}^{n+1} + T_{i-1,j}^{n+1}}{(\Delta X)^2} + \frac{T_{i,j+1}^{n+1} - 2T_{i,j}^{n+1} + T_{i,j-1}^{n+1}}{(\Delta Y)^2} \right) $$

  • Unconditionally stable
  • Requires solving a coupled system at each time level

3. Crank-Nicolson Method

$$ \begin{aligned} \frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t} = \frac{\alpha}{2} \Big[ & \left( \frac{T_{i+1,j}^n - 2T_{i,j}^n + T_{i-1,j}^n}{(\Delta X)^2} + \frac{T_{i,j+1}^n - 2T_{i,j}^n + T_{i,j-1}^n}{(\Delta Y)^2} \right) + \\ & \left( \frac{T_{i+1,j}^{n+1} - 2T_{i,j}^{n+1} + T_{i-1,j}^{n+1}}{(\Delta X)^2} + \frac{T_{i,j+1}^{n+1} - 2T_{i,j}^{n+1} + T_{i,j-1}^{n+1}}{(\Delta Y)^2} \right) \Big] \end{aligned} $$

  • Second-order accurate in time and space
  • Unconditionally stable
  • System involves five unknowns per equation, not tridiagonal

Solving the Algebraic System

For 2D problems, efficient solution methods include:

  • Jacobi Iteration
  • Gauss-Seidel Iteration – uses updated values immediately
  • SOR (Successive Over-Relaxation) – accelerates Gauss-Seidel with relaxation factor \( \omega \)
  • ADI (Alternating Direction Implicit) – splits 2D problem into 1D steps
    • Step 1: Solve along X (Y treated explicitly)
    • Step 2: Solve along Y (X treated explicitly)
  • Multigrid Method – uses coarse-fine grids to accelerate convergence

Boundary Conditions

  • Dirichlet: Specified temperature at boundary
  • Neumann: Specified heat flux, e.g., insulated wall \( \frac{\partial T}{\partial n} = 0 \)
  • Convective (Robin): Newton’s law:

    $$ h(T_\infty - T_w) = -k \frac{\partial T}{\partial n} $$

Conclusion

The Finite Difference Method (FDM) is a versatile approach for 2D heat conduction. Explicit schemes are easy to implement but often unstable. Implicit methods like Crank-Nicolson and ADI offer better stability and efficiency, especially when paired with fast iterative solvers.