Question

Consider transient two-dimensional heat conduction in a rectangular region that is to be solved by the explicit method. If all boundaries of the region are either insulated or at specified temperatures, express the stability criterion for this problem in its simplest form.

Short answer

Answer: The stability criterion for this problem is given by the inequality: $$\Delta t \le \frac{1}{2\alpha \left(\frac{1}{(\Delta x)^2} + \frac{1}{(\Delta y)^2} \right)}$$ where $$\Delta t$$ is the time step size, $$\Delta x$$ and $$\Delta y$$ are the spatial step sizes in the x and y directions, respectively, and $$\alpha$$ is the material's thermal diffusivity.

Step-by-step solution

Write down the two-dimensional heat conduction equation

The transient two-dimensional heat conduction equation, without heat generation, is given by: $$\frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right)$$ Where \(T\) is the temperature, \(t\) is the time, \(x, y\) are the spatial coordinates, and \(\alpha\) is the material's thermal diffusivity.

Discretize the equation using finite difference method

Using the forward difference for time and central difference for spatial coordinates, we can discretize the heat conduction equation as follows: $$ \frac{T^{p+1}_{i,j} - T^{p}_{i,j}}{\Delta t} = \alpha \left( \frac{T^{p}_{i+1,j} - 2\cdot T^{p}_{i,j} + T^{p}_{i-1,j}}{(\Delta x)^2} + \frac{T^{p}_{i,j+1} - 2\cdot T^{p}_{i,j} + T^{p}_{i,j-1}}{(\Delta y)^2} \right)$$ Where \(T^{p}_{i,j}\) is the temperature at point \((i,j)\) at time level \(p\), \(\Delta t\) is the time step size, \(\Delta x\) and \(\Delta y\) are the spatial step sizes in the \(x\) and \(y\) directions, respectively.

Obtain the explicit equation for the next temperature step

We can solve the above equation for \(T^{p+1}_{i,j}\), which gives the explicit formula for the next temperature step: $$T^{p+1}_{i,j} = T^{p}_{i,j} + \alpha \Delta t \left( \frac{T^{p}_{i+1,j} - 2\cdot T^{p}_{i,j} + T^{p}_{i-1,j}}{(\Delta x)^2} + \frac{T^{p}_{i,j+1} - 2\cdot T^{p}_{i,j} + T^{p}_{i,j-1}}{(\Delta y)^2} \right)$$

Derive the stability criterion for the explicit method

For stability, we must satisfy the Von Neumann stability criterion which, in this context, states that: $$ \alpha \Delta t \left(\frac{1}{(\Delta x)^2} + \frac{1}{(\Delta y)^2}\right) \le \frac{1}{2}$$ So the stability criterion for the transient two-dimensional heat conduction problem, considering the explicit method and assuming a rectangular region with insulated or specified temperature boundaries, can be expressed in its simplest form as: $$\Delta t \le \frac{1}{2\alpha \left(\frac{1}{(\Delta x)^2} + \frac{1}{(\Delta y)^2} \right)}$$