Express the general stability criterion for the explicit method of solution of transient heat conduction problems.

The explicit finite difference method is a numerical method used to solve partial differential equations, including those that describe transient heat conduction problems. The heat equation for a one-dimensional transient heat conduction problem can be written as: \(\frac{\partial T}{\partial t} = \alpha\frac{\partial^2 T}{\partial x^2}\) where \(T\) is the temperature, \(t\) is the time, \(\alpha\) is the thermal diffusivity, and \(x\) is the spatial coordinate. We discretize the temperature field \(T\) using a finite difference grid in both time and space: \(T_{i,j} = T(x_i, t_j)\) Now, we approximate the derivatives in the heat equation using finite differences: \(\frac{T_{i,j+1} - T_{i,j}}{\Delta t} = \alpha\frac{T_{i-1,j} - 2T_{i,j} + T_{i+1,j}}{\Delta x^2}\) Solving for \(T_{i,j+1}\), we get the explicit method: \(T_{i,j+1} = T_{i,j} + \frac{\alpha \Delta t}{\Delta x^2}(T_{i-1,j} - 2T_{i,j} + T_{i+1,j})\)

Von Neumann stability analysis is a technique used to determine numerical stability in finite difference methods. For an explicit method to be stable, the amplification factor needs to be less than or equal to 1. Let the temperature at time step \(j\) be expressed as a Fourier series: \(T_{i,j} = \sum_{k} \xi_j(k) e^{ikx_i}\) Substituting this expression of \(T_{i,j}\) into the explicit finite difference method, we can define the amplification factor \(G(k)\) as the ratio between \(\xi_{j+1}(k)\) and \(\xi_j(k)\): \(G(k) = \frac{\xi_{j+1}(k)}{\xi_j(k)}\) Solving for \(G(k)\), we get: \(G(k) = 1 + \frac{\alpha\Delta t}{\Delta x^2}(e^{-ik\Delta x} - 2 + e^{ik\Delta x})\) For numerical stability, we need \(|G(k)| \le 1\). Simplifying the expression for \(|G(k)|\): \(|G(k)| = |1 + \frac{\alpha\Delta t}{\Delta x^2}(e^{-ik\Delta x} - 2 + e^{ik\Delta x})|\) Using the identity \(|e^{ix}| = 1\) and applying the triangle inequality, we get: \(|G(k)| \le 1 + \frac{\alpha\Delta t}{\Delta x^2}(1 - 2 + 1) = 1 - 4\frac{\alpha\Delta t}{\Delta x^2}\) To satisfy the stability criterion (\(|G(k)| \le 1\)), we must have: \(1 - 4\frac{\alpha\Delta t}{\Delta x^2} \le 1\) Rearranging this inequality, we get the stability criterion for the explicit method of solution of transient heat conduction problems: \(\frac{\alpha\Delta t}{\Delta x^2} \le \frac{1}{2}\)