What are the two basic methods of solution of transient problems based on finite differencing? How do heat transfer terms in the energy balance formulation differ in the two methods?

Finite differencing is a numerical technique used to solve partial differential equations (PDEs) by approximating derivatives with difference equations. This technique is particularly useful for solving transient (time-dependent) problems, where the solution evolves over time.

The two basic methods of solution of transient problems based on finite differencing are the explicit method and the implicit method. Explicit method: In this method, the value of the dependent variable at a future time step is calculated using the values of the variables at the current time step. This makes the explicit method easy to implement, but it can be unstable if the time step is too large. Implicit method: In contrast to the explicit method, the implicit method calculates the values of the dependent variable at the future time step by considering both the current and the future time steps. This makes the implicit method generally more stable, but it requires solving a system of linear equations, which can be more computationally intensive.

In the energy balance formulation of transient problems, the primary difference between the explicit and implicit methods lies in how the heat transfer terms are treated. For the explicit method, the heat transfer terms at the current time step are used to approximate the energy balance equation. This means that the equation is fully explicit and depends only on the values from the current time step. Mathematically, this can be represented as: \[T^{n+1}_i = T^n_i + \Delta t \left[\frac{\alpha}{(\Delta x)^2}(T^n_{i+1} - 2T^n_i + T^n_{i-1}) \right]\] For the implicit method, the heat transfer terms at the future time step are used in the energy balance equation, resulting in an implicit equation that depends on the values from both the current and future time steps. Mathematically, this is represented as: \[T^{n+1}_i = T^n_i + \Delta t \left[\frac{\alpha}{(\Delta x)^2}(T^{n+1}_{i+1} - 2T^{n+1}_i + T^{n+1}_{i-1}) \right]\] In summary, the major difference between the heat transfer terms in the explicit and implicit methods is that the explicit method uses the heat transfer terms at the current time step while the implicit method uses the terms at the future time step. This leads to different stability and computational characteristics between the two methods, as discussed in Step 2.