Question

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?

Short answer

Question: Explain the differences in heat transfer terms between the explicit and implicit methods in solving transient problems in finite difference systems. Answer: The differences in heat transfer terms between the explicit and implicit methods in solving transient problems in finite difference systems lie in their time and space differencing approaches. In the explicit method, forward time and central space differencing are used, allowing for future values to be calculated explicitly based on the current values. In the implicit method, backward time and central space differencing are employed, resulting in a system of linear equations that involve both current and future time step values.

Step-by-step solution

Understand the explicit method

In the explicit method, the finite difference approximation of the dependent variable (e.g., temperature) at a future time step is calculated using the information from the current time step. The algorithm allows us to determine the future values of the variable explicitly based on the current values.

Compute heat transfer terms in Explicit method

In the explicit method, the heat transfer terms in the energy balance equation are approximated using forward time and central space differencing. For example, in a 1D heat conduction problem with the heat equation: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\) The heat transfer terms in the explicit method would be: \(\frac{T^{n+1}_{i}-T^{n}_{i}}{\Delta t} = \alpha \frac{T^n_{i+1}-2T^n_{i}+T^n_{i-1}}{(\Delta x)^2}\) Here, \(T^n_i\) represents temperature at the current time step \(n\) and position \(i\), and \(T^{n+1}_i\) denotes the temperature at the next time step \((n+1)\).

Understand the implicit method

The implicit method, also known as the backward Euler method, updates the dependent variable (e.g., temperature) at a future time step using the information from both the current and future time steps. In this method, we solve a system of linear equations to obtain the future values of the variables.

Compute heat transfer terms in Implicit method

In the implicit method, the heat transfer terms in the energy balance equation are approximated using backward time and central space differencing. Using the 1D heat conduction problem, the heat transfer terms in the implicit method would be: \(\frac{T^{n+1}_{i}-T^n_{i}}{\Delta t} = \alpha \frac{T^{n+1}_{i+1}-2T^{n+1}_{i}+T^{n+1}_{i-1}}{(\Delta x)^2}\) Here, both the temperature at the next time step \((n+1)\) and the current time step \(n\) are used to compute the heat transfer terms.

Compare the heat transfer terms in the explicit and implicit methods

The key difference between the heat transfer terms in the explicit and implicit methods lies in the time and space differencing approaches. In the explicit method, we use forward time and central space differencing, which allows us to find the solution for the next time step explicitly based on the current values. In the implicit method, we employ backward time and central space differencing, resulting in a system of linear equations that involve both current and future time step values. In summary, both the explicit and implicit methods solve transient problems in finite difference systems using different approaches to heat transfer terms in the energy balance equation. The explicit method calculates future values explicitly using current time step values, while the implicit method requires solving a system of linear equations involving both current and future time step values.