Question

The implicit method is unconditionally stable and thus any value of time step \(\Delta t\) can be used in the solution of transient heat conduction problems. To minimize the computation time, someone suggests using a very large value of \(\Delta t\) since there is no danger of instability. Do you agree with this suggestion? Explain.

Short answer

Answer: While the implicit method is unconditionally stable, allowing for the use of larger time steps without causing instability, using a very large time step can lead to a loss of accuracy in the solution. It is important to balance time step size, stability, accuracy, and computation time to obtain the most efficient and accurate solution for a specific problem. Thus, using a very large time step may not always be the best approach.

Step-by-step solution

Stability of implicit method

The implicit method is unconditionally stable, which means it will produce a stable solution regardless of the time step size chosen. So, on the stability aspect, there is no limitation on the size of the time step we can use.

Accuracy of solutions

However, stability alone does not guarantee accurate solutions. The choice of a very large \(\Delta t\) may lead to a significant reduction in the accuracy of the results, as the solutions at each time step may deviate considerably from the actual behavior of the system.

Relation between time step size and computation time

In general, a larger time step size will result in fewer time steps needed to reach a certain point in time, thus potentially reducing computation time. However, if the time step is too large, the inaccuracies in the solution might require additional corrective actions or smaller time steps later, which could increase the overall computational effort.

Conclusion

While the unconditionally stable nature of the implicit method allows for the use of larger time steps without the danger of instability, using a very large value of \(\Delta t\) can lead to a loss of accuracy in the solution. It is important to balance the time step size, stability, accuracy, and computation time to obtain the most efficient and accurate solution for a specific problem. Therefore, simply using a very large time step is not always the best approach.