Question

To make a more accurate calculation of the previous problem the subdivision in layers can be made finer, say in 40 layers. What effect does that have on the time step?

Short answer

Answer: When the number of layers is increased to 40, the time step size will decrease.

Step-by-step solution

Set up a proportionality equation

Since we assume that the time step is linearly related to the number of layers, we can write a simple proportionality equation as follows: \[ \dfrac{\text{Initial Time Step}}{\text{Final Time Step}} = \dfrac{\text{Initial Number of Layers}}{\text{Final Number of Layers}} \] As the question states, the final number of layers is 40 layers.

Calculate the final time step

To find out the effect of increasing the number of layers on the time step, we'll solve the equation we came up with in the previous step for the final time step.\ \[ \text{Final Time Step} = \dfrac{\text{Initial Time Step} \times \text{Initial Number of Layers}}{\text{Final Number of Layers}} \] Since we don't have actual numerical values for the initial time step and the initial number of layers, we cannot calculate the exact final time step value. However, we can still find out how the ratio of initial time step to final time step changes with respect to the ratio of initial number of layers to the final number of layers.

Analyze the effect of changing the number of layers on time step

From the relation we arrived at in Step 2, we can observe that the time step is inversely proportional to the number of layers. This means that if we increase the number of layers, the time step will decrease, and if we decrease the number of layers, the time step will increase. Hence, making the subdivision in layers finer (increasing the number of layers to 40) will result in a decrease in the time step size.