Question

What is a practical way of checking if the discretization error has been significant in calculations?

Short answer

Answer: A practical way to check for discretization error is by performing calculations with varying discretization steps (e.g., different grid sizes or mesh resolutions) and comparing the results. If the error is significant, we would expect to see considerable differences in the results as we refine the discretization. Calculate the discretization ratio and the relative error between the coarser and finer discretizations. If the relative error is small or decreases as the discretization is refined, it is a good indication that the discretization error is not significant. If not, further refinement or analysis may be needed.

Step-by-step solution

Understand discretization error

Discretization error occurs when a continuous function is approximated by a discrete one. This is common in numerical methods, such as finite difference and finite element methods, where a continuous domain is split into discrete points or elements. The error is introduced when the discrete approximation does not capture the exact behavior of the continuous function.

Use multiple discretization steps

A practical way to check for discretization error is to perform calculations with varying discretization steps (e.g., different grid sizes or mesh resolutions) and then compare the results. If the error is significant, we would expect to see considerable differences in the results as we refine the discretization.

Compute the discretization ratio

Choose two different discretization steps and compute their ratio, typically denoted as \(r\). For example, if we use a grid size of \(h_1\) for the first simulation and \(h_2\) for the second, where \(h_2\) is a finer discretization, then the discretization ratio is \(r = \frac{h_1}{h_2}\).

Analyze the results

Compare the results obtained from the simulations with different discretization steps. This can be done by computing the relative error of the results, defined as \(\epsilon = \frac{|S_1 - S_2|}{|S_1|}\), where \(S_1\) and \(S_2\) are the solutions (or some measure of the solution) from the coarser and finer discretizations, respectively.

Evaluate the significance of the error

If the relative error is small or decreases as the discretization is refined, it is a good indication that the discretization error is not significant. However, if the relative error is large or does not decrease as the discretization is refined, it indicates that the discretization error may be significant, and further refinement or analysis of the model is needed.