Question

Can the global (accumulated) discretization error be less than the local error during a step? Explain.

Short answer

Explain your answer. Answer: Yes, it is possible for the global discretization error to be smaller than the local error during a step in a numerical method. This can occur when local errors compensate for each other to some extent due to their distribution in different directions. The specific behavior of the numerical method and the errors it introduces during its operation will determine if this occurs.

Step-by-step solution

Define global and local discretization errors

The global discretization error is the accumulated error of a method when solving a differential equation numerically from the initial point up to a certain point. It is caused by the way the numerical method approximates the exact solution. On the other hand, the local discretization error is the amount by which the method deviates from the exact solution within just one small step of the numerical method.

Analyze local and global discretization errors' interplay

In general, the global error increases as the number of steps increases. However, every step introduces a new local error, which is compounded into the accumulated error. These local discretization errors can accumulate in the same direction or they can be distributed in different directions. This means that there might be cases where the errors compensate each other partially, which makes the total accumulated global discretization error smaller than the local error in one of the steps.

Provide an explanation

During each time step, a local discretization error may be introduced due to various factors such as truncation errors in the method, inaccuracies in the input data, or round-off errors. If the local errors are such that they tend to cancel each other (for example, one positive error is followed by a similar negative error), the global discretization error will be less than the sum of the local errors. There may also be cases where the global error at a specific time is less than the local error during one step because some of the previous local errors have compensated each other. In conclusion, it is possible for the global discretization error to be less than the local error during a step, due to the fact that local errors can compensate for each other to some extent. However, this depends on the specific behavior of the numerical method and the errors it introduces during its operation.