Question

What is the cause of the discretization error? How does the global discretization error differ from the local discretization error?

Short answer

Answer: The main difference between global and local discretization errors is that local discretization error focuses on the error associated with a single step in the discretization process, while global discretization error is concerned with the overall error across the entire problem domain.

Step-by-step solution

Definition: Discretization Error

Discretization error is the difference between the exact solution of a continuous problem (such as a differential equation) and the approximate solution obtained by discretizing the problem (converting it into a discrete problem, such as a difference equation). The error occurs because the discretization process involves approximating the continuous function by a discrete representation. The discretization error depends on factors like the method used for discretization, the step size or grid spacing, and the smoothness of the function being approximated.

Local Discretization Error

Local discretization error refers to the error associated with a single step in the discretization process. In the context of numerical methods for solving differential equations, this error corresponds to the difference between the exact and approximate solution at a specific point in the discrete domain (usually between two grid points). The local error can be influenced by the step size, the method used for discretization, and the nature of the function being approximated. Reducing the step size or using higher-order methods can help reduce the local discretization error.

Global Discretization Error

Global discretization error is the cumulative effect of local discretization errors over the entire domain of the problem. It measures the overall difference between the exact and approximate solutions and is affected by factors such as the method used, the step size, and the problem's nature (e.g., the smoothness of the function being approximated). The global error also depends on the stability and convergence of the numerical method, as the error may either accumulate or dissipate as the computation proceeds. To minimize the global discretization error, the numerical method should be chosen wisely, and proper step size should be used by considering the problem's characteristics.

Comparison: Global vs Local Discretization Error

Local discretization error focuses on the error associated with a single step in the discretization process, while global discretization error is concerned with the overall error across the entire problem domain. Reducing the local error may not necessarily result in a decrease in global error, as the latter is influenced by the stability and convergence properties of the numerical method used. Similarly, a small global error does not guarantee a small error at every point in the domain, as the local error might still be larger in some areas.