The order of error of a numerical method is a measure of the rate at which the error in the computed solution of a problem decreases as the step size or grid size is decreased. It is typically denoted as O(h^n), where h is the step size and n is a positive integer.
The order of error is determined by analyzing the Taylor series expansion of the numerical method and identifying the highest power of h that appears in the error term. This power corresponds to the order of error.
The order of error provides important information about the accuracy and efficiency of a numerical method. A higher order of error indicates that the method is more accurate and requires fewer computations to achieve a desired level of accuracy.
Yes, the order of error can vary for different problems and numerical methods. It depends on the complexity of the problem and the behavior of the numerical method for that particular problem. Some methods may have a higher order of error for certain problems and a lower order of error for others.
The order of error is directly related to the convergence of a numerical method. A higher order of error indicates faster convergence, meaning that the computed solution will approach the true solution at a faster rate as the step size is decreased. However, a higher order of error does not guarantee convergence, as other factors such as stability and consistency also play a role in the convergence of a numerical method.