What is the best way to determine the accuracy of a numerical solution?

2 Error measures

The second step to determine the accuracy of a numerical solution is to quantify the error using appropriate measures. There are different ways to measure the error, depending on the type and nature of the problem. Some common error measures are absolute error, relative error, percentage error, and norm error. Absolute error is the difference between the exact and the numerical solution at a given point or variable. Relative error is the ratio of the absolute error to the exact solution at a given point or variable. Percentage error is the relative error multiplied by 100. Norm error is the measure of the error over the entire domain or system, using a specific norm, such as the L1, L2, or L-infinity norm.

3 Error reduction

The third step to determine the accuracy of a numerical solution is to reduce the error if it is too large or unacceptable for the problem. There are different ways to reduce the error, depending on the type and nature of the problem. Some common error reduction techniques are grid refinement, order increase, precision increase, and convergence acceleration. Grid refinement is the process of increasing the number of elements or points in the domain to reduce the discretization error. Order increase is the process of using higher-order methods or schemes to reduce the discretization error. Precision increase is the process of using higher-precision computer systems or algorithms to reduce the round-off error. Convergence acceleration is the process of using techniques or methods to speed up the convergence of iterative methods and reduce the iteration error.

4 Convergence criteria

The fourth step to determine the accuracy of a numerical solution is to check whether the solution has converged or not. Convergence means that the solution approaches a limit as the number of elements, points, iterations, or steps increases. There are different ways to check the convergence, depending on the type and nature of the problem. Some common convergence criteria are residual, tolerance, and error estimate. Residual is the difference between the left-hand side and the right-hand side of an equation or a system of equations. Tolerance is a predefined value that indicates the acceptable level of error or residual. Error estimate is a calculated value that indicates the upper bound of the error or the rate of convergence.

5 Error analysis

The fifth step to determine the accuracy of a numerical solution is to analyze the error using graphical or numerical tools. Error analysis is the process of understanding the behavior and characteristics of the error, such as its magnitude, distribution, pattern, and trend. Error analysis can help to identify the sources and types of error, to compare different methods or solutions, and to evaluate the accuracy and reliability of the solution. Some common error analysis tools are error plots, error tables, error indicators, and error norms.

6 Error control

The final step to determine the accuracy of a numerical solution is to control the error using adaptive or feedback methods. Error control is the process of adjusting the parameters or variables of the problem or the method to achieve a desired level of accuracy or convergence. Error control can help to optimize the efficiency and performance of the numerical solution, by avoiding unnecessary computations or resources. Some common error control methods are adaptive grid, adaptive order, adaptive precision, and adaptive iteration.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?