How do you assess the accuracy and stability of a numerical solution for a fluid flow problem?

1 Sources of error

The first step in assessing the accuracy and stability of a numerical solution is to identify the sources of error that may affect it. There are three main types of error that can occur in numerical methods: truncation error, round-off error, and discretization error. Truncation error arises from the approximation of the continuous mathematical model by a finite series or a finite difference. Round-off error results from the finite precision of the computer arithmetic and the representation of numbers. Discretization error stems from the approximation of the continuous domain and boundary conditions by a discrete grid or mesh. These errors can be reduced by using higher-order methods, increasing the precision, and refining the grid or mesh, but they cannot be eliminated completely.

2 Convergence analysis

The second step in assessing the accuracy and stability of a numerical solution is to perform a convergence analysis. Convergence analysis is the process of checking whether the numerical solution approaches the exact or analytical solution as the grid or mesh size, the time step, or the truncation order are reduced. Convergence analysis can be done by comparing the numerical solution with the analytical solution, if available, or by comparing the numerical solution at different levels of refinement. Convergence analysis can also be done by calculating the error norms, such as the L2 norm or the L-infinity norm, which measure the difference between the numerical and analytical solutions. Convergence analysis can help you determine the order of accuracy of your method and the optimal grid or mesh size and time step for your problem.

3 Stability analysis

The third step in assessing the accuracy and stability of a numerical solution is to perform a stability analysis. Stability analysis is the process of checking whether the numerical solution remains bounded and does not grow without bound or oscillate wildly as the time or iteration progresses. Stability analysis can be done by applying a stability criterion, such as the Courant-Friedrichs-Lewy (CFL) condition or the von Neumann condition, which relate the time step, the grid or mesh size, and the numerical scheme. Stability analysis can also be done by inspecting the numerical solution for spurious modes, such as numerical diffusion, dispersion, or dissipation, which may alter the physical behavior of the fluid flow. Stability analysis can help you ensure that your method is consistent and stable for your problem.