CFD: How to Deal with Errors and Uncertainties

In order to develop trust and confidence in CFD simulations, one needs to understand, at best, to evaluate and control the errors and uncertainty associated with CFD simulations. The definitions of the terms error and uncertainty, based on the AIAA guidelines (American Institute of Aeronautics and Astronautics, 1998) on CFD, are: 

UNCERTAINTY – “A potential deficiency in any phase or activity of the modeling process that is due to the lack of knowledge."

ERROR - “A recognizable deficiency in any phase or activity of modeling and simulation that is not due to lack of knowledge.” 

The word potential is highlighted in the definition of uncertainty, thereby implying that one may or may not encounter it. The lack of knowledge justifies its place, which is bereft of the understanding of the physical flow processes that goes into the modeling procedure. Turbulence modeling is a very good example of uncertainty by itself, as there’s a lot to the phenomenon of turbulence that is still not understood clearly. Uncertainty can be classified as input and physical.

Input: caused due to erroneous boundary conditions or incomplete information on the material fluid properties etc.

Physical: caused due to the failure to account for characteristics such as chemical reactions, incompressibility, turbulence, etc.

The definition of error implies that the deficiency is identifiable upon examination. Errors can also be classified as acknowledged or unacknowledged:

Acknowledged: constitutes the errors, which can be identified and removed or chosen to be kept to achieve a finite level of accuracy.

1. Physical approximation error:  It constitutes not only the lack of understanding of the phenomenon behind the system but also the way in which one approaches them. The ‘ill posed’ problems often give blown-up solutions or, at best incorrect results.
   
2. Computer round-off errors: These develop with the representation of floating-point numbers on the computer and the accuracy, typically stored with 16, 32, or 64 bits,  at which numbers are stored. Since information received by the computers are binary, this becomes much more significant for decimals of a higher order.
   
3. Iterative/Convergence errors: Iteration or convergence errors occur due to the difference between a fully converged solution of a finite number of grid points and a solution that has not fully achieved convergence.  Convergence errors, therefore, can occur either because of the user's being too impatient to allow the solution algorithm to complete its progress to the final converged solution or because of the user's applying too large convergence tolerances to halt the iteration process when the CFD solution may still be considerably far from its converged state.
   
4. Discretization errors: Discretization errors are those errors that occur from the representation of the governing flow equations and other physical models as algebraic expressions in a discrete domain of space and time. These equations in the spatial form are commonly represented by finite-difference, finite-volume, and finite-element methods. The level of discretization error is dependent on the features of the flow as resolved by the grid. Errors may develop due to the representation of discontinuities, e.g., shocks on a grid.  
   
5. Interpolation errors: These are the errors that come about in the zonal interfaces where the solution of one zone is approximated on the boundary of the other zone.
   
6. Truncation error: Truncation error is defined as the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation, i.e., the truncation error terms are those of the expansion which are not used in the discretized equation. It is a function of the grid quality and flow gradients. If the order of the leading term of the truncation error is of second order, it is known as a numerical or artificial viscosity. A positive viscous term will indicate that errors will be damped, whereas a negative viscous term will indicate that errors will grow (blow-up).
   
7. Dispersive & Dissipative errors: Dispersive errors cause oscillations in the solution, which can be fixed by adding artificial dissipation to decrease the size of the dispersive errors. Dissipation error terms cause a smoothing of gradients, observed when a scheme is converted from higher-order to lower-order.
   

Unacknowledged: It constitutes programming or usage errors that have no set procedures for finding them and may continue within the code or simulation. 

1. Programming errors: These types of errors are discovered by systematically performing verification studies of subroutines, reviewing lines of code, and performing validation studies.
   
2. Usage errors: These are due to the application of the code in a less-than-accurate or improper manner. Usage errors may actually show up as modeling and discretization errors. The user sets the models, grid, algorithm, and inputs used in a simulation, which then establishes the accuracy of the simulation. Usage errors can exist in CAD creation, grid generation, and post-processing software.
   

Errors are an integral part of the simulation process, and the ones discussed above will likely creep into the computation with or without our control. Hence the most important recommendation is to be critical of the results produced by a CFD code and use not only physical insight but also all the available knowledge on numerical analysis to get reliable results efficiently.