Some very important terms any CFD user should know.
Accuracy; This can be thought of as three issues:
• Modelling accuracy; How well do the differential equations represent the physical system. For single phase laminar flow this is not usually an issue, but may be once models are introduced for turbulence, etc.
• Discretization accuracy; How well does the discretized solution (the collection of velocity, pressure, temperature, etc values at grid nodes) represent the true solution of the original differential equations.
• Solver Accuracy; How close does the matrix solver get to the true solution of the discretized system.
• Most CFD schemes employ an iterative solution procedure to solve the resulting system of discretized algebraic equations. Stability in this context refers to the convergence (or otherwise) of this process.
• In time-dependent problems stability refers to whether the method produces a bounded solution (assuming the exact solution should remain bounded).
• A stable scheme thus ensures that small errors (which inevitably appear in a numerical solution) do not get magnified.
• Stability of a scheme can be analysed analytically for very simple equations, but there are few such results for non-linear coupled systems.
• In practise, stability often places a restriction on the time step that can be used, or the level of under-relaxation applied.
Stability vs. Accuracy
• In general, there is often a trade-off between accuracy and stability.
• A numerical scheme that is very diffusive, for example, can be very stable because it is effectively adding too much “viscosity” to the problem.
• However, by doing so it may be smoothing out steep gradients, and will not, therefore, be very accurate.
• Understanding these effects, and how to get the right balance between the two, is a crucial aspect of CFD.
• If the discretization scheme is consistent, then it should formally become exact as the grid spacing/time step tends to zero.
• Truncation errors (see later posts) are generally proportional to (∆x)n or (∆t)n for some n (which depends on the discretization scheme); a consistent scheme will have n > 0.
• The equations being solved arise from physical conservation laws. A conservative numerical scheme will retain this property on both a local (cell) and global (domain) level.
• For example, in a steady state problem there should be a balance between mass inflow and outflow over each cell, and over the entire domain.
• Ensures that the numerical solution lies within physical bounds.
• For example, in a heat conduction problem the minimum and maximum temperatures should occur on the domain boundaries. A bounded scheme would not produce spurious maxima/minima within the domain.
• Higher order discretization schemes can often produce unbounded solutions in the form of undershoots and overshoots, which can sometimes lead to stability and convergence problems.