CFD

• Numerical solutions are wrong until proven to be correct
• The best way to prove that numerical results are correct is by reproducing problems for which there is an analytical solution - demonstrating that the method matches the exact solution
• Interpolation


  • Filling in the gaps between discrete data points
  • Allows integration and differentiation of data
• Lagrange polynomials


  Curves of arbitrary order that pass through all data points
• Splines


  Curves that meet at the same value and have the same first and second derivatives at the joining points
• Least squares


  Fitting a lower order polynomial to noisy data by minimising the mean square error
• Numerical integration


  1. Fit a curve to the data

  2. Integrate the area under the curve
• Trapezoidal rule


  Numerical integration using straight lines between data points
• Simpson's rule


  Numerical integration using quadratic Lagrangian polynomials fit to groups of 3 points
• Different computational operations take different amounts of time, with additions/subtractions/multiplications being very fast, divisions being slower, and more complex functions being even slower
• Order of accuracy


  Tells you how quickly the numerical solution approaches the exact solution as the sample spacing ∆x decreases
• The order of accuracy of a numerical method does not change with the order of the derivative
• Second order accurate scheme


  Error is proportional to ∆x^2
• First order accurate scheme

  Error is proportional to ∆x^1
• As the grid spacing is halved, the error in a second order scheme decreases by a factor of 4, while in a first order scheme it decreases by a factor of 2
• Stability


  A numerical scheme must be stable, otherwise errors will grow exponentially and the simulation will crash
• Consistency


  As the grid spacing tends to zero, the approximation will become exact
• Verification


  Ensuring the numerical method is derived and implemented correctly
• Validation


  Checking if the governing equation is the correct choice to model the actual physics of the problem
• Error norms (L1, L2, L∞)


  Used to quantify the errors between the numerical solution and the exact solution
• Computing the observed order of accuracy


  1. Calculate the error norms for multiple grid sizes

  2. Plot the error norms against the grid spacing on a log-log plot

  3. The slope of the line is the observed order of accuracy
• For problems with two spatial dimensions, the error norm definition must be modified to account for the additional dimension
• Error norms


  Denoted Lp where p is an integer ranging from 1,2...∞, the most common choices are the L1, L2 and L∞ norms
• L1 error


  Mean absolute error
• L2 error


  Root mean square error
• L∞ error


  Maximum error
• For problems with two spatial dimensions, the error norm definition must be changed
• Observed order of accuracy


  The actual order of accuracy of the implementation of the algorithm, tests for bugs in the algorithm derivation or coding errors
• The real error consists of many terms with many different derivatives, so the model of the error only gives the exact convergence rate as the number of points becomes very high
• Grid resolutions should vary by a factor of 2 from one calculation to the next when estimating the error
• At low grid resolutions, the observed value of n can differ dramatically from the expected due to the solution not being well resolved
• At high grid resolutions, the error asymptotes towards the expected order of accuracy for the numerical scheme used
• The heat equation is an exception where first order accuracy in time is acceptable because the time step size is constrained by stability to be proportional to the square of the grid spacing
• Forward in Time, Central in Space (FTCS) discretisation of the heat equation


  1. Approximate uxx using central differences

  2. Approximate ut using forward Euler

  3. Rearrange to get un+1_i = un_i + c^2*Δt/Δx^2 * (un_i+1 - 2un_i + un_i-1)
• Stability criterion for FTCS heat equation

  Δt ≤ Δx^2/(2c^2)
• Backward in Time, Central in Space (BTCS) discretisation of the heat equation

  1. Approximate uxx using central differences at the next time level

  2. Rearrange to get un+1_i = un_i + c^2*Δt/Δx^2 * (un+1_i+1 - 2un+1_i + un+1_i-1)
• Stability of BTCS heat equation

  Unconditionally stable, no restriction on Δt
• The linear advection equation describes the advection (transport/flow) of a quantity at a constant velocity
• Explicit discretisation of linear advection equation


  1. Approximate Yt using forward Euler

  2. Approximate Yx using first-order upwind
• such a straightforward analytical solution, yet has a similar behaviour to fluid dynamics, makes it an ideal starting point to understand the numerical discretisation of more complicated systems of equations like the Navier-Stokes equation