Numerical methods in fluid mechanics

Discretization

The central process in CFD is the process of

• we are solving the correct equations (consistency property)
• that the error can be decreased as we increase the number of degrees of freedom (stability and convergence).

Once these two criteria are established, the power of computing machines can be leveraged to solve the problem in a numerically reliable fashion. Various discretization schemes have been developed to cope with a variety of issues. The most notable for our purposes are:

Finite difference method

Finite difference replace the infinitesimal limiting process of derivative calculation:

${\displaystyle \lim _{\Delta x\to 0}f'(x)={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

with a finite limiting process, i.e.

${\displaystyle f'(x)={\frac {f(x+\Delta x)-f(x)}{\Delta x}}+O(\Delta x)}$

The term ${\displaystyle O(\Delta x)}$ gives an indication of the magnitude of the error as a function of the mesh spacing. In this instance, the error is halved if the grid spacing, _x is halved, and we say that this is a first order method. Most FDM used in practice are at least second order accurate except in very special circumstances. Finite Difference method is still the most popular numerical method for solution of PDEs because of their simplicity, efficiency and low computational cost. Their major drawback is in their geometric inflexibility which complicates their applications to general complex domains. These can be alleviated by the use of either mapping techniques and/or masking to fit the computational mesh to the computational domain.

Finite element method

The finite element method was designed to deal with problem with complicated computational regions. The PDE is first recast into a variational form which essentially forces the mean error to be small everywhere. The discretization step proceeds by dividing the computational domain into elements of triangular or rectangular shape. The solution within each element is interpolated with a polynomial of usually low order. Again, the unknowns are the solution at the collocation points. The CFD community adopted the FEM in the 1980s when reliable methods for dealing with advection dominated problems were devised.

Spectral method

Both finite element and finite difference methods are low order methods, usually of 2nd − 4th order, and have local approximation property. By local we mean that a particular collocation point is affected by a limited number of points around it. In contrast, spectral method have global approximation property. The interpolation functions, either polynomials or trigonomic functions are global in nature. Their main benefits is in the rate of convergence which depends on the smoothness of the solution (i.e. how many continuous derivatives does it admit). For infinitely smooth solution, the error decreases exponentially, i.e. faster than algebraic. Spectral methods are mostly used in the computations of homogeneous turbulence, and require relatively simple geometries. Atmospheric model have also adopted spectral methods because of their convergence properties and the regular spherical shape of their computational domain.

Finite volume method

Forward Euler approximation

${\displaystyle {\frac {u^{n+1}-u^{n}}{\Delta t}}\approx \kappa u^{n}}$