Spectral Methods

The major problem with any numerical solution of a differential equation is accurate calculations of derivatives. This is why nearly all early turbulence DNS's utilized spectral methods, which are extremely accurate and non-dissipative tools for calculating derivatives of discrete data sets. If crafted correctly, such methods enjoy exponential convergence to a highly accurate solution. A spectral method actually approximates a real-space function with a series sum of orthogonal functions. Mathematically, this looks like:

$$f(x_j) = \sum^{N-1}_{n=0} \hat{f_n} \phi_{n,j} \hspace{1cm} j =\{0,1,\cdots,N-1\}$$ (1)

The most common choice for the orthogonal functions is the Fourier series. This may seem to complicating things, but actually this helps to calculate the spatial derivative of f since Fourier series functions (complex exponentials) are easy to differentiate. Again, mathematically this results in:

$$f_j = \sum^{N-1}_{n=0} \hat{f_n} e^{i\omega_n j} \hspace{1cm... ... f_j = \frac{1}{N} \sum^{N-1}_{n=0} \hat{f_n} e^{-i\omega_n j}$$ (2)

which are the inverse Fourier transformation and the Fourier transformation, respectively. Now, we can differentiate f :

$$\frac{\partial f}{\partial x_j} = \sum^{N-1}_{n=0} \underbrace{i \omega_n \hat{f_n}}_{\hat{g_n}} e^{i \omega_n j}$$ (3)

Therefore, in order to calculate the derivative of f, (1) calculate the Fourier transform, (2) compute new Fourier coefficients, and (3) calculate the inverse Fourier transform of the new series (with calculated coefficients from (2)). This method is quite time consuming, but with the help of the Fast Fourier Transforms (FFT), the method becomes $O(N\log N)$ instead of O(N²). However, several important stipulations must be observed when using this method to solve the Navier-Stokes equation:

1.

The orthogonal functions should be well behaved and continuous to reduce Gibb's phenomena as much as possible (i.e. to recover pointwise exponential accuracy at all points including the discontinuities).

2.

Grid spacing must be on the order of the Kolmogorov scale of the flow (smallest scale within the flow).

3.

Aliasing errors (false translation of new modes into the domain) due to convective terms should be removed. They can cause either numerical instability or excessive turbulence decay.

One of the big disadvantages of this method is that it is not yet clear how these procedures can be extended to curvilinear grids which is so common in aerodynamics.