Pseudospectral Methods

Pseudospectral Methods

Change of Interval

To can change the limits of the integration (in order to apply Quadrature), we introduce $\tau \in [-1,+1]$ as a new independent variable and perform a change of variable for $t$ in terms of $\tau$, by defining:

\[\tau = \frac{2}{t_{{N}_{t}}-t_0}t - \frac{t_{N_t}+t_0}{t_{N_t}-t_0}\]

Polynomial Interpolation

Select a set of $N_t+1$ node points:

\[\mathbf{\tau} = [\tau_0,\tau_1,\tau_2,.....,\tau_{N_t}]\]

A unique polynomial $P(\tau)$ exists (i.e. $\exists! P(\tau)$) of a maximum degree of $N_t$ where:

\[f(\tau_k)=P(\tau_k),\;\;\;k={0,1,2,....N_t}\]

But, between the intervals, we must approximate $f(\tau)$ as:

\[f(\tau) \approx P(\tau)= \sum_{k=0}^{N_t}f(\tau_k)\phi_k(\tau)\]

with $\phi_k(•)$ are basis polynomials that are built by interpolating $f(\tau)$ at the node points.

Approximating Derivatives

We can also approximate the derivative of a function $f(\tau)$ as:

\[\frac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}=\dot{f}(\tau_k)\approx\dot{P}(\tau_k)=\sum_{i=0}^{N_t}D_{ki}f(\tau_i)\]

With $\mathbf{D}$ is a $(N_t+1)\times(N_t+1)$ differentiation matrix that depends on:

Now we have an approximation of $\dot{f}(\tau_k)$ that depends only on $f(\tau)$!

Approximating Integrals

The integral we are interested in evaluating is:

\[\int_{t_0}^{t_{N_t}}f(t)\mathrm{d}t=\frac{t_{N_t}-t_0}{2}\int_{-1}^{1}f(\tau_k)\mathrm{d}\tau\]

This can be approximated using quadrature:

\[\int_{-1}^{1}f(\tau_k)\mathrm{d}\tau\sum_{k=0}^{N_t}w_kf(\tau_k)\]

where $w_k$ are quadrature weights and depend only on:

Legendre Pseudospectral Method

Define an N order Legendre polynomial as:

\[L_N(\tau) = \frac{1}{2^NN!}\frac{\mathrm{d}^n}{\mathrm{d}\tau^N}(\tau^2-1)^N\]
\[:nowrap:$$$$\begin{equation} \tau_k = \left \{ \begin{aligned} &-1, && \text{if}\ k=0 \\ &\text{kth}\;\text{root}\;of \dot{L}_{N_t}(\tau), && \text{if}\ k = {1, 2, 3, .. N_t-1}\\ &+1\, && \text{if}\ k = N_t \end{aligned} \right. \end{equation}\]