Spectral Properties of the Weighted Jacobi Propagator

Some insight into why the relaxation process converges can be gained by looking at properties of the matrix $P_{\omega}$ which propagates the field to the next step of iteration: $P_{\omega} \phi_i^t = \phi_i^{t+1}$. Here, following Ref. [18], we analyze the so-called weighted Jacobi method, which is a direct realization of Eq. 13. We assume that the function values are pinned at zero on the boundaries and we set $\rho = 0$ for this discussion. Therefore, the final result of the iterations will be $\phi=0$ everywhere. The matrix propagator can be written as:

$$P_{\omega} = I - \frac{\omega}{2} \left[ \begin{array} {rrr... ... & . & . & . & . & .\ 0 & . & . & . & . & .\end{array}\right]$$ (14)

The eigenfunctions of $P_{\omega}$ are simply Fourier sin waves (or modes) labeled by the index k, with $1 \leq k \leq N-1$, where N is the number of grid points. The corresponding eigenvalues are given by:

$$\lambda_k (P_{\omega}) = 1 -2 \omega {\mbox{sin}}^2 (\frac{k \pi} {2N} ) , \ \ 1 \leq k \leq N-1.$$ (15)

The small k modes have long wavelengths. The error in the function (before it has fully converged) can be expanded in the sin waves, and the kth mode of the error is reduced by an amount proportional to $\lambda_k^t$ after t iterations. Thus, for this weighted Jacobi method it is necessary that $0 < \omega < 1$ for the method to converge. The spectral radius is the eigenvalue with largest magnitude (closest to one), and it is clear that the convergence degrades as we go to higher resolution grids, or equivalently to larger domains. Specifically, the longest wavelength eigenvalue is:

$$\lambda_1 = 1 - 2 \omega {\mbox{sin}} ^2 (\frac{\pi}{2N}) = 1 - 2 \omega {\mbox{sin}} ^2 (\frac{\pi h}{2}).$$ (16)

By expanding the sin squared function, we see that the eigenvalue is 1 - O(h²); as $\lambda_1 \rightarrow 1$ that mode takes a longer and longer time to converge. One is then left with a problem common to all real space relaxation solvers, critical slowing down (CSD). The (important!) long wavelength portions of the solution become very difficult to handle for large systems.