This work further examines features of the waveform relaxation method for solving large scale systems of ordinary differential equations. The main issue under examination is the convolution successive overrelaxation (CSOR) waveform algorithm for a linear initial value problem of the form \[ B u_t + A u = f ,\quad u(0)=u_0 , \tag{1} \] where \(A\) and \(B\) are square matrices, with \(B\) nonsingular. While the SOR waveform algorithm is the natural extension for (1) of the classic SOR method for linear systems, CSOR goes a step further by substituting a convolution for the multiplication of the correction terms by a parameter. When applied to the discretisation of the heat equation with spatial mesh-size \(h\), the asymptotic convergence factor of CSOR is \(1-O(h)\), for small \(h\), while for the standard SOR waveform method the factor is \(1-O(h^2)\). This apparent theoretical advantage of CSOR makes the results presented in the paper especially interesting.
Both the continuous and discretised in time CSOR approaches are examined and the existence under suitable hypotheses of convolution kernels for them is reviewed, as well as their convergence properties. For (1) with \(B=I\), the optimal convolution kernel with point relaxation is explicitly determined.
One of the main results in the paper is that for such systems, if a strictly stable multistep is chosen for the time discretisation of (1), then the corresponding optimal kernels converge pointwise to the optimal continuous kernel, as the time step tends to \(0\). The quantitative behaviour of this result is examined in the case of the heat equation, with standard approximations both in space and time. In the closing section, results are proved that extend those regarding the existence of convolutions kernels and their convergence properties are established as well.