Automatic Grid Control in Adaptive BVP Solvers

Modern adaptive techniques in two-point boundary value problems generate the mesh by constructing a function that maps a uniform grid to the desired nonuniform grid. This paper describes a new control algorithm for constructing a grid density function $phi(x)$, such that the local mesh width $Delta x_{j+1/2}=x_{j+1}-x_j$ is computed as $Delta x_{j+1/2} = varepsilon_N / varphi_{j+1/2}$. Here $varepsilon_N$ is the accuracy control parameter corresponding to $N$ interior points, while ${varphi_{j+1/2}}_0^N$ is a discrete approximation to $phi(x)$ accounting for mesh width variation. Feedback control theory is applied to generate a new density from the previous one. Further, digital filters may be employed to process the error estimate as well as the step density, and causal digital filters can be used in the mesh refining step.