Time Stepping Via One-Dimensional Padé Approximation

Numerical evolution methods of rational form for reaction diffusion equations

“…Also, We now turn to a class of schemes that is simpler and less expensive per time-step than other explicit schemes, namely one-dimensional Pade approximations, which make use of rational functions. This approach was first developed in [5], where…”

“…First-order stability polynomials The optimal stability polynomial, which pro vides maximum j3 for given s, was obtained in [39] to be P,(z) = T, (1 + £) , ,3= 2s 2 . In Figure 1.9, the stability region S was plotted for P 2 and P 5 . We notice that in…”

“…This factor also served as motivation to develop an explicit scheme which is highly stable for the purposes of the present experiment, and is associated with lower computational costs since it is explicit. Pade time stepping (PTS), described in [5], provides such a scheme. It is based on Pade approximation (see [9]) for time stepping each component of an ODE system.…”

Block splitting of exponential operators via Padé approximation and BSPTS schemes

“…PTS schemes, which are as simple and inexpen sive per time-step as other explicit methods, possess, in many cases, properties of stability similar to those offered by implicit approaches. However, as noted in [4], PTS schemes are not unconditionally stable in general, and in certain cases they are even less stable than some other popular explicit finite difference schemes.…”

“…A new class of explicit schemes, Fade time-stepping (PTS) schemes has been re cently created and introduced [4] for numerically evolving solutions of ordinary differential equations problems. PTS schemes, which are as simple and inexpen sive per time-step as other explicit methods, possess, in many cases, properties of stability similar to those offered by implicit approaches.…”