Parallel adaptive and time-stabilizing schemes for constant-coefficient parabolic PDE's

New asynchronous and time stabilizing finite-difference methods for constant coefficient parabolic PDEs are presented. These schemes remove the synchronization overhead and are suitable for shared-memory, message-passing, single- or multi-user multi-processors. They generalize previous schemes by allowing adaptive and non-constant time increments. They are characterized by the time-stabilizing property: the derivatives of their time-level with respect to the spatial coordinates are bounded as the number of time-steps increases, and they vanish as the spatial mesh-size approaches zero. Therefore, unlike others, our schemes are suitable for multi-processors with non-identical processors or unbalanced workload, provided that the differences among their speeds/loads are not too large. The consistency, stability, convergence and the time-stabilizing property of our methods are analyzed in detail. When implemented on the shared-memory multi-user Sequent Balance machine, they show to provide an excellent efficiency and in certain cases it is tripled,, compared to other existing schemes. However, their truncation error is of lower order (O(Δx_l)nstead of O(Δx_l2)). Hence, because of the dimensional additivity of the truncation error, these schemes are less accurate especially for multi-dimensional problems. For such problems, if high accuracy is sought, in general one should use higher order schemes. Nevertheless, under certain circumstances, when the spatial increments are relatively large, the performance of our asynchronous schemes (i.e., both their efficiency and accuracy) may be about the same as compared to their synchronous counterparts, and sometimes even better.