The numerical solution of time-dependent ordinary and partial differential equations presents a number of well known difficulties-including, possibly, severe restrictions on time-step sizes for stability in explicit procedures, as well as need for solution of challenging, generally nonlinear systems of equations in implicit schemes. In this note we introduce a novel class of explicit methods based on use of one-dimensional Padé approximation. These schemes, which are as simple and inexpensive per time-step as other explicit algorithms, possess, in many cases, properties of stability similar to those offered by implicit approaches. We demonstrate the character of our schemes through application to notoriously stiff systems of ODEs and PDEs. In a number of important cases, use of these algorithms has resulted in orders-of-magnitude reductions in computing times over those required by leading approaches.

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Journal of Scientific Computing
School of Mathematics and Statistics

Amundsen, D, & Bruno, O. (Oscar). (2007). Time stepping via one-dimensional padé approximation. Journal of Scientific Computing, 30(1), 83–115. doi:10.1007/s10915-005-9021-4