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A singly diagonally implicit two-step peer triple with global error control for stiff ordinary differential equations

Kulikov, Gennady Yu; Weiner, R.

SIAM Journal on Scientific Computing, 37(3) (2015), A1593–A1613

This paper elaborates a new numerical method for treating stiff ordinary differential equations (ODEs) which constitute a basic simulation tool in many areas of study. For its efficiency, this technique is grounded in singly diagonally implicit two-step peer schemes designed recently. The latter means that linear systems with the same coefficient matrix are solved to advance a step of the method and to evaluate its global error as well. Moreover, in contrast to many other numerical methods with global error control, we monitor and regulate the scaled global error, which corresponds better with practitioners' needs. In other words, we present and study one peer triple which is able to solve stiff problems and to ensure that the true error of the derived numerical solution does not exceed the user-supplied bound at the same time. This implies that local and global error estimation and control mechanisms are also discussed here. Finally, efficiency of our numerical tool is checked in comparison to efficiency of stiff built-in MATLAB ODE solvers on various test problems.