Numerical positivity for Runge-Kutta methods

Over the last years, a great effort has been done to develop Runge-Kutta methods preserving qualitative properties of the exact solution like monotonicity of positivity.Some results are available in the literature that ensure these properties under certain stepsize restictions. However, these results are given for the exact numerical solution whereas in practice the numerical solution available is only an approximation of it. For example, when implicit Runge-Kutta methods are used, the numerical solution obtained comes out from the inexact numerical resolution of nonlinear systems. The aim of this work is the study of the effective stepsize restrictions for positivity when implicit Runge-Kutta schemes are used. To achive this goal, we consider separately the problem of finding positive stage value predictors, and the analysis of the iterative scheme used, in this case Newton method

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances