Runge-Kutta software defect control for boundary value ODEs. Mono-implicit Runge-Kutta (MIRK) formulas are implicit Runge-Kutta methods, for which the stages are defined explicitly in terms of $y_0$ and $y_1$. They can therefore be implemented very efficiently for the numerical solution of boundary value problems.par This publication presents new MIRK schemes with dense output, and discusses several implementation issues (modified Newton iteration, use of defect control instead of the standard global error control, mesh selection). Extensive numerical tests with a newly developed code MIRKDC are reported, and comparisons with the well-known code COLNEW are given.

References in zbMATH (referenced in 36 articles , 1 standard article )

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  1. McLachlan, Robert I.; Offen, Christian: Preservation of bifurcations of Hamiltonian boundary value problems under discretisation (2020)
  2. McLachlan, Robert I.; Offen, Christian: Symplectic integration of boundary value problems (2019)
  3. Putkaradze, Vakhtang; Rogers, Stuart: Constraint control of nonholonomic mechanical systems (2018)
  4. Famelis, Ioannis Th.; Tsitouras, Ch.: Quadratic shooting solution for an environmental parameter prediction problem (2015)
  5. Mazzia, Francesca; Cash, Jeff R.; Soetaert, Karline: Solving boundary value problems in the open source software R: package bvpSolve (2014)
  6. El-Mistikawy, Tarek M. A.: Modular analysis of sequential solution methods for almost block diagonal systems of equations (2013)
  7. Enright, W. H.; Muir, P. H.: New interpolants for asymptotically correct defect control of BVODEs (2010)
  8. Amodio, Pierluigi; Settanni, Giuseppina: Variable step/order generalized upwind methods for the numerical solution of second order singular perturbation problems (2009)
  9. Amodio, P.; Sgura, I.: High order generalized upwind schemes and numerical solution of singular perturbation problems (2007)
  10. Cash, Jeff R.: The numerical solution of nonlinear two-point boundary value problems using iterated deferred correction -- a survey (2006)
  11. Cash, J. R.; Mazzia, F.; Sumarti, N.; Trigiante, D.: The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems (2006)
  12. Enright, W. H.: Software for ordinary and delay differential equations: Accurate discrete approximate solutions are not enough (2006)
  13. Mazzia, Francesca; Sestini, Alessandra; Trigiante, Donato: BS linear multistep methods on non-uniform meshes (2006)
  14. Shampine, L. F.; Muir, P. H.; Xu, H.: A user-friendly Fortran BVP solver (2006)
  15. Amodio, Pierluigi; Sgura, Ivonne: High-order finite difference schemes for the solution of second-order BVPs (2005)
  16. Cash, J. R.; Mazzia, F.: A new mesh selection algorithm, based on conditioning, for two-point boundary value codes (2005)
  17. Shampine, L. F.: Solving ODEs and DDEs with residual control (2005)
  18. Cash, J. R.: A survey of some global methods for solving two-point BVPs (2004)
  19. Cash, J. R.; Moore, D. R.: High-order interpolants for solutionsof two-point boundary value problems using MIRK methods (2004)
  20. Fairweather, Graeme; Gladwell, Ian: Algorithms for almost block diagonal linear systems (2004)

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