PMIRKDC: a parallel mono-implicit Runge-Kutta code with defect control for boundary value ODEs We describe parallel software, PMIRKDC, for solving boundary value ordinary differential equations (BVODEs). This software is based on the package, MIRKDC, which employs mono-implicit Runge-Kutta schemes within a defect control algorithm. The primary computational costs involve the treatment of large, almost block diagonal (ABD) linear systems. The most significant feature of PMIRKDC is the replacement of sequential ABD software, COLROW, with new parallel ABD software, RSCALE, based on a parallel block eigenvalue rescaling algorithm. Other modifications involve parallelization of the setup of the ABD systems and solution interpolants, and defect estimation. Numerical results show almost linear speedups.

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  1. McLachlan, Robert I.; Offen, Christian: Preservation of bifurcations of Hamiltonian boundary value problems under discretisation (2020)
  2. Corless, Robert M.; Kaya, C. Yalçın; Moir, Robert H. C.: Optimal residuals and the Dahlquist test problem (2019)
  3. Keskin, Ali Ümit: Boundary value problems for engineers. With MATLAB solutions (2019)
  4. McLachlan, Robert I.; Offen, Christian: Symplectic integration of boundary value problems (2019)
  5. Putkaradze, Vakhtang; Rogers, Stuart: Constraint control of nonholonomic mechanical systems (2018)
  6. Famelis, Ioannis Th.; Tsitouras, Ch.: Quadratic shooting solution for an environmental parameter prediction problem (2015)
  7. Fabien, Brian C.: Parallel indirect solution of optimal control problems (2014)
  8. Mazzia, Francesca; Cash, Jeff R.; Soetaert, Karline: Solving boundary value problems in the open source software R: package bvpSolve (2014)
  9. Corless, Robert M.; Fillion, Nicolas: A graduate introduction to numerical methods. From the viewpoint of backward error analysis (2013)
  10. El-Mistikawy, Tarek M. A.: Modular analysis of sequential solution methods for almost block diagonal systems of equations (2013)
  11. Enright, W. H.; Muir, P. H.: New interpolants for asymptotically correct defect control of BVODEs (2010)
  12. Amodio, Pierluigi; Settanni, Giuseppina: Variable step/order generalized upwind methods for the numerical solution of second order singular perturbation problems (2009)
  13. Amodio, P.; Sgura, I.: High order generalized upwind schemes and numerical solution of singular perturbation problems (2007)
  14. Amodio, Pierluigi; Romanazzi, Giuseppe: Algorithm 859: BABDCR -- a Fortran 90 package for the solution of bordered ABD linear systems. (2006)
  15. 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)
  16. Mazzia, Francesca; Sestini, Alessandra; Trigiante, Donato: BS linear multistep methods on non-uniform meshes (2006)
  17. Cash, J. R.; Mazzia, F.: A new mesh selection algorithm, based on conditioning, for two-point boundary value codes (2005)
  18. Shampine, L. F.: Solving ODEs and DDEs with residual control (2005)
  19. Cash, J. R.; Moore, D. R.: High-order interpolants for solutionsof two-point boundary value problems using MIRK methods (2004)
  20. Shampine, L. F.; Muir, P. H.: Estimating conditioning of BVPs for ODEs (2004)

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