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. Famelis, Ioannis Th.; Tsitouras, Ch.: Quadratic shooting solution for an environmental parameter prediction problem (2015)
  2. Mazzia, Francesca; Cash, Jeff R.; Soetaert, Karline: Solving boundary value problems in the open source software R: package bvpSolve (2014)
  3. Corless, Robert M.; Fillion, Nicolas: A graduate introduction to numerical methods. From the viewpoint of backward error analysis (2013)
  4. El-Mistikawy, Tarek M.A.: Modular analysis of sequential solution methods for almost block diagonal systems of equations (2013)
  5. Enright, W.H.; Muir, P.H.: New interpolants for asymptotically correct defect control of BVODEs (2010)
  6. Amodio, Pierluigi; Settanni, Giuseppina: Variable step/order generalized upwind methods for the numerical solution of second order singular perturbation problems (2009)
  7. Amodio, P.; Sgura, I.: High order generalized upwind schemes and numerical solution of singular perturbation problems (2007)
  8. Amodio, Pierluigi; Romanazzi, Giuseppe: Algorithm 859: BABDCR -- a Fortran 90 package for the solution of bordered ABD linear systems. (2006)
  9. 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)
  10. Mazzia, Francesca; Sestini, Alessandra; Trigiante, Donato: BS linear multistep methods on non-uniform meshes (2006)
  11. Cash, J.R.; Mazzia, F.: A new mesh selection algorithm, based on conditioning, for two-point boundary value codes (2005)
  12. Shampine, L.F.: Solving ODEs and DDEs with residual control (2005)
  13. Shampine, L.F.; Muir, P.H.: Estimating conditioning of BVPs for ODEs (2004)
  14. Aceto, Lidia; Mazzia, Francesca; Trigiante, Donato: The performances of the code TOM on the Holt problem (2003)
  15. Muir, P.H.; Pancer, R.N.; Jackson, K.R.: PMIRKDC: a parallel mono-implicit Runge-Kutta code with defect control for boundary value ODEs (2003)
  16. Horgan, Cornelius O.; Saccomandi, Giuseppe; Sgura, Ivonne: A two-point boundary value problem for the axial shear of hardening isotropic incompressible nonlinearly elastic materials (2002)
  17. Kierzenka, Jacek; Shampine, Lawrence F.: A BVP solver based on residual control and the Maltab PSE (2001)
  18. Muir, P.H.; Adams, M.: Mono-implicit Runge-Kutta-Nyström methods with application to boundary value ordinary differential equations (2001)
  19. Amodio, P.; Cash, J.R.; Roussos, G.; Wright, R.W.; Fairweather, G.; Gladwell, I.; Kraut, G.L.; Paprzycki, M.: Almost block diagonal linear systems: sequential and parallel solution techniques, and applications. (2000)
  20. Enright, W.H.: Accurate approximate solution of partial differential equations at off-mesh points (2000)

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