A Numerical Study of Global Error and Defect Control Schemes for BVODEs                                                   
                                                                     
                                                                     
                                             
A Numerical Study of Global Error and Defect Control Schemes for BVODEs 

Jason J. Boisvert, Paul H. Muir, and Raymond J. Spiteri

Abstract

  Boundary value ordinary differential equations (BVODEs) are systems
  of ODEs with boundary conditions imposed at two or more distinct
  points.  The global error (GE) of a numerical solution to a BVODE is
  the difference between that numerical solution and the exact
  solution. The defect is the amount by which the numerical solution
  fails to satisfy the ODEs and boundary conditions.  The BVODE
  solver, BVP_SOLVER, computes a numerical solution whose (estimated)
  defect satisfies a given user tolerance but it can also provide an a
  posteriori estimate of the GE using Richardson extrapolation
  (RE). Using a modified version of BVP_SOLVER, we present, in this
  report, numerical experiments comparing four strategies for a
  posteriori GE estimation of a defect controlled numerical solution,
  based on (i) RE, (ii) the direct use of a higher order (HO)
  discretization formula (a mono-implicit Runge-Kutta (MIRK) formula),
  (iii) the use of a higher order discretization formula (a MIRK
  formula) within a deferred correction (DC) framework, and (iv) the
  product of the defect estimate and an estimate of the BVODE
  conditioning constant (CO).  We also present numerical experiments
  investigating a (further) modified version of BVP_SOLVER that
  provides options for (i) defect control (DefC), (ii) GE control
  (GEC), and combinations thereof: (iii) a sequential combination
  control (SCC) in which we first compute a defect controlled solution
  and then, using this solution, continue on to compute a GE
  controlled solution, and (iv) a parallel combination control (PCC)
  in which we simultaneously control estimates of the defect and the GE.