Superconvergent interpolants for collocation methods applied to mixed-order BVODEs

ABSTRACT

Continuous approximations to boundary value problems in ordinary differential equations (BVODEs), constructed using collocation at Gauss points, are more accurate at the mesh points than at off-mesh points. From these approximations, it is possible to construct improved continuous approximations by extending the high accuracy that is available at the mesh points to off-mesh points. One possibility is the bootstrap approach, which improves the accuracy of the approximate solution at the off-mesh points in a sequence of steps until the accuracy at the mesh points and off-mesh points is consistent. A bootstrap approach for systems of mixed-order BVODEs is developed to improve approximate solutions produced by COLNEW, a Gauss-collocation-based software package. An implementation of this approach is discussed and numerical results presented which confirm that the improved approximations satisfy the predicted error bounds and are relatively inexpensive to construct.

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Boundary value problems

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation
      G.1.2 Approximation
          Subjects: Spline and piecewise polynomial approximation

General Terms:
Algorithms, Performance, Theory

Keywords:
Hermite-Birkhoff interpolation, bootstrap, collocation