High-Order interpolants for solutionsof two-point boundary value problems using MIRK methods

In this paper, high-order interpolants are presented for constructing continuous solutionsto a system of two-point boundary value differential equations between widely spaced but accurate dependent variable values. These interpolants are local and symmetric, requiring data only within a single mesh interval and they require a small number of right-hand side evaluations of the defining ODE system to achieve the required order of accuracy. Internal derivative information in the mono-implicit Runge-Kutta formulae is exploited to reduce the number of additional right-hand side evaluations necessary to define the interpolant to the required order of accuracy. When the underlying ODE system is second order, very economical and accurate interpolants are found. All of the interpolants are suitable for grid refinement algorithms in automatic adaptive two-point boundary value packages such as TWPBVP.