A family of uniformly accurate order Lobatto-Runge-Kutta collocation methods

we consider the construction of an interpolant for use with lobatto-runge-kutta collocation methods. the main aim is to derive single symmetric continuous solution(interpolant) for uniform accuracy at the step points as well as at the off-step points whose uniform order six everywhere in the interval of consideration. we evaluate the continuous scheme at different off-step points to obtain multi-hybrid schemes which if desired can be solved simultaneously for dense approximations. the multi-hybrid schemes obtained were converted to lobatto-runge-kutta collocation methods for accurate solution of initial value problems. the unique feature of the paper is the idea of using all the set of off-step collocation points as additional interpolation points while symmetry is retained naturally by integration identities as equal areas under the various segments of the solution graph over the interval of consideration. we show two possible ways of implementing the interpolant to achieve the aim and compare them on some numerical examples.