  The paper aims at developing low-storage implicit Runge-Kutta methods which
are easy to implement and achieve higher-order of convergence for both the
velocity and pressure in the finite volume formulation of the incompressible
Navier-Stokes equations on a static collocated grid. To this end, the effect of
the momentum interpolation, a procedure required by the finite volume method
for collocated grids, on the differential-algebraic nature of the
spatially-discretized Navier-Stokes equations should be examined first. A new
framework for the momentum interpolation is established, based on which the
semi-discrete Navier-Stokes equations can be strictly viewed as a system of
differential-algebraic equations of index 2. The accuracy and convergence of
the proposed momentum interpolation framework is examined. We then propose a
new method of applying implicit Runge-Kutta schemes to the time-marching of the
index 2 system of the incompressible Navier-Stokes equations. Compared to the
standard method, the proposed one significantly reduces the numerical
difficulties in momentum interpolations and delivers higher-order pressures
without requiring additional computational effort. Applying stiff-accurate
diagonal implicit Runge-Kutta (DIRK) schemes with the proposed method allows
the schemes to attain the classical order of convergence for both the velocity
and pressure. We also develop two families of low-storage stiff-accurate DIRK
schemes to reduce the storage required by their implementations. Examining the
two dimensional Taylor-Green vortex as an example, the spatial and temporal
accuracy of the proposed methods in simulating incompressible flow is
demonstrated.
