  In this paper, we introduce a class of high order immersed finite volume
methods (IFVM) for one-dimensional interface problems. We show the optimal
convergence of IFVM in H1 and L2 norms. We also prove some superconvergence
results of IFVM. To be more precise, the IFVM solution is superconvergent of
order p+2 at the roots of generalized Lobatto polynomials, and the flux is
superconvergent of order p+1 at generalized Gauss points on each element
including the interface element. Furthermore, for diffusion interface problems,
the convergence rates for IFVM solution at the mesh points and the flux at
generalized Gaussian points can both be raised to 2p. These superconvergence
results are consistent with those for the standard finite volume methods.
Numerical examples are provided to confirm our theoretical analysis.
