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