  In this paper, we investigate the application of the maximum principle
preserving (MPP) parametrized flux limiters to the high order finite volume
scheme with Runge-Kutta time discretization for solving convection dominated
problems. Such flux limiter was originally proposed in [Xu, Math. Comp., 2013]
and further developed in [Xiong et. al., J. Comp. Phys., 2013] for finite
difference WENO schemes with Runge-Kutta time discretization for convection
equations. The main idea is to limit the temporal integrated high order
numerical flux toward a first order MPP monotone flux. In this paper, we
generalize such flux limiter to high order finite volume methods solving
convection-dominated problems, which is easy to implement and introduces little
computational overhead. More importantly, for the first time in the finite
volume setting, we provide a general proof that the proposed flux limiter
maintains high order accuracy of the original WENO scheme for linear advection
problems without any additional time step restriction. For general nonlinear
convection-dominated problems, we prove that the proposed flux limiter
introduces up to $O(\Delta x^3+\Delta t^3)$ modification to the high order
temporal integrated flux in the original WENO scheme without extra time step
constraint. We also numerically investigate the preservation of up to ninth
order accuracy of the proposed flux limiter in a general setting. The advantage
of the proposed method is demonstrated through various numerical experiments.
