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For time-dependent PDEs, the numerical schemes can be renderedbound-preserving without losing conservation and accuracy, by a post processingprocedure of solving a constrained minimization in each time step. Such aconstrained optimization can be formulated as a nonsmooth convex minimization,which can be efficiently solved by first order optimization methods, if usingthe optimal algorithm parameters. By analyzing the asymptotic linearconvergence rate of the generalized Douglas-Rachford splitting method, optimalalgorithm parameters can be approximately expressed as a simple function of thenumber of out-of-bounds cells. We demonstrate the efficiency of this simplechoice of algorithm parameters by applying such a limiter to cell averages of adiscontinuous Galerkin scheme solving phase field equations for 3D demandingproblems. Numerical tests on a sophisticated 3D Cahn-Hilliard-Navier-Stokessystem indicate that the limiter is high order accurate, very efficient, andwell-suited for large-scale simulations. For each time step, it takes at most$20$ iterations for the Douglas-Rachford splitting to enforce bounds andconservation up to the round-off error, for which the computational cost is atmost $80N$ with $N$ being the total number of cells.

A simple and efficient convex optimization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system