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

翻译：对于含时偏微分方程，可通过在每个时间步求解一个约束最小化问题的后处理过程，使数值格式在不损失守恒性和精度的前提下保持界限制。此类约束优化可表述为非光滑凸最小化问题，若采用最优算法参数，则可通过一阶优化方法高效求解。通过分析广义Douglas-Rachford分裂方法的渐近线性收敛速率，可将最优算法参数近似表示为越界单元数量的简单函数。我们通过将该限制器应用于求解三维高难度相场问题的间断伽辽金格式的单元平均值，证明了这种简单算法参数选择的高效性。针对复杂的三维Cahn-Hilliard-Navier-Stokes系统的数值测试表明，该限制器具有高阶精度、极高效性，且适用于大规模计算。在每个时间步，Douglas-Rachford分裂方法至多迭代20次即可将界限制和守恒性约束满足到舍入误差水平，其计算代价至多为80N，其中N为单元总数。