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：对于时间依赖型偏微分方程，可通过在每个时间步求解约束最小化的后处理过程，使数值格式在不损失守恒性和精度的前提下保持界限性。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，则可利用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，最优算法参数可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方程的不连续Galerkin格式的单元平均值，验证了这种简单参数选取策略的效率。针对复杂三维Cahn-Hilliard-Navier-Stokes系统的数值测试表明，该限制器具有高阶精度、极高效率，且适用于大规模数值模拟。在每个时间步中，Douglas-Rachford分裂方法最多仅需20次迭代即可将界约束和守恒性保持至舍入误差水平，其计算成本不超过$80N$（其中$N$为单元总数）。