While constraints arise naturally in many physical models, their treatment in mathematical and numerical models varies widely, depending on the nature of the constraint and the availability of simulation tools to enforce it. In this paper, we consider the solution of discretized PDE models that have a natural constraint on the positivity (or non-negativity) of the solution. While discretizations of such models often offer analogous positivity properties on their exact solutions, the use of approximate solution algorithms (and the unavoidable effects of floating -- point arithmetic) often destroy any guarantees that the computed approximate solution will satisfy the (discretized form of the) physical constraints, unless the discrete model is solved to much higher precision than discretization error would dictate. Here, we introduce a class of iterative solution algorithms, based on the unigrid variant of multigrid methods, where such positivity constraints can be preserved throughout the approximate solution process. Numerical results for one- and two-dimensional model problems show both the effectiveness of the approach and the trade-off required to ensure positivity of approximate solutions throughout the solution process.

翻译：尽管约束条件在许多物理模型中自然产生，但根据约束的性质以及实施约束的模拟工具的可用性，其在数学和数值模型中的处理方法差异很大。本文考虑具有解的正性（或非负性）自然约束的离散化偏微分方程模型的求解问题。虽然此类模型的离散化方法通常在其精确解上保持类似的正性性质，但近似求解算法（以及浮点运算不可避免的影响）往往会破坏计算近似解满足物理约束（的离散形式）的任何保证，除非离散模型以远高于离散化误差所需的精度进行求解。本文引入一类基于多重网格方法单网格变体的迭代求解算法，可在近似求解过程中保持此类正性约束。一维和二维模型问题的数值结果既展示了该方法的有效性，也说明了在整个求解过程中确保近似解正性所需付出的代价。