一种用于椭圆型偏微分方程的保正性单网格方法
A positivity-preserving unigrid method for elliptic PDEs
AI总结：
针对带正性约束的离散椭圆型PDE，本文提出基于多重网格单网格变体的迭代算法，使近似求解全过程保持非负，并通过一维、二维数值实验验证其有效性与计算权衡。
摘要：
尽管约束在许多物理模型中自然出现，但它们在数学模型和数值模型中的处理方式差异很大，这取决于约束的性质以及是否有可用于强制满足该约束的仿真工具。在本文中，我们考虑求解离散化的偏微分方程（PDE）模型，这类模型对解的正性（或非负性）具有天然约束。虽然此类模型的离散化通常在其精确解上提供类似的正性性质，但使用近似求解算法（以及不可避免的浮点运算影响）往往会破坏计算所得近似解满足（离散形式的）物理约束的任何保证，除非将离散模型求解到远高于离散误差所要求的精度。在这里，我们提出一类基于多重网格方法的单网格变体的迭代求解算法，在该算法中，此类正性约束可以在整个近似求解过程中得到保持。针对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，也展示了为确保近似解在整个求解过程中保持正性所需付出的权衡。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
