A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving
A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving
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DOI：
10.1016/j.cma.2022.114585
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发表时间：
2021-07
影响因子：
7.2
通讯作者：
Q. Cheng;Jie Shen
中科院分区：
文献类型：
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作者：
Q. Cheng;Jie Shen
提出了一种新的拉格朗日乘子方法来构造抛物型方程的保正格式。新方法引入了时空拉格朗日乘子，以加强与Karush-Kuhn-Tucker（KKT）条件的积极性。然后，我们使用预测-校正方法来构造一类正性方案：用通用的半隐式或隐式方案作为预测步骤，并且执行正性的校正步骤可以以可忽略的成本实现。我们还提出了一个修改，使我们能够构建的计划，除了积极性保存，也是质量守恒。这种新的方法不限于任何特定的空间离散化，并可以与各种时间离散计划相结合。我们建立稳定性结果，我们的第一和第二阶计划下的一般设置，并提出了大量的数值结果来验证新的方法。
We propose a new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations. The new approach introduces a space–time Lagrange multiplier to enforce the positivity with the Karush–Kuhn–Tucker (KKT) conditions. We then use a predictor–corrector approach to construct a class of positivity schemes: with a generic semi-implicit or implicit scheme as the prediction step, and the correction step, which enforces the positivity, can be implemented with negligible cost. We also present a modification which allows us to construct schemes which, in addition to positivity preserving, is also mass conserving. This new approach is not restricted to any particular spatial discretization and can be combined with various time discretization schemes. We establish stability results for our first- and second-order schemes under a general setting, and present ample numerical results to validate the new approach.