Positivity‐preserving, flux‐limited finite‐difference and finite‐element methods for reactive transport
Positivity‐preserving, flux‐limited finite‐difference and finite‐element methods for reactive transport
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反应输运的保正性、通量有限的有限差分和有限元方法
DOI：
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发表时间：
2003
期刊：
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通讯作者：
G. Carey
中科院分区：
文献类型：
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作者：
R. MacKinnon;G. Carey
针对反应输运问题，提出了一类新的保正、限通量有限差分和Petrov-Galerkin （PG）有限元方法。这些方法与经典的TVD通量限制格式相似，主要区别在于通量限制器约束的设计是为了保持涉及扩散和反应的问题的正性。在有限元公式中，我们还考虑了集总质量矩阵和一致质量矩阵形式中数值正交对保正性的影响。对后一种格式的分析表明，只有当通量限制格式是隐式的并且满足时间步长的附加下界条件时，才能保证所得到的差分方程的保正解。我们证明了这个条件也适用于线性扩散方程的标准伽辽金线性有限元近似。数值实验证明了该方法的性能，并证实了有和没有非线性反应的输运问题的时间步长、网格间距和通量限制的理论条件。版权所有©2003 John Wiley & Sons， Ltd
A new class of positivity‐preserving, flux‐limited finite‐difference and Petrov–Galerkin (PG) finite‐element methods are devised for reactive transport problems.The methods are similar to classical TVD flux‐limited schemes with the main difference being that the flux‐limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite‐element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity‐preserving property. Analysis of the latter scheme shows that positivity‐preserving solutions of the resulting difference equations can only be guaranteed if the flux‐limited scheme is both implicit and satisfies an additional lower‐bound condition on time‐step size. We show that this condition also applies to standard Galerkin linear finite‐element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time‐step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd.