变分不等式方法用于强制对流传输方程的非负约束
Variational inequality approach to enforce the non-negative constraint for advection-diffusion equations
AI总结：
本文提出基于变分不等式的方法，用于强制对流传输方程的非负约束，同时适用于扩散方程和非对称对流传输方程，并展示了其在大规模问题中的并行性能。
摘要：
预测模拟对于许多地下应用的成功至关重要，获得准确的非负解对于这些数值模拟中的传输方程至关重要。本文提出了一种基于变分不等式（VI）的计算框架，可用于强制重要的数学属性（如最大原理）和物理约束（如非负约束）。我们证明该框架不仅适用于扩散方程，也适用于非对称对流传输方程。该框架的一个优点是它可以与任何弱形式的对流传输方程结合使用，包括单场形式，这些形式在计算上具有吸引力。特别强调了VI方法在大规模和异质问题中的并行和算法性能。还展示了在某些条件下QP和VI是等价的。使用PETSc库中的最新QP和VI求解器在各种稳态2D和3D基准上进行测试，并展示了QP和VI求解器的可扩展性比较研究。然后通过模拟异质多孔介质中流体的混溶置换将所提框架扩展到瞬态问题，并说明了在这些耦合问题中强制最大原理的重要性。我们的数值实验表明，VI确实是一种在大规模计算环境中强制最大原理和非负约束的有效方法。还提供了Firedrake项目文件以及关于计算机实现的讨论，以帮助读者理解所提框架。
英文摘要：
Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection-diffusion equations. An attractive feature of the proposed framework is that it works with with any weak formulation for the advection-diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework.
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