论文标题
达西定律，术语
Darcy's Law with a Source term
论文作者
论文摘要
我们介绍了JKO方案的一种新型变体，以使用依赖压力的源术语来近似Darcy定律。通过引入一个隐式控制源术语的新变量，即使总质量随时间变化，我们的方案仍然能够使用标准的Wasserstein-2-Metric。利用我们方案的双重公式，我们表明离散的时间近似满足了连续解决方案预期的许多有用属性，例如比较原理和均匀的$ l^1 $ equiciContinuity。即使在不存在增长期限的情况下，这些属性也是新的。最后，我们表明我们的离散近似收敛到相应的PDE系统的解决方案，包括具有一般非线性源项的肿瘤生长模型。
We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform $L^1$-equicontinuity. Many of these properties are new even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.
