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.

中文翻译：

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具有源项的达西定律

我们引入了 JKO 方案的一种新颖变体，以使用与压力相关的源项来近似达西定律。通过引入一个隐式控制源项的新变量，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。利用我们方案的对偶公式，我们表明时间离散近似满足连续解的许多有用属性，例如比较原理和均匀 $L^1$-equicontinuity。即使在没有增长项的众所周知的情况下，这些属性中的许多也是新的。最后，我们展示了我们的离散近似收敛到相应 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。