The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and derived forms such as the projection method of Taira and Colonius (J. Comput. Phys., 2007), have been useful for simulating flow physics in problems with moving interfaces on stationary grids. However, in their interface treatment, these methods do not distinguish one side from the other, but rather, apply the motion constraint to both sides, and the associated interface force is an inseparable mix of contributions from each side. In this work, we define a discrete Heaviside function, a natural companion to the familiar discrete Dirac delta function (DDF), to define a masked version of each field on the grid which, to within the error of the DDF, takes the intended value of the field on the respective sides of the interface. From this foundation we develop discrete operators and identities that are uniformly applicable to any surface geometry. We use these to develop extended forms of prototypical partial differential equations, including Poisson, convection-diffusion, and incompressible Navier-Stokes, that govern the discrete masked fields. These equations contain the familiar forcing term of the IBM, but also additional terms that regularize the jumps in field quantities onto the grid and enable us to individually specify the constraints on field behavior on each side of the interface. Drawing the connection between these terms and the layer potentials in elliptic problems, we refer to them generically as immersed layers. We demonstrate the application of the method to several representative problems, including two-dimensional incompressible flows inside a rotating cylinder and external to a rotating square.

Peskin (J. Comput. Phys., 1977) 的浸入边界法 (IBM) 以及诸如 Taira 和 Colonius 的投影方法 (J. Comput. Phys., 2007) 等派生形式对于模拟流动物理非常有用在固定网格上移动界面的问题中。然而，在它们的界面处理中，这些方法并没有将一侧与另一侧区分开来，而是对两侧施加运动约束，相关的界面力是来自每一侧的不可分割的组合。在这项工作中，我们定义了一个离散 Heaviside 函数，它是熟悉的离散狄拉克 delta 函数 (DDF) 的自然伴侣，用于定义网格上每个字段的掩码版本，在 DDF 的误差范围内，取预期值界面两侧的字段。在此基础上，我们开发了统一适用于任何表面几何形状的离散算子和恒等式。我们使用这些来开发扩展形式的原型偏微分方程，包括泊松、对流扩散和不可压缩的 Navier-Stokes，它们控制着离散的掩蔽场。这些方程包含熟悉的 IBM 强迫项，但也包含附加项，这些项将场量跳跃到网格上进行正则化，并使我们能够单独指定界面每一侧的场行为约束。绘制这些术语与椭圆问题中层势之间的联系，我们将它们统称为浸入层。我们演示了该方法在几个有代表性的问题中的应用，