The ghost-cell immersed boundary methods (IBMs) are widely used to implement boundary conditions on non-body fitted grids. It has been previously shown that they have two major drawbacks. Firstly, these methods tend to have a maximum stencil size larger than 1, yielding non-band matrices. Secondly, for a maximum stencil size of 2 the ghost-cell linear IBM [1] have a first-order convergence rate for Neumann immersed boundary conditions. To address these two shortcomings and in the pursuit of increased accuracy and increased order of convergence the current article proposes the linear/quadratic square shifting methods for the ghost-cell IBM for Cartesian grids. The linear square shifting method guarantees a maximum stencil size of 1 and also improves the accuracy and convergence while the quadratic square shifting method improves the accuracy and convergence while maintaining the same stencil size of 2 as the original linear method [1]. The quadratic ghost-cell method [2], [3] further improves the accuracy and convergence whilst maintaining a maximum stencil size of 3 while the currently proposed quadratic square shifting method makes it possible to increase the Lagrange polynomial interpolation order whilst maintaining a maximum stencil of 2 resulting in an improved order of convergence for Neumann immersed boundary conditions. The proposed methods are evaluated by considering their accuracy and convergence thanks to a comprehensive verification and validation process. Firstly, the canonical verification 2D and 3D Poisson test problems for various analytical solutions and immersed boundaries are considered and are followed by various verification and validation test cases for the Navier-Stokes governing equations, with and without heat transfer, in 2D and 3D.

鬼单元浸没边界方法（IBMs）被广泛用于在非人体拟合网格上实现边界条件。先前已证明它们具有两个主要缺点。首先，这些方法的最大模版尺寸往往大于1，从而产生非带状矩阵。其次，对于最大模版大小为2的鬼单元线性IBM [1]对于Neumann浸入边界条件具有一阶收敛速度。为了解决这两个缺点，并在追求更高的准确性和更高的收敛阶数的情况下，当前文章提出了适用于笛卡尔网格的重影单元IBM的线性/二次方平移方法。线性平方移位方法可确保最大的模具尺寸为1，并且还提高了精度和收敛性，而二次方形移位方法则在保持与原始线性方法相同的2的模板尺寸的同时，提高了精度和收敛性。二次幻像元方法[2]，[3]进一步提高了精度和收敛性，同时保持最大模版大小为3，而当前提出的二次平方移位方法使得可以在保持最大模版的同时增加Lagrange多项式插值阶数的2导致Neumann浸入边界条件的收敛阶得到改善。通过全面的验证和确认过程，可以通过考虑其准确性和收敛性来评估所提出的方法。首先，