The initial condition problem for a binary neutron star system requires a Poisson equation solver for the velocity potential with a Neumann-like boundary condition on the surface of the star. Difficulties that arise in this boundary value problem are: (a) the boundary is not known a priori, but constitutes part of the solution of the problem; (b) various terms become singular at the boundary. In this work, we present a new method to solve the fluid Poisson equation for irrotational/spinning binary neutron stars. The advantage of the new method is that it does not require complex fluid surface fitted coordinates and it can be implemented in a Cartesian grid, which is a standard choice in numerical relativity calculations. This is accomplished by employing the source term method proposed by Towers, where the boundary condition is treated as a jump condition and is incorporated as additional source terms in the Poisson equation, which is then solved iteratively. The issue of singular terms caused by vanishing density on the surface is resolved with an additional separation that shifts the computation boundary to the interior of the star. We present two-dimensional tests to show the convergence of the source term method, and we further apply this solver to a realistic three-dimensional binary neutron star problem. By comparing our solution with the one coming from the initial data solver cocal, we demonstrate agreement to approximately 1%. Our method can be used in other problems with non-smooth solutions like in magnetized neutron stars.

双中子星系统的初始条件问题需要一个泊松方程求解器，求解速度势，在恒星表面上具有类似诺依曼的边界条件。在这个边界值问题中出现的困难是：（a）边界不是先验的，但构成问题解决方案的一部分；(b) 各种术语在边界处变为单数。在这项工作中，我们提出了一种求解无旋/自旋双中子星的流体泊松方程的新方法。新方法的优点是不需要复杂的流体表面拟合坐标，可以在笛卡尔网格中实现，这是数值相对论计算中的标准选择。这是通过采用 Towers 提出的源项方法来实现的，其中边界条件被视为跳跃条件，并作为附加源项合并到泊松方程中，然后迭代求解。由表面密度消失引起的奇异项问题通过额外的分离得到解决，该分离将计算边界转移到恒星的内部。我们提出了二维测试来显示源项方法的收敛性，我们进一步将此求解器应用于现实的三维双中子星问题。通过将我们的解决方案与来自初始数据求解器 cocal 的解决方案进行比较，我们证明了大约 1% 的一致性。我们的方法可用于其他非光滑解的问题，如磁化中子星。我们证明同意约 1%。我们的方法可用于其他非光滑解的问题，如磁化中子星。我们证明同意约 1%。我们的方法可用于其他非光滑解的问题，如磁化中子星。