We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

我们提出了两种准确有效的算法，用于求解不可压缩、无旋欧拉方程，该方程具有二维自由表面，背景流动在周期性、多重连接的流体域上，包括静止障碍物和可变底部地形。一种方法是根据表面速度势制定的，而另一种方法是发展涡流片强度。两种方法都采用周期性柯西积分形式的层势来计算自由表面的法向速度，与自由表面和边界的任意参数化兼容，并允许围绕每个障碍物进行循环，从而产生多值速度势但单值流函数。我们证明了得到的第二类 Fredholm 积分方程是可逆的，可能是在物理驱动的有限秩校正之后。在角度-弧长设置中，我们展示了如何避免与空间周期性不兼容的曲线重建错误。我们使用所提出的方法来研究由在平坦或可变底部边界上方的几个椭圆障碍物周围流动产生的重力毛细波。在每种情况下，自由表面最终都会在飞溅奇点中自相交或与边界碰撞。我们还展示了如何以光谱精度评估整个流体的速度和压力，包括自由表面和固体边界附近。为了评估时间演化的准确性，我们监测能量守恒和傅里叶模式的衰减，并将两种方法的数值结果相互比较。我们为离散线性系统实现了几个求解器并比较了它们的性能。最快的方法是使用图形处理单元 (GPU) 来构建矩阵并执行广义最小残差法 (GMRES) 的迭代。