标题： 水波在障碍物和地形上的背景流数值算法 显示英文标题

标题： Numerical Algorithms for Water Waves with Background Flow over Obstacles and Topography

摘要： 我们提出了两种精确且高效的算法，用于在二维情况下求解具有自由表面的不可压缩、无旋欧拉方程，在周期性、多连通流体域上存在背景流，该域包括静止障碍物和可变底部地形。 一种方法是基于表面速度势来表述，而另一种方法则是演化涡量片强度。 这两种方法都采用周期化柯西积分形式的层势来计算自由表面的法向速度，与任意参数化的自由表面和边界兼容，并允许每个障碍物周围有环量，这导致速度势为多值，但流函数为单值。 我们证明了得到的第二类弗雷德霍姆积分方程是可逆的，可能需要经过一个物理上有动机的有限秩修正。 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误差。 我们使用所提出的方法研究了流体绕几个椭圆障碍物上方的平坦或可变底部边界产生的重力毛细波。 在每种情况下，自由表面最终都会在溅射奇点处自我相交或与边界碰撞。 我们还展示了如何在整个流体中以谱精度计算速度和压力，包括在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态的衰减，并将两种方法的数值结果进行比较。 我们实现了几种离散线性系统的求解器，并比较了它们的性能。 最快的方法使用图形处理单元（GPU）来构建矩阵并执行广义最小残差方法（GMRES）的迭代。 显示英文摘要

摘要： 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).