We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier-Stokes equations with a free surface for the accurate simulation of nonlinear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit 4th order Runge-Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge-Kutta stages. This result in the emergence of a Poisson pressure problem that constitute a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric $p$-multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the $p$-multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results.

翻译：本文提出了一种基于带自由表面的不可压缩Navier-Stokes方程的高阶精确计算流体动力学模型，用于时域内非线性色散水波的精确模拟。空间离散化采用垂直方向的Chebyshev多项式与水平方向的Fourier基函数，从而可利用快速Chebyshev变换和Fourier变换高效计算空间导数。时间离散通过广义低存储显式四阶Runge-Kutta方法实现；为保持质量守恒并达到高阶精度，需在所有Runge-Kutta阶段满足速度-压力耦合条件。这导致泊松压力问题的出现，该问题构成了质量守恒的几何守恒律。针对产生的泊松问题，本文提出采用基于几何$p$-多重网格方案的加速迭代求解器进行高效求解——该方法充分利用空间离散中的高阶多项式基函数，从而区别于传统的低阶数值格式。我们在数值波浪水槽背景下进行了数值实验以验证该方案：结果表明$p$-多重网格加速数值方案能有效解决作为计算瓶颈的泊松问题；模型可实现预期的谱收敛性；并能模拟非平坦底床上的波浪传播，与实验结果高度吻合。