High-order velocity and pressure boundary conditions are presented in Eulerian incompressible smoothed particle hydrodynamics (ISPH). While the high-order convergence of Eulerian ISPH has been demonstrated by the authors for periodic internal flows using Gaussian kernels this was limited by first to second-order accuracy for cases with solid boundaries. Since the SPH interpolation method is numerically robust there is potential for obtaining high-order accuracy in topologically complex domains with robust high-order accurate boundary conditions. In this paper high-order finite-difference extrapolation methods at solid boundaries are developed in Eulerian ISPH to allow for enforcement of the Dirichlet boundary condition for velocity and the Neumann boundary condition for pressure with high-order accuracy. Convergence up to fourth-order is demonstrated for 2-D Taylor-Couette flow and 3-D simulations of Taylor-Couette cellular flow structures are used to demonstrate accuracy and robustness. The order of accuracy may be extended to even higher-order using the analysis presented. Compact fourth-order Wendland-type kernels have also been derived to reduce the particle support region thereby lowering computational effort without loss of high-order convergence. The proposed formulation is therefore entirely high order.

高阶速度和压力边界条件在欧拉不可压缩平滑粒子流体动力学（ISPH）中给出。尽管作者已经证明了使用高斯核的周期性内部流动对欧拉ISPH的高阶收敛性，但对于具有固体边界的情况，这受到了一阶到二阶精度的限制。由于SPH插值方法在数值上是鲁棒的，因此具有在具有鲁棒的高阶精确边界条件的拓扑复杂域中获得高阶精度的潜力。在本文中，在欧拉ISPH中开发了在固体边界上的高阶有限差分外推方法，从而可以以高阶精度实施速度的Dirichlet边界条件和压力的Neumann边界条件。对2-D Taylor-Couette流动展示了高达四阶的收敛性，并且使用Taylor-Couette细胞流动结构的3-D仿真来演示准确性和鲁棒性。使用提供的分析，可以将精度的阶数扩展到更高阶。还已经导出了紧凑的四阶Wendland型内核以减少粒子支持区域，从而降低了计算工作量，而不会损失高阶收敛性。因此，提出的配方完全是高级的。还已经导出了紧凑的四阶Wendland型内核以减少粒子支持区域，从而降低了计算工作量，而不会损失高阶收敛性。因此，提出的配方完全是高级的。还已经导出了紧凑的四阶Wendland型内核以减少粒子支持区域，从而降低了计算工作量，而不会损失高阶收敛性。因此，提出的配方完全是高级的。