High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time (FOT) and second-order-in-time (SOT) Hermite schemes are developed. Hermite methods use the solution and multiple derivatives as unknowns and achieve space-time orders of accuracy $2m-1$ (FOT) and $2m$ (SOT) for methods using $(m+1)^d$ degree of freedom per node in $d$ dimensions. The compatibility boundary conditions (CBCs) are based on taking time derivatives of the boundary conditions and using the governing equations to replace the time derivatives with spatial derivatives. These resulting constraint equations augment the Hermite scheme on the boundary. The solvability of the equations resulting from the compatibility conditions are analyzed. Numerical examples demonstrate the accuracy and stability of the new schemes in two dimensions.

翻译：本文提出了曲线域上波动方程的高阶精确 Hermite 方法。边界处理采用中心化相容性条件，而非更标准的单侧近似。文中发展了时间一阶（FOT）和时间二阶（SOT）的 Hermite 格式。Hermite 方法将解及其多阶导数作为未知量，在 $d$ 维空间中每个节点使用 $(m+1)^d$ 个自由度时，分别达到 $2m-1$（FOT）和 $2m$（SOT）的时空精度阶。相容性边界条件（CBCs）基于对边界条件取时间导数，并利用控制方程将时间导数替换为空间导数。由此产生的约束方程增强了边界上的 Hermite 格式。文中分析了由相容性条件导出的方程的可解性。数值算例在二维情况下验证了新格式的精确性与稳定性。