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：本文提出了在曲线域上求解波动方程的高阶精确 Hermite 方法。边界处理采用中心化的相容性条件，而非更标准的单侧近似。我们发展了时间一阶（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，对于在 $d$ 维空间中每个节点使用 $(m+1)^d$ 个自由度的格式，其时空精度阶分别达到 $2m-1$（FOT）和 $2m$（SOT）。相容性边界条件（CBCs）基于对边界条件求时间导数，并利用控制方程将时间导数替换为空间导数。这些导出的约束方程增强了边界处的 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。