We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to one.

我们开发了一种数值方法，以节能的方式在交错曲线网格上以协变形式求解声波方程。协方差基础分解的使用导致旋转不变方案，该方案在旋转网格上的表现优于笛卡尔基础分解。离散化基于高阶按部分求和（SBP）算子，并且保留了逆度量张量的对称性和正定性。为了提高准确性并降低计算成本，我们还导出了度量张量的修正离散化，从而导致条件稳定的离散化。得出产生点状条件的边界，可以对其进行评估以检查修改后的离散化的稳定性。