Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

•We derive a provably stable summation-by-parts finite difference method for the acoustic wave equation on general curvilinear staggered grids.•All of the velocity components and pressure field are staggered in the grid.•Rotational invariance is preserved by decomposing the velocity field with respect to the covariant basis.•Two alternatives are provided for discretizing the metric tensor such that the discrete kinetic energy becomes positive.•A modified discretization of the metric tensor is proposed that improves accuracy and efficiency. An efficient approach is derived for ensuring that the modified discretization is stable on a given mesh. 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.