Selling's decomposition and the anisotropic wave equation

The design of efficient numerical schemes for anisotropic partial differential equations, featuring privileged directions in general not aligned with the coordinate axes, raises challenging discretization problems which can often be addressed with the tools of algorithmic geometry. We introduce and study a novel scheme for the anisotropic scalar wave equation, discretized on Cartesian grids, and prove that it has lower numerical dispersion than its classical counterparts in anisotropic settings. The scheme is based on Selling's decomposition of symmetric positive definite matrices, of which we construct a three-dimensional variant with smooth coefficients, used to establish second or fourth order convergence rates depending on the scheme variant. Numerical experiments illustrate our results.