Stability of an explicit high-order spectral element method for acoustics in heterogeneous media based on local element stability criteria

This paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high-order spectral elements. The global stability criterion is taken as a minimum over local element stability criteria, obtained through the solution of element-borne eigenvalue problems. First, an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and/or rapidly fluctuating medium using asymptotic analysis. This criterion is only dependent upon the maximum velocity at the vertices of the mesh elements, and not on the velocity at the interior nodes of the high-order elements. Second, in a more general setting, bounds are derived using statistics of the coefficients of the elemental dispersion matrices. Different bounds are presented, discussed, and compared. Several numerical experiments show the accuracy of the proposed criteria in one-dimensional test cases as well as in more realistic large-scale three-dimensional problems.