Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable

Abstract

We investigate three effects of the variable geometric terms that arise when approximating linear conservation laws on curved elements with a provably stable skew-symmetric variant of the discontinuous Galerkin spectral element method (DGSEM). We show for a constant coefficient system that the non-constant coefficient problem generated by mapping a curved element to the reference element is stable and has energy bounded by the initial value as long as the discrete metric identities are satisfied. Under those same conditions, the skew-symmetric approximation is also constant state preserving and discretely conservative, just like the original DGSEM.