"Nonlinear Stability of the Compressible Navier-Stokes Eqns: Nonconforming Interfaces"
Nonlinearly stable (entropy stable) discretizations of arbitrary order exist for the compressible Navier-Stokes (NS) equations for all diagonal norm, tensor-product, summation-by-parts (SBP) operators. The NS equations are discretized in strong conservation form, and a novel choice of nonlinear fluxes ensures pointwise conservation of mass, momentum, energy and a nonlinear entropy function that guarantees L2 stability of the (semi-)discrete solution. The stability estimates are sharp and do not rely on common assumptions of "integral exactness,", or "added dissipation." The discrete operators are fully consistent with the Lax-Wendroff theorem. Thus, captured shocks converge to weak solutions provided physical dissipation is sufficient at shocks.
A high-level overview of the SBP entropy stability literature is given. Then, recent progress is reported on developing entropy-stable (SS) discontinuous spectral collocation formulations for hexahedral elements. This effort extends previous work on entropy stability to include p-refinement at curvilinear nonconforming interfaces. A generalization of existing entropy stability theory is required to accommodate the nuances of the nonconforming curvilinear coupling.
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