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Shell Lecture Series - 2/05, 3:00PM, Duncan Hall 1064

Mark H. Carpenter


"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.

Shell Lectures Series are made possible through the generous support of the Shell Oil Company.

Department of Computational and Applied Mathematics
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