Entropy Stability Theory For Difference Approximations of Quasilinear Problems

Friday, May 14, 2004 - 9:30am - 10:20am
Keller 3-180
Eitan Tadmor (University of Maryland)
We provide a general overview on the entropy stability of difference approximations in the context of quasilinear conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms.

As our main tool we use a comparison principle, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes. To this end, we introduce closed-form expressions for new (families) of new entropy-conservative schemes, keeping the 'perfect differencing' of the underlying differential form. In particular, entropy stability is enforced on rarefactions while keeping sharp resolution of shock discontinuities.

A comparison with the numerical viscosities associated with these entropy-conservative schemes provides a useful framework for the construction and analysis of existing and new entropy stable scheme. We employ this framework for a host of first- and second-order accurate schemes. The comparison approach yields precise characterizations of entropy stable semi-discrete schemes for both scalar problems and multi-dimensional systems of equations. We extend these results to the fully discrete case, where the question of stability is settled under optimal CFL conditions using a complementary approach based on homotopy arguments.