We extend recent results by Pisier on$K$-subcouples, i.e. subcouples of an interpolation couple that preserve the$K$-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

We introduce and analyze lower ($Ricci$) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾$K$ $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .

No abstract uploaded!

In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution [X. Liu, S. Zhang, H. Zhang and C-W. Shu, A new class of central compact schemes with spectral-like resolution I: Linear schemes, Journal of Computational Physics 248 (2013) 235-256]. In our approach, we compute the flux derivatives on the cell-nodes by the physical fluxes on the cell nodes and numerical fluxes on the cell centers. To acquire the numerical fluxes on the cell centers, we perform a weighted hybrid interpolation of an upwind interpolation and a central interpolation. Through systematic analysis and numerical tests, we show that our nonlinear compact scheme has high order, high resolution and low dissipation, and has the same ability to capture strong discontinuities as regular weighted essentially non-oscillatory (WENO) schemes. It is a good choice for the simulation of multiscale problems with shock waves.

No abstract uploaded!