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.

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This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of an operator that is defined in this paper. Let $X=\mathscr{X}y$ to be the differential polynomial associated with $\mathscr{X}$, the order of $\mathscr{X}$, $\mathrm{ord}(\mathscr{X})$, is defined as the order of a differential ideal $\Lambda$ of differential polynomials that is a nontrivial expansion of the ideal $\{X\}$ and with the lowest order. In this paper, we prove that there are only four possible values for the order of a differential operator, $0$, $1$, $2$, $3$, or $\infty$. Furthermore, when the order is finite, the expansion $\Lambda$ is generated by $X$ and a differential polynomial $A$, which can be obtained through a rational solution of a partial differential equation that is given explicitly in this paper. When the order is infinite, the expansion $\Lambda$ is just the unit ideal. In additional, if, and only if, the order of $\mathscr{X}$ is $0$, $1$, or $2$, the polynomial differential equation associating with $\mathscr{X}$ has Liouvillian first integrals. Examples and connections with Godbilon-Vey sequences are also discussed.

In the context of recently proposed holographic dualities between higher spin theories in AdS3 and 1+1-dimensional CFTs with W-symmetry algebras, we revisit the definition of higher spin black hole thermodynamics and the dictionary between bulk fields and dual CFT operators. We build a canonical formalism based on three ingredients: a gauge-invariant definition of conserved charges and chemical potentials in the presence of higher spin black holes, a canonical definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned with the asymptotic symmetry algebra. We show that our canonical formalism shares the same formal structure as the so-called holomorphic formalism, but differs in the definition of charges and chemical potentials and in the bulk-to-boundary dictionary. Most importantly, we show that it admits a consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail and generalize our construction to theories based on the hs[\lambda] algebra, and on the sl(N,R) algebra for any choice of sl(2,R) embedding.

In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction-convection-diffusion equations. The equation under consideration involves a small diffusivity $\varepsilon$ and a large reaction coefficient $\sigma$, leading to high P\'eclet number and high Damk\"{o}hler number. In addition to giving error estimates of the approximations in $L^2$ and $H^1$ norms, we explicitly establish the dependence of error bounds on the diffusivity, the $L^{\infty}$ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh P\'eclet number and a large mesh Damk\"{o}hler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods.