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Abstract Several local (z ≲ 0.2) metal-poor dwarf active galactic nuclei (AGNs) have remarkably similar properties to those of high-redshift little red dots (LRDs), and are proposed to be local analogs of LRDs. We use long-term photometric and spectroscopic observations of three local LRDs spanning ∼20 yr to measure variability in their rest-frame optical continua and broad Hαemission lines. Using Zwicky Transient Facility light curves over a rest-frame ∼5 yr baseline, ther-band intrinsic rms variability is ${(9\times 10^{-5})}_{-9\times {10}^{-5}}^{+0.014}$ mag (J1022+0841), 0.025 ± 0.004 mag (J1025+1402) and 0.020 ± 0.005 mag (J1047+0739), indicating low intrinsic variability (<3%–4% at 3σ). These variability amplitudes are much lower than those for dwarf AGNs and more massive quasars. We constrain the broad Hαvariability to be less than a few percent, without significant profile changes, over a rest-frame baseline of ∼15 yr in J1025+1402 and J1047+0739, respectively. Local LRDs have exceptionally large Hαequivalent widths and Hα/Hβratios that far exceed the Case B recombination value, in contrast with normal broad-line AGNs. In the context of recent theoretical models of LRDs as dense gas-enshrouded massive black holes with super-Eddington accretion, our results suggest that the photosphere emission is long-term stable and the broad Balmer lines do not arise primarily by photoionization. This scenario is consistent with the lack of variability, large Hα/Hβratios, little dust extinction, and expected high gas density. Virial black hole mass estimates using broad Hαassuming photoionization are therefore highly questionable for LRDs. 

Abstract In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.