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We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ $(k \ge 0)$ are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge--Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results.

In this paper, we design a new class of central compact schemes based on the cell-centered compact schemes of Lele {[}S.K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992) 16-42{]}. These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points (cell boundaries) and the cell-centers. In our approach, instead of using a compact interpolation to compute the values on cell-centers, the physical values on these half grid points are stored as independent variables and updated using the same scheme as the physical values on the grid points. This approach increases the memory requirement but not the computational costs. Through systematic Fourier analysis and numerical tests, we observe that the schemes have excellent properties of high order, high resolution and low dissipation. It is an ideal class of schemes for the simulation of multi-scale problems such as aeroacoustics and turbulence.

We show that if X is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC coincides with the lower dimension of the condensation set C, while the Assouad dimension of EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension of E, then the upper regularity dimension of any measure supported on EC is strictly larger than the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity dimension fails. 1