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Two-dimensional interaction between two Taylor vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially nonoscillatory finite difference scheme. The main purpose of this study is to reveal the mechanism of sound generation in two-dimensional interaction of two Taylor vortices. Based on an extensive parameter study on the evolution of the vorticity field, we classify the interaction of two Taylor vortices into four types. The first type is the interaction of two counter-rotating vortices with similar strengths. The second type is the interaction of two co-rotating vortices without merging. The third type is the merging of two co-rotating vortices. The fourth type is the interaction of two vortices with a large difference in their strengths or scales. The mechanism of sound generation is analyzed.

In this paper, we focus on error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu. We show that, with ${P}^k$ (piecewise polynomials of degree $k$) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree $(2k)$, error estimates of $O(h^{k+1/2})$ are obtained for general monotone fluxes, and optimal estimates of $O(h^{k+1})$ are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree $(2k+1)$, error estimates of $O(h^k)$ are obtained for general monotone fluxes, and $O(h^{k+1/2})$ are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis.

For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in Liu and Yu (2014) [23]when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.

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