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A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows. (English) Zbl 1427.76189
Summary: We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss-Lobatto-Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge-Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for $$Re = 400-5000$$ and flows around an impulsively started cylinder for $$Re = 550-9500$$. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.

##### MSC:
 76M28 Particle methods and lattice-gas methods 76M22 Spectral methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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