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adaptive ADER advection airfoil algorithm applied approximation asymptotic asymptotic analysis Boltzmann equation boundary conditions calculation cell cellular automata coefficients compressible flow conservation laws constant convective defined denotes density derivatives diffusion discrete displacement distribution edge error estimate error indicator Euler equations example Figure finite difference finite element method finite volume method flow field formulation geometry given gradient grid-free hand side higher order hyperbolic implemented incompressible integral interface interpolation inviscid JCASim kinetic lattice Boltzmann lattice Boltzmann method linear low Mach number LSKUM macroscopic Mathematics matrix mesh Navier-Stokes equations nodes numerical flux numerical method numerical scheme numerical solution obtained optimization order scheme overlap parameters particles pressure problem reconstruction refinement resp Reynolds number rotation saturation equation semi-Lagrangian shock wave simulation solved solver source term space stencil step structure technique transonic transonic flow turbine turbulent unstructured grids upwind scheme values variables variation vector viscosity WENO