Fifth order AWENO finite difference scheme with adaptive numerical diffusion for Euler equations

In solving hyperbolic conservation laws using the fifth-order characteristic-wise alternative WENO finite-difference scheme (AWENO) with Z-type affine-invariant nonlinear Ai-weights, the classical local Lax–Friedrichs flux (LLF) is modified with an adaptive numerical diffusion (ND) coefficient to form an adaptive LLF flux (LLF-M). The adaptive ND coefficient depends nonlinearly on the local scale-independent smoothness measures on the pressure, density, dilation, and vorticity. The feature sensor combines the well-known Durcos’ sensor on the dilation and vorticity of the velocity field and Jameson's sensor on the density and pressure. Based on the measure of the feature sensor, the ND coefficient of the LLF-M flux is designed to transit smoothly and quickly from a set minimum to the maximum, the local spectral radius of the eigenvalues of the Jacobian of the flux. Hence, the modified AWENO scheme improves the resolution of small-scale structures due to a substantial reduction of excessive dissipation while capturing discontinuities essentially non-oscillatory (ENO-property). The performance of the improved AWENO scheme is validated in one- and two-dimensional benchmark problems with discontinuities and smooth small-scale structures. The results show that the new adaptive LLF-M flux improves resolution, captures fine-scale structures, and is robust in a long-term simulation.