Abstract

An adaptive H-/p-refinement strategy using a novel sensor is devised and tested in a block-spectral compressible Euler code equipped with adaptive-mesh refinement (AMR) and high-order flux-reconstruction numerics. At each Gauss quadrature point (or solution point) within each spectral block (or mesh element) the discrete velocity jump ΔU = ∂U/∂y1Δy1+∂V/∂y2Δy2+∂W/∂y3Δy3 is calculated and normalized by the local speed of sound, a. The grid spacing, Δxi, is calculated in each direction as the distance between auxiliary Gauss-Lobatto points, staggered relative to the solution points. The polynomial order is increased from p=0 to p=pmax in regions of weak compression, (ΔU/a)crit<ΔU/a<0 and kept at p=pmax in regions of flow expansion ΔU/a≥0, while staying at the H=0 base mesh level. Regions experiencing strong compressions, i.e. ΔU/a<(ΔU/a)crit, are H-refined up to H=Hmax where Hmax is applied at the location of maximum compression, ΔU/a=min(ΔU/a) in the domain, while keeping p=0 to guarantee robustness and monotonicity of the solution in the H refined region. The critical value of (ΔU/a)crit= -0.06 is found to effectively separate smooth and non-smooth solution regions, supported by a 1D detonation initiation test case in ideal gas and a shock-to-detonation transition in high explosives. Using this value, the Sod shock tube, Shu-Osher problem, double Mach reflection and a 2D detonation in a high-explosive are simulated with the proposed adaptive H-/p-refinement. In the Sod shock tube case, p-refinement resolves the (weak) contact discontinuity while H-refinement enhances the grid resolution in the shock exploiting the monotonicity of the p=0 reconstruction. For the Shu-Osher problem, p-refinement captures the small-scale oscillations trailing the shock that would be otherwise attenuated, while H-refinement triggered by the ΔU-sensor appropriately tracks the shock. In the double Mach reflection problem, H-refinement confines the numerical diffusion around the reflected shock while p-refinement recaptures many physical features trailing the shock. Finally, in the 2D high-explosive detonation case, H-refinement follows the leading shock and resolves the curvature of the detonation wave, while p-refinement adds resolution to the trailing reaction zone. Finally, the proposed methodology is tested in a detonation-wave propagation test case in high-explosives with numerical predictions comparing favorably against experiments.

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