Simple and robust h-adaptive shock-capturing method for flux reconstruction framework

In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adaptive technique avoids the use of redundant meshes in smooth regions,while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning.The compatibility of these two significant features relies on a novel limiter designed for mesh refinements.It ensures the positivity of solutions on all newly created cells.Therefore,the proposed method is completely positivity-preserving and thus highly robust.It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance.The performance of the proposed method is examined in various numerical experiments.When solving Euler equations,the technique of Local Artificial Diffusivity(LAD)is additionally coupled to damp oscillations.More importantly,when solving Navier-Stokes equations,the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly.The implementation of the method becomes rather simple.

Interface flux reconstruction method based on optimized weight essentially non-oscillatory scheme

Aimed at the computational aeroacoustics multi-scale problem of complex configurations discretized with multi-size mesh, the flux reconstruction method based on modified Weight Essentially Non-Oscillatory（WENO） scheme is proposed at the interfaces of multi-block grids.With the idea of Dispersion-Relation-Preserving（DRP） scheme, different weight coefficients are obtained by optimization, so that it is in WENO schemes with various characteristics of dispersion and dissipation. On the basis, hybrid flux vector splitting method is utilized to intelligently judge the amplitude of the gap between grid interfaces. After the simulation and analysis of 1D convection equation with different initial conditions, modified WENO scheme is proved to be able to independently distinguish the gap amplitude and generate corresponding dissipation according to the grid resolution. Using the idea of flux reconstruction at grid interfaces, modified WENO scheme with increasing dissipation is applied at grid points, while DRP scheme with low dispersion and dissipation is applied at the inner part of grids. Moreover, Gauss impulse spread and periodic point sound source flow among three cylinders with multi-scale grids are carried out. The results show that the flux reconstruction method at grid interfaces is capable of dealing with Computational Aero Acoustics（CAA） multi-scale problems.

An efficient inverse approach for reconstructing time-and space-dependent heat flux of participating medium

The decentralized fuzzy inference method(DFIM)is employed as an optimization technique to reconstruct time-and space-dependent heat flux of two-dimensional(2D)participating medium.The forward coupled radiative and conductive heat transfer problem is solved by a combination of finite volume method and discrete ordinate method.The reconstruction task is formulated as an inverse problem,and the DFIM is used to reconstruct the unknown heat flux.No prior information on the heat flux distribution is required for the inverse analysis.All retrieval results illustrate that the time-and spacedependent heat flux of participating medium can be exactly recovered by the DFIM.The present method is proved to be more efficient and accurate than other optimization techniques.The effects of heat flux form,initial guess,medium property,and measurement error on reconstruction results are investigated.Simulated results indicate that the DFIM is robust to reconstruct different kinds of heat fluxes even with noisy data.

A high-order scheme based on lattice Boltzmann flux solver for viscous compressible flow simulations

In this paper,a high-order scheme based on the lattice Boltzmann flux solver(LBFS)is proposed to simulate viscous compressible flows.The flux reconstruction(FR)approach is adopted to implement the spatial discretization.The LBFS is employed to compute the inviscid flux by using the local reconstruction of the lattice Boltzmann equation solutions from macroscopic flow variables.Meanwhile,a switch function is used in LBFS to adjust the magnitude of the numerical viscosity.Thus,it is more beneficial to capture both strong shock waves and thin boundary layers.Moreover,the viscous flux is computed according to the local discontinuous Galerkin method.Some typical compressible viscous problems,including manufactured solution case,lid-driven cavity flow,supersonic flow around a cylinder and subsonic flow over a NACA0012 airfoil,are simulated to demonstrate the accuracy and robustness of the proposed FR-LBFS.

A Priori Subcell Limiting Approach for the FR/CPR Method on Unstructured Meshes

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction(FR/CPR)methods on twodimensional unstructured quadrilateralmeshes.Firstly,a modified indicator based on modal energy coefficients is proposed to detect troubled cells,where discontinuities exist.Then,troubled cells are decomposed into nonuniform subcells and each subcell has one solution point.A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted(NNW)interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR method.Numerical investigations show that the proposed subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.

Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR(correction procedure or collocation penalty via reconstruction).The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure.In addition to being simple and economical,it unifies several existing methods including discontinuous Galerkin,staggered grid,spectral volume,and spectral difference.To discretize the diffusion terms,we use the BR2(Bassi and Rebay),interior penalty,compact DG(CDG),and I-continuous approaches.The first three of these approaches,originally derived using the integral formulation,were recast here in the CPR framework,whereas the I-continuous scheme,originally derived for a quadrilateral mesh,was extended to a triangular mesh.Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out.Finally,results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.