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The present paper deals with a new improvement of hierarchical multi-dimensional limiting process for resolving the subcell distribution of high-order methods on two-dimensional mixed meshes. From previous studies, the multi-dimensional limiting process (MLP) was hierarchically extended to the discontinuous Galerkin (DG) method and the flux reconstruction/correction procedure via reconstruction (FR/CPR) method on simplex meshes. It was reported that the hierarchical MLP (hMLP) shows several remarkable characteristics such as the preservation of the formal order-of-accuracy in smooth region and a sharp capturing of discontinuities in an efficient and accurate manner. At the same time, it was also surfaced that such characteristics are valid only on simplex meshes, and numerical Gibbs–Wilbraham oscillations are concealed in subcell distribution in the form of high-order polynomial modes. Subcell Gibbs–Wilbraham oscillations become potentially unstable near discontinuities and adversely affect numerical solutions in the sense of cell-averaged solutions as well as subcell distributions. In order to overcome the two issues, the behavior of the hMLP on mixed meshes is mathematically examined, and the simplex-decomposed P1-projected MLP condition and smooth extrema detector are derived. Secondly, a troubled-boundary detector is designed by analyzing the behavior of computed solutions across boundary-edges. Finally, hMLP_BD is proposed by combining the simplex-decomposed P1-projected MLP condition and smooth extrema detector with the troubled-boundary detector. Through extensive numerical tests, it is confirmed that the hMLP_BD scheme successfully eliminates subcell oscillations and provides reliable subcell distributions on two-dimensional triangular grids as well as mixed grids, while preserving the expected order-of-accuracy in smooth region.

Hierarchical multi-dimensional limiting process (MLP) is improved and extended for flux reconstruction or correction procedure via reconstruction (FR/CPR) on unstructured grids. MLP was originally developed in finite volume method (FVM) and it provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn approximation or reconstruction. The resulting algorithm is referred to as the hierarchical MLP and facilitates detailed capture of flow structures while maintaining formal order-of-accuracy in a smooth region and providing accurate non-oscillatory solutions across a discontinuous region. This algorithm was developed within modal DG framework, but it can also be formulated into a nodal framework, most notably the FR/CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by a projection procedure and the hierarchical MLP limiting step. Extensive numerical analyses and computations, ranging from two-dimensional to three-dimensional fluid systems, have demonstrated that the proposed limiting approach yields outstanding performances in capturing compressible inviscid and viscous flow features.

As the second extension of the direct reconstruction method (DRM), DRM for the surface integration of the discontinuous Galerkin (DG) weak formulation is presented on multi-dimensional high-order mixed meshes. From the previous study, DRM was successfully extended to the volume integration of the DG method, and produced a significant reduction in both the computational cost and memory overhead, particularly for high-order solution approximations on 3-D high-order meshes. DRM has the potential to reduce the computational cost and memory overhead further by treating the surface integration term consistently. To realize this, we design a linear dependency detector and formulate an adaptive orthonormalization process from the modified Gram-Schmidt (MGS) process. DRM is then applied to the restricted function space of the surface integration in an adaptive and optimal manner. The resulting methods, namely the DRM surface integration with the brute force points (BFP) and shape function points (SFP) methods, perform very well on high-order mixed-curved meshes. In addition, the adaptive MGS process is robust and accurate in extracting linear faces/edges from a given set of mixed-curved meshes. Various benchmark tests from the compressible Euler and Navier-Stokes equations verify the impact of DRM for 3-D large-scale computations on mixed-curved meshes. As an example, the DRM framework applied to the 3-D surface and volume integrations with DG-P3 approximation on P3-mesh may yield O ( 10 ) × speed-up, compared to conventional quadrature-based methods.

The present paper deals with an efficient and accurate limiting strategy for the multi-dimensional hyperbolic conservation laws on unstructured grids. The multi-dimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The basic idea of the proposed limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured meshes are obtained. Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the newly developed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Numerous test cases are presented to validate the basic features of the proposed approach.

On the basis of the Eu's generalized hydrodynamic (GH) theories for diatomic single species gas and monatomic multi-species gas, an axisymmetric GH computational model for multi-species gas containing monatomic and diatomic molecules is developed for the numerical simulation of hypersonic rarefied gas flows. The multi-species GH computational model includes monatomic and diatomic species of O"2,N"2, NO, O, N. The mass diffusion flux of the gas mixture is included in the GH constitutive relation. In addition, the physical relationship between the mass diffusion and heat fluxes is added to the evolution equation set. The multi-species GH theory includes the rotational nonequilibrium effect of diatomic molecules by introducing excess normal stress associated with the bulk viscosity. An efficient multi-species GH numerical solver for axisymmetric rarefied flows is then developed by adopting various numerical techniques, such as an adequate nonlinear equation solver for the GH constitutive relation, an accurate flux splitting scheme, multi-grid convergence acceleration and slip-wall boundary conditions. For validation, the proposed computational model is applied to hypersonic rarefied flows over a space shuttle nose, a sphere and a reentry body as well as 1D shock structure. By comparing the results of the multi-species GH model with those of the Navier-Stokes equation and the DSMC, the accuracy and physical consistency of the GH computational model are critically examined.

The present papers deal with numerical methods toward the accurate and efficient computations of multi-dimensional steady/unsteady compressible flows. In Part I, a new spatial discretization technique is introduced to reduce excessive numerical dissipation in a non-flow-aligned grid system. Through the analysis of TVD limiters, a criterion is proposed to predict cell-interface states accurately both in smooth region and in discontinuous region. According to the criterion, a new way of re-evaluating the cell-interface convective flux in AUSM-type methods is developed. The resultant flux reduces numerical dissipation remarkably in multi-dimensional flows. Also, the monotonicity of AUSM-type methods is achieved by modifying the pressure splitting function directly based on the governing equations and the detection of sonic transition point with respect to a cell-interface. It is noted that the newly formulated AUSM-type flux for Multi-dimensional flows, named M-AUSMPW+, possesses many improved properties in term of accuracy, computational efficiency, monotonicity and grid independency. Through numerous test cases from contact and shock discontinuities, vortex flow, shock wave/boundary-layer interaction to viscous shock tube problems, M-AUSMPW+ proves to be efficient and about twice more accurate than conventional upwind schemes. The three-dimensional implementation of M-AUSMPW+ is expected to provide accuracy and efficiency improvement furthermore.

This paper deals with the development of an improved Roe scheme that is free from the shock instability and still preserves the accuracy and efficiency of the original Roe's Flux Difference Splitting (FDS). Roe's FDS is known to possess good accuracy but to suffer from the shock instability, such as the carbuncle phenomenon. As the first step towards a shock-stable scheme, Roe's FDS is compared with the HLLE scheme to identify the source of the shock instability. Through a linear perturbation analysis on the odd-even decoupling problem, damping characteristic is examined and Mach number-based functions f and g are introduced to balance damping and feeding rates, which leads to a shock-stable Roe scheme. In order to satisfy the conservation of total enthalpy, which is crucial in predicting surface heat transfer rate in high-speed steady flows, an analysis of dissipation mechanism in the energy equation is carried out to find out the error source and to make the proposed scheme preserve total enthalpy. By modifying the maximum-minimum wave speed, I the problem of expansion shock and numerical instability in the expansion region is also remedied without sacrificing the exact capturing of contact discontinuity. Various numerical tests concerned with the shock instability are performed to validate the robustness of the proposed scheme. Then, viscous flow test cases ranging from transonic to hypersonic regime are calculated to demonstrate the accuracy, robustness, and other essential features of the proposed scheme.

In order to eliminate or minimize the numerical error by shock waves due to grid distribution in multidimensional hypersonic flows, a new grid reconstruction scheme, the shock-aligned grid technique (SAGT), is proposed. The error due to shock waves in a non-shock-aligned grid system magnifies in proportion to the Mach number and propagates on the downstream side of the flow field to contaminate sensitive aerodynamic coefficients or flow quantities. SAGT, combined with the AUSMPW+ scheme proposed in Part I of the present work, not only provides an accurate solution but also reduces the grid dependency of a numerical scheme without a substantial increase in computational cost. In addition, SAGT is robust and flexible enough to deal with complex flow problems involving shock interaction and reflection and equilibrium and nonequilibrium effects. Extensive numerical tests from a hypersonic blunt body flow to hypersonic nonequilibrium flows validate the accuracy, efficiency, robustness, and convergence characteristics of SAGT.

This paper deals with the development of an improved gas-kinetic BGK scheme for inviscid and viscous flow fields. As the first step toward efficient calculation, particle distribution functions in the general solution of the BGK model are simplified to the extent that the essential features of the standard gas-kinetic BGK scheme are not lost. Then, improved schemes are suggested, which overcome difficulties that may arise in the applications of BGK-type schemes to compressible viscous flow calculations. A Prandtl number correction method is also developed to allow the present schemes to work for arbitrary Pr number. For steady state problems, convergence acceleration techniques suitable for the present schemes are developed in the framework of an implicit time integration. Various numerical experiments ranging from one-dimensional shock tubes to viscous turbulent flows are performed to demonstrate accuracy, robustness, and other essential features of the present method.

Starting from the BGK model of the Boltzmann equation, we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes. The proposed numerical approach is composed of two steps?an initial reconstruction step and a gas evolution step. In the initial reconstruction step, an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge. In the gas evolution step, the local integral solution of the BGK model is used to compute numerical fluxes at a cell interface. This approach provides an alternative to Riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in Godunov-type schemes. A classich-refinement adaptive procedure is implemented to increase the spatial resolution of high-speed unsteady flow characteristics such as shock waves, contact discontinuities, or expansion waves with minimal computational costs and memory overheads. This procedure involves mesh enrichment/coarsening steps to either insert nodes on an edge center in high-gradient regions or delete nodes in over-resolved regions. Numerical results of several test cases for unsteady compressible inviscid flows are presented. In order to verify the accuracy and robustness of the current numerical approach, the computed results are compared with analytical solutions, experimental data, the results of structured mesh calculations, and the results obtained by widely used flux splitting methods.