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For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax–Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21] . In this paper, based on the same time-stepping method and the second-order GKS flux function [42] , a fourth-order gas-kinetic scheme is constructed for the Euler and Navier–Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24] , the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge–Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function provides a dynamic process of evolution from the kinetic scale particle free transport to the hydrodynamic scale wave propagation, which provides the physics for the non-equilibrium numerical shock structure construction to the near equilibrium NS solution. As a result, with the implementation of the fifth-order WENO initial reconstruction, in the smooth region the current two-stage GKS provides an accuracy of O ( ( Δ x ) 5 , ( Δ t ) 4 ) for the Euler equations, and O ( ( Δ x ) 5 , τ 2 Δ t ) for the NS equations, where τ is the time between particle collisions. Many numerical tests, including difficult ones for the Navier–Stokes solvers, have been used to validate the current method. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer to the hypersonic viscous heat conducting flow. Following the two-stage time-stepping framework, the third-order GKS flux function can be used as well to construct a fifth-order method with the usage of both first-order and second-order time derivatives of the flux function. The use of time-accurate flux function may have great advantages on the development of higher-order CFD methods. [ABSTRACT FROM AUTHOR]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

Abstract: The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero. [Copyright &y& Elsevier]

Abstract: This paper presents a discontinuous Galerkin BGK (DGBGK) method for both viscous and inviscid flow simulations under a DG framework with a gas-kinetic flux and WENO limiters. In the DGBGK method, the construction of the flux in the DG method is based on the particle transport and collisional mechanism which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation. Due to the connection between the gas-kinetic BGK model and the Euler as well as the Navier–Stokes equations, both viscous and inviscid flow equations can be simulated by a unified formulation. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition. In the current method, the time accuracy is achieved by the direct integration of both time-dependent flux function at a cell interface and the flow variables inside each element. Numerical examples in one and two space dimensions are presented to illustrate the robustness and accuracy of the present scheme. [Copyright &y& Elsevier]

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities. [ABSTRACT FROM AUTHOR]

This paper presents the development of a third-order compact gas-kinetic scheme (GKS) for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme utilizes a time-dependent gas distribution function at a cell interface to not only calculate the fluxes needed for updating the cell-averaged flow variables but also to evaluate the flow variables at the cell interface. This leads to the evolution of cell-averaged gradients of flow variables. The success of this scheme heavily relies on the initial data reconstruction techniques, with an emphasis on their application to the tetrahedral mesh. Employing a conventional second-order unlimited least-square reconstruction directly on the cell-averaged flow variables of von Neumann neighbouring cells can introduce linear instability into the scheme. However, by using the updated cell-averaged gradients, the GKS with a third-order compact smooth reconstruction remains linearly stable under a large CFL number when applied to a tetrahedral mesh. To enhance the robustness of the high-order compact GKS for capturing a discontinuous solution, we propose a novel two-step multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. This innovative approach overcomes the stability issues associated with a second-order compact reconstruction by incorporating a pre-reconstruction step. Additionally, it simplifies the third-order non-linear reconstruction process by adding a single large stencil to those used in the second-order one. A high-order wall boundary condition is achieved by fusing the constrained least-square technique with the WENO procedure, where a quadratic element is used in the reconstruction for cells with a curved boundary. Numerical tests involving both the second-order and third-order compact GKS are presented, encompassing both inviscid and viscous flows at both low and high speeds. The results demonstrate that the proposed third-order compact scheme possesses robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations. • A third-order compact gas-kinetic scheme on 3D unstructured tetrahedral mesh is constructed. • A two-step multi-resolution weighted essentially non-oscillatory reconstruction is proposed. • High-order curved wall boundary treatment is obtained from the constrained least-square technique and the WENO procedure. • The scheme can use a much larger time step than that used in other conventional compact schemes. [ABSTRACT FROM AUTHOR]

• A fourth- up to sixth-order compact scheme is constructed on unstructured mesh. • Accurate solutions of compressible Euler and Navier-Stokes equations are obtained. • MSMD improves temporal accuracy and efficiency of the scheme. • The scheme presents excellent performance from shock interaction to acoustic wave propagation. Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Ji et al. (2020) [40]), in this paper a higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averaged flow variables and their cell-averaged derivatives, the nonlinear high-order compact reconstruction can be obtained for these variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination of non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in GKS is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function provides the Navier-Stokes (NS) flux function directly without separating the inviscid and viscous terms, which simplifies the numerical discretization of the NS equations directly on unstructured mesh. Based on the time-accurate flux solver, the two-stage fourth-order time discretization can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same order Runge-Kutta time stepping method. The current high-order GKS can uniformly capture acoustic and shock waves without identifying troubled cells and implementing additional limiting procedure. In addition, the fourth- up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme in most flow problems. The fourth-order GKS on unstructured mesh will be used in the computations from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems have advantages in the development of high-order schemes. [ABSTRACT FROM AUTHOR]

• A third-order compact scheme is constructed on unstructured mesh. • The solutions of compressible Euler and Navier-Stokes equations are obtained. • MSMD improves temporal accuracy and efficiency of the scheme. • The third-order compact scheme has the same robustness as the second-order one. As an extension of a fourth-order compact gas kinetic scheme (GKS) on structured mesh [24] , this work is about the development of a third-order compact GKS on unstructured mesh for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution, the time-dependent gas distribution function at a cell interface in GKS provides not only the flux function and its time derivative, but also the time accurate flow variables there at the next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables can be obtained as well using the updated flow variables at the closed cell interfaces around that cell through the divergence theorem. Therefore, with the cell-averaged flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. Following the reconstruction technique in [64] , a new HWENO reconstruction on triangular mesh is designed in the current scheme. Combined with the two-stage temporal discretization and second-order time accurate flux function, a third-order compact scheme on unstructured mesh has been constructed. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular mesh. The robustness and accuracy of the scheme have been validated through many cases, including strong shocks in the hypersonic viscous flow and smooth Navier-Stokes solution. [ABSTRACT FROM AUTHOR]