Lower-upper Symmetric Gauss-Seidel Research Articles

Improvements for matrix-type implicit temporal scheme regarding entropy fix and local time step

The matrix version of Symmetric Successive Over Relaxation (matrix-SSOR) scheme has been proved to be more efficient than the standard Lower-Upper Symmetric Gauss-Seidel (LU-SGS), but less robust for high-speed flows. In order to ulteriorly improve the convergence rate as well as numerical stability of matrix-SSOR, two improvements regarding entropy fix and local time step have been proposed and validated. Firstly, an augmented entropy fix method is imposed on the inviscid Jacobian matrix and proved to be effective in two high-speed flows, in which the key parameter in entropy fix is discussed and found to be insensitive within appropriate range of values. Since the time step also has great effects on the numerical stability and convergence rate, a modified cell residual adapted local time step method with consideration of the residual history is developed, which is found to be effective for increasing the convergence rate when the matrix-SSOR is applied, but invalid when the LU-SGS is used. The proposed modified local time step method is also insensitive to the key parameter within appropriate range of values. The two modifications can be conveniently implanted into analogous matrix-type implicit schemes to improve the numerical performance.

A massively parallel implicit 3D unstructured grid solver for computing turbulent flows on latest distributed memory computational architectures

An implicit unstructured grid density-based solver–PRAVAHA based on a parallel variant of the lower upper symmetric Gauss-Seidel (LUSGS) method is developed to compute large-scale engineering problems. A four-layered parallel algorithm is designed to efficiently compute three-dimensional turbulent flows on massively parallel modern multiple instruction multiple data-stream (MIMD) computational hardware. This data parallel approach achieves multiple layers of parallelism including continuity of flow solution, transfer of solution gradients, and calculation of drag/lift/solution residuals, right up to the innermost implicit LUSGS solver sub-routine, which is relatively less explored in the literature. Domain decomposition is performed using the METIS software based on multi-level graph partitioning algorithms. Non-blocking message-passing interface library functions are used to manage inter-processor communication through explicit message passing, efficiently. Super-linear scalability of the parallel solver is established on the current state-of-the-art supercomputing facility, the 838 teraflops PARAM seva on up to 6144 cores. Linear or even super-linear speedup on problems of significant size is observed even on ad-hoc parallel computing platforms like workstations and multi-node clusters, for turbulent flow simulations.

Numerical analyses of aeroacoustic characteristics of tiltrotor considering the aerodynamic interaction by the fuselage in hover

Numerical investigations are performed to investigate the interaction effect between the rotor and fuselage on the aerodynamic and aeroacoustic characteristics of the XV-15 tiltrotor during hovering flight. A hybrid method, which combines a Computational Fluid Dynamics (CFD) solver and an aeroacoustics simulation code, has been developed in order to accurately capture the complex interaction flowfield and acoustic pressure of the tiltrotor. To simulate the interaction of the fuselage on the aeroacoustic source of the rotor, a CFD solver based on a sliding mesh algorithm and unsteady Reynolds-Averaged Navier-Stokes equations is utilized. Additionally, to enhance the accuracy of the interaction simulation, the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method is adopted. A specialized code, based on the Ffowcs Williams-Hawkings (FW-H) equations and propagation time, is developed to simulate the rotor noise. Then, the numerical investigation on the aerodynamic and aeroacoustic interaction between a full-scale tiltrotor and fuselage (including the wing) is conducted. In addition, the study explores various tiltrotor configurations to analyze mechanisms for suppressing interaction noise. The results indicate that the aerodynamic loading of tiltrotor fluctuates severely interacted with the fuselage. This, in turn, causes an increase in noise by approximately 5 dB. What's more, decreasing the fountain effect would have a positive effect on reducing the range of variation in loading, and it could ultimately suppress the interaction noise.

A computational strategy for nonlinear coupled constitutive relations of rarefied nonequilibrium flows on unstructured grids

The Navier-Stokes-Fourier (NSF) equations are no longer valid in high-Knudsen and high-Mach flows, where the local thermodynamically equilibrium assumption fails, while the nonlinear coupled constitutive relations (NCCR) have been proven to be an efficient approach for rarefied non-equilibrium flows under the structured finite-volume-method computational fluid dynamic (FVM-CFD) solver. In this paper, by adding the time derivatives of non-conserved variables to the NCCR model, a computational strategy for NCCR model under the unstructured FVM-CFD framework is first proposed for further engineering applications. An upwind flux-splitting scheme with the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit time-marching scheme is employed. The linear stability of one-dimensional modified NCCR model in time and space is analyzed subsequently, which indicates the modified NCCR model is unconditionally stable. The accuracy and computational performance of the proposed NCCR solution are assessed by a feat of the numerical tests for hypersonic flow over a blunt cylinder without a wake, supersonic flow past a circular cylinder with a wake, and hypersonic flow around a planetary probe. Computational results show that the newly-developed NCCR solution can get solutions in better agreement with the direct simulation Monte Carlo (DSMC) than NSF results. Moreover, the newly-developed NCCR solution has a better property of convergence than the undecomposed NCCR algorithm under the unstructured FVM-CFD framework. These superior advantages of the present computational strategy are expected to make the NCCR method a promising engineering tool for modeling rarefied non-equilibrium flows.

Optimization strategies for multi‐block structured CFD simulation based on Sunway TaihuLight

AbstractDecomposition and solver are the main performance bottlenecks of multi‐block structured CFD simulation involving complex industrial configurations such as aero‐engine, shock‐boundary layer interactions, turbulence modeling and so on. In this article, we proposed several optimization strategies to improve the computing efficiency of multi‐block structured CFD simulation based on Sunway TaihuLight super computing system, including: (1) a load balancing decomposition approach combined with recursive segmentation of undirected graphs and block mapping for multi‐structured blocks, (2) two‐level parallelism that utilizes MPI + OpenACC2.0* hybrid parallel paradigms with various performance optimizations such as data preprocessing, reducing unnecessary loops of subroutine calls, collapse, and tile syntax, memory access optimization between the main memory and local data memory (LDM), and (3) a carefully orchestrated pipeline and register communication strategy between computing processor elements (CPEs) to tackle the dependence of LU‐SGS (Lower‐Upper Symmetric Gauss–Seidel). Numerical simulations were conducted to evaluate the proposed optimization strategies. The results showed that our parallel implementation provides high load balance and efficiency, achieving a speedup of 8× + for one loop step, and a speed up of 2× + for strong correlation kernels.

Acceleration of the data-parallel lower-upper relaxation time-integration method on GPU for an unstructured CFD solver

The acceleration of CFD solvers on GPUs allows for large speedups, which enable more complex and detailed simulations. However, not all numerical methods can be efficiently executed in a massively parallel fashion. The Runge–Kutta (RK) and Lower-Upper Symmetric Gauss–Seidel (LU-SGS) time-integration methods are widely used on GPUs and multicore CPUs, respectively; however, the RK method suffers from low convergence speed, while the LU-SGS method is not adapted to a many-core environment. In this paper, we propose to accelerate the matrix-free version of the Data-Parallel Lower-Upper Relaxation (DP-LUR) time-integration method on the unstructured solver FaSTAR, developed by JAXA. The DP-LUR method outperformed the RK and LU-SGS methods on an NVIDIA Tesla V100 GPU when tested on the ONERA M6 and NASA CRM geometries, and reached up to 3.83% of the performance peak of one device.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

Time-Accurate Solution of Unsteady Flows in an Implicit Solver Using Block LUSGS Method

Implicit solvers based on the lower-upper symmetric Gauss–Seidel (LUSGS) method are widely used in Computational Fluid Dynamics (CFD) codes to obtain steady-state solutions using large false-transient time steps. However, the method destroys time accuracy and thus cannot be directly used for true transient problems. For such problems, the LUSGS method has to be used with a dual time-stepping algorithm. In this paper, a modified method that preserves time accuracy and is called the block LUSGS (B-LUSGS) solver is employed to compute unsteady flows using larger time steps. This method is compared with the mentioned dual time-stepping technique used for transient problems. Numerical solutions obtained for Stokes' second problem simulation underline the superior temporal accuracy of the block LUSGS method in computing unsteady flows, even at CFL numbers close to 30. Simulations of 2D laminar and 3D turbulent vortex shedding downstream of a circular cylinder further validate the modified method in accurately predicting unsteady separated flows at a CFL number of 100. Comparisons of overall simulation CPU times show that the B-LUSGS method is at least six times computationally cheaper than the dual time-stepping algorithm that uses the traditional LUSGS solver. Abbreviations: BDF, Backward differentiation formula; CFD, Computational fluid dynamics; CFL Courant–Friedrichs–Lewy; CPU, Central processing unit; DTS, Dual time-stepping; LES, Large-eddy simulation; LUSGS, Lower-upper symmetric Gauss–Seidel; NASA, National Aeronautics and Space Administration; PDE, Partial differential equation; RAM, Random access memory; RANS, Reynolds-averaged Navier–Stokes; SA, Spalart–Allmaras; SST, Shear stress transport.

Unified algorithm of chemically reacting flows at all speeds

Based on traditional 3-D multi-component NS(Navier-stokes) equations, with chemical reaction Preconditioning method was analyzed and the preconditioning matrix was applied for the pseudo-time derivative term of the equations. Preconditioned AUSM+-up scheme was used for space discretization, with implicit time integration LU-SGS (Lower-Upper Symmetric Gauss-Seidel). These two schemes were combined to develop a unified algorithm for flows at all speeds. In addition, Numerical experiments of flows from low speed to high speed, with and without chemical reaction, were performed in this paper. By comparing the results of simulations with experiments showed the algorithm utilized the imposed preconditioning matrix, obtained better results. It was fairly well demonstrated that the improved methods can be applied efficiently for chemical reacting flow at all speeds.

A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clustering

Jacobian clustering is proposed to reduce the computational cost associated with matrix operations encountered in the Newton iteration in fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry. The iterative solver is based on the Lower-Upper Symmetric Gauss-Seidel (LUSGS) algorithm and sparse matrix technique. The evaluation and sparse Lower-Upper (LU) factorization of the diagonal block of the system Jacobian are performed within clusters rather than individual cells for Computational Fluid Dynamics (CFD) simulations. Cells that are close in the state space are clustered to provide the averaged states for calculating the Jacobians of chemical source terms and transport fluxes. The cells retrieve the factorized sparse matrices from the belonging clusters to perform the necessary iterations. For the purpose of clustering, the spatial dependency of transport Jacobian in the diagonal block is eliminated. To further reduce the computational cost, the sparsity of chemical Jacobian is augmented by removing the insignificant matrix elements. The method is tested in various one-dimensional hydrocarbon flames with both the second order Crank-Nicolson scheme and a third order implicit Runge-Kutta scheme. Various chemical mechanisms with 9 to 111 species are used to test the performance of the iterative solver. Fast convergence of Newton iteration is achieved, and the formal order of accuracy is demonstrated with Jacobian clustering. The overall costs of evaluation and factorization of the block diagonal Jacobian are negligible compared to the cost of calculating transport fluxes and chemical source terms. The averaged costs of Jacobian evaluation, LU factorization and Newton iteration, all increase only linearly with the number of chemical species. The fully coupled, fully implicit Crank-Nicolson scheme with Jacobian clustering shows 4 to 42 times speedup in computational time compared to the decoupled implicit scheme with Strang operator splitting. Jacobian clustering is promising to increase the computational efficiency of high order fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry.

Efficient same-dimensional implicit time advancement parallel scheme and optimization methods for the iteration parameters using a graphics-processing unit

Many studies have focused on the acceleration of computational fluid dynamics (CFD) using multicore hardware such as graphics-processing units (GPUs) in the field of parallel computing. In GPU acceleration, CFD parallel granularity generally refers to the point granularity parallelization as a unit of the grid. An implicit time advancement scheme is more efficient and faster than an explicit time advancement scheme for CFD. However, for commonly used implicit schemes, such as the lower-upper symmetric Gauss–Seidel (LUSGS) scheme, the parallel dimensionality is reduced, resulting in a highly time-consuming procedure. In this paper, the data-parallel upper-lower relaxation (DPLUR) scheme based on Jacobi iteration is used and then implemented on a GPU. Numerical experiments are carried out and show that the computing speed of point granularity parallelization using the DPLUR scheme, especially implemented on a GPU, is much higher than that of dimensionality reduction using the LUSGS scheme. Moreover, the influence of different Jacobi inner iteration steps (JIIS) on the convergence time is discussed, and two JIIS optimization algorithms are proposed according to the characteristics of convergence. On the basis of the memory access form, a DPLUR red–black (DPRB) scheme with more stable and faster convergence than the conventional DPLUR scheme is developed. Finally, some standard cases are adopted to verify the effectiveness of DPRB schemes with the JIIS optimization algorithm.

An implicit high-order radial basis function-based differential quadrature-finite volume method on unstructured grids to simulate incompressible flows with heat transfer

A high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method is presented in this work to efficiently simulate incompressible flows with heat transfer on unstructured mesh. The velocity and temperature fields are solved by locally using the lattice Boltzmann flux solver and the high-order finite volume method. Specifically, the proposed highly accurate finite volume method utilizes a high-order Taylor polynomial to approximate the solution within every control cell. Spatial derivatives are the corresponding coefficients in the polynomial, and they are approximated by the meshless radial basis function-based differential quadrature (RBFDQ) method. The diffusive and convective fluxes at each cell interface are simultaneously evaluated through local reconstruction of lattice Boltzmann solution using D2Q9 lattice velocity model. To efficiently calculate the solution with high-order accuracy, an implicit time-marching method incorporating the lower-upper symmetric Gauss-Seidel (LU-SGS) and the explicit first stage, singly-diagonally implicit Runge-Kutta (ESDIRK) approaches is devised. The proposed method is comprehensively validated by a series of numerical experiments containing both steady-state and time-dependent heat transfer problems with/without curved boundaries at a wide variety of Rayleigh numbers and Grashof numbers. The obtained results demonstrate a high degree of accuracy and reliability of the proposed method for complex flows on unstructured mesh. In comparison with the classical second-order method, the proposed high-order method has better computational efficiency when comparable results are achieved.

A hierarchical wavefront method for LU-SGS

The lower–upper Symmetric-Gauss–Seidel (LU-SGS) method is one of typical implicit methods, especially for an application that requires high convergence and accuracy. However, the data-dependencies among grid points in the LU-SGS method prevent parallelization and vectorization, both of which are important for modern computing systems. As modern computing systems consist of multiple nodes, each of which is equipped with multi-core processors with a vector processing capability, to take advantages of these computing systems, both parallel computing and vector computing are mandatory. This paper proposes a hierarchical wavefront method that combines the wavefront and the hyperplane methods in order to achieve both parallelization and vectorization of the LU-SGS method with block-structured meshes. The evaluation results show that the hierarchical wavefront method is 4.9 times faster than the conventional wavefront method and 24.2 times faster than the hyperplane method.

On the application of higher-order Backward Difference (BDF) methods for computing turbulent flows

The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes.

An implicit lattice Boltzmann flux solver for simulation of compressible flows

The lattice Boltzmann flux solver (LBFS) is a novel Boltzmann-typed solver under the finite volume framework, which has many advantages over traditional lattice Boltzmann equation (LBE) solvers. However, the existing versions of LBFS are explicit in temporal discretization. Therefore, they are of low computational efficiency and are mostly applied to the simulation of flows around simple geometries. In this paper, an implicit lattice Boltzmann flux solver for compressible flows is proposed, which can efficiently march in time with a large CFL number, and therefore facilitates the simulation of flows around complex geometries in practical engineering applications. Aiming to obtain the time-accurate results of unsteady flows, the implicit LBFS is further combined with the dual time-stepping technique to efficiently simulate unsteady flows, in which a sub-iteration is applied on the pseudo time level. In the sub-iteration, the Jacobian matrix is preconditioned by the lower-upper symmetric Gauss-Seidel (LUSGS) method and solved by the generalized minimal residual method (GMRES). The accuracy and efficiency of the proposed solver are demonstrated by various simulations in a wide Mach number regime. To further demonstrate its performance in practical engineering applications, the proposed solver is also used to simulate the flow over complex aircraft models.

Efficient high-order radial basis-function-based differential quadrature-finite volume method for incompressible flows on unstructured grids.

This paper presents an efficient high-order radial basis-function-based differential quadrature-finite volume method for incompressible flows on unstructured grids. In this method, a high-order polynomial based on the Taylor series expansion is applied within each control cell to approximate the solution. The derivatives in the Taylor series expansion are approximated by the mesh-free radial basis-function-based differential quadrature method. The recently proposed lattice Boltzmann flux solver is applied to simultaneously evaluate the inviscid and viscous fluxes at the cell interface by the local solution of the lattice Boltzmann equation. In the present high-order method, a premultiplied coefficient matrix appears in the time-dependent term, reflecting the implicit nature. The implicit time-marching techniques, i.e., the lower-upper symmetric Gauss-Seidel and the explicit first stage, singly diagonally implicit Runge-Kutta schemes, are incorporated to efficiently solve the resultant ordinary differential equations. Several numerical examples are tested to validate the accuracy, efficiency, and robustness of the present method on unstructured grids. Compared with the k-exact method, the present method enjoys higher accuracy and better computational efficiency.

Dynamic Drag Reduction Effects of Aerospikes and Aerodisks

In order to investigate the dynamic characteristics of blunt aircraft mounted with aerospikes and aerodisks in large-amplitude force-pitching, the Roe spatial scheme and the lower-upper symmetric Gauss-Seidel (LU-SGS) method with dual time step are employed for discretization of unsteady Navier-Stokes (N-S) equations. A parametric investigation on the flow fields is conducted by altering the pitching period, aerospike length, and aerodisk diameter consequently via a variable-controlling procedure. Dynamic characteristics of aerodynamic drag as well as the visualization of unsteady flow fields are achieved, and the results show that the aerodynamics of hypersonic aircraft under the condition of large-amplitude force-pitching vibration have hysteresis characteristics affected by periods of force-pitching vibration. In addition, when changing aerospike length and aerodisk diameter, the variation tendency of drag reduction efficiency is determined by the pitching angle of the oscillation process.

An implicit time spectral method using adaptive stablization

An adaptive time spectral viscosity (TSV) approach is proposed to stabilize the time-spectral method (TSM) by dealiasing its solution. Instead of being assigned uniformly in space, the cut-off wave number for TSV is determined locally by spectrum analysis of the flow variables. On each grid cell, the significant low-frequency modes are adaptively separated from the TSV targeted high-frequency ones according to local flow physics. The space-time Lower-Upper Symmetric Gauss-Seidel(ST-LU-SGS) implicit scheme is used to solve the modified time-spectral equations. TSV operator is treated implicitly in the same way as the time spectral operator to facilitate fast convergence. Computational results of inviscid and turbulent flows over a pitching NACA0012 airfoil are presented for validation of the adaptive TSV approach. For the turbulent flow case, the proposed approach is also applied to the Spalart–Allmaras (S-A) turbulence model equation. The accuracy and the convergence rate of the adaptive TSV approach is demonstrated in both test cases.

Implicit Time-Spectral Method for Unsteady Reynolds-Averaged Navier–Stokes Computations of Turbulent Flows

An efficient implicit Fourier time-spectral method is developed for the unsteady Reynolds-averaged Navier–Stokes (URANS) computations of turbulent flows. In addition to the time-spectral URANS equations, the time-spectral Spalart–Allmaras turbulence model equation is also solved by the space–time lower-upper symmetric Gauss–Seidel (LU-SGS) implicit scheme. The efficient LU-SGS sweeps are directly extended to the time domain without introducing additional approximation error, delayed implicit temporal coupling, and additional algorithmic steps. The ability of the algorithm is verified through the test cases of the pitching NACA0012 airfoil and pitching ONERA M6 wing. Dominant Fourier modes of the drag coefficient and its components are identified. Special attention is paid to the nonlinear behavior of the surface pressure and skin friction coefficients in the shock-free area, especially the region downstream the shock where the influence of turbulence is significant. For the test case of a pitching ONERA M6 wing, difference in local flow unsteadiness for the two branches of the -type shock is observed. The phenomenon is further discussed through study of the shock motion range and the upstream Mach number for each shock branch.

Reconstructed discontinuous Galerkin methods for compressible flows based on a new hyperbolic Navier-Stokes system

A new first-order hyperbolic system (FOHS) is formulated for the compressible Navier-Stokes equations. The resulting hyperbolic Navier-Stokes system (HNS), termed HNS20G in this paper, introduces the gradients of density, velocity, and temperature as auxiliary variables. Efficient, accurate, compact and robust reconstructed discontinuous Galerkin (rDG) methods are developed for solving this new HNS system. The newly introduced variables are recycled to obtain the gradients of the primary variables. The gradients of these gradient variables are reconstructed based on a newly developed variational formulation in order to obtain a higher order polynomial solution for these primary variables without increasing the number of degrees of freedom. The implicit backward Euler method is used to integrate solution in time for steady flow problems, while the third-order explicit first stage singly diagonally Runge-Kutta (ESDIRK) time marching method is implemented for advancing solutions in time for unsteady flows. The flux Jacobian matrices are obtained with an automatic differentiation toolkit TAPENADE. The approximate system of linear equations is solved with either symmetric Gauss-Seidel (SGS) method or general minimum residual (GMRES) algorithm with a lower-upper symmetric Gauss-Seidel (LU-SGS) preconditioner. A number of test cases are presented to assess accuracy and performance of the newly developed HNS+rDG methods for both steady and unsteady compressible viscous flows. Numerical experiments demonstrate that the developed HNS+rDG methods are able to achieve the designed order of accuracy for both primary variables and the their gradients, and provide an attractive and viable alternative for solving the compressible Navier-Stokes equations.