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This paper introduces a novel hybrid numerical method, SAUSM, designed for accurate and robust simulation of compressible flows governed by the Euler equations. While the AUSM + scheme provides proper resolution of smooth flow features, it is susceptible to anomalies, particularly the carbuncle phenomenon near strong shock discontinuities. Conversely, the AUFS scheme offers inherent stability in capturing shocks; however, it lacks the accuracy of AUSM + in smooth regions. The proposed SAUSM method combines AUSM + and AUFS through an adaptive weighting function, facilitating a seamless transition between the schemes. This approach preserves the accuracy of AUSM + in smooth regions while ensuring robust shock-capturing capabilities near discontinuities. The effectiveness of the SAUSM method is rigorously demonstrated through a comprehensive suite of progressively complex test cases. Numerical experiments demonstrate SAUSM's proficiency in resolving intense shock patterns and discontinuities without introducing anomalies. In the selected test cases, SAUSM agrees with reference solutions and effectively mitigates anomalies observed in AUSM + , including kinked Mach stems. In the challenging test case involving hypersonic blunt body flow over a cylinder, SAUSM adapts dissipation effectively by utilizing its adaptive weighting function to generate smooth pressure distributions, thereby eliminating the carbuncle instability linked to AUSM + when applied to a high aspect ratio grid. The consistent formulation of flux splitting and the adaptive weighting in SAUSM prevent excessive dissipation away from discontinuities, thus preserving accuracy comparable to that of exact Riemann solvers. Consequently, SAUSM emerges as a promising and innovative approach to accurately and robustly simulate a wide range of compressible Euler flows. The comprehensive results obtained from the validation tests firmly establish SAUSM as a highly effective general-purpose technique for computational fluid dynamics in academic research. [ABSTRACT FROM AUTHOR]

The HLLC approximate Riemann solver improves upon the HLL Riemann solver by resolving contact discontinuities. This is a particularly desirable property for multi-material codes in which problems usually contain material interfaces. However, the HLLC solver is known to suffer from the carbuncle phenomenon, a numerical instability most apparent at grid-aligned shocks in multi-dimensional simulations. Many problems of interest, including high energy-density physics applications, require the accurate resolution of both material interfaces and hydrodynamic shocks. A variety of methods have been developed to cure this instability, with varying degrees of complexity. The objective of this work is to describe a simple approach to modify the HLLC Riemann solver and prevent the carbuncle instability. The method is then demonstrated for assorted two-dimensional test problems known to exhibit the shock instability. The performance of the new solver is compared with that of the standard HLL and HLLC Riemann solvers. [ABSTRACT FROM AUTHOR]

The numerical shock instability problem significantly limits the application of the finite volume method in hypersonic flow simulations, especially for high-order schemes, which will encounter more severe shock instability problems. In the previous work, the MSAT (matrix stability analysis tool) was introduced to quantitatively analyze the numerical shock instability problem. However, the utilization of the three-point reconstruction stencil in the original version of MSAT presents challenges in conducting investigations into high-order schemes. The current work provides an extension of MSAT by extending the reconstruction stencil to five points, a challenging but necessary endeavor for high-order schemes. This extension enables MSAT to provide analyses for schemes with high-order reconstruction methods. Furthermore, the updated MSAT can be utilized to investigate the impact of characteristic variable reconstruction on the shock instability problem. Results demonstrate the reliability of the updated MSAT and reveal novel characteristics in the shock instability of high-order schemes. Consequently, the updated MSAT will contribute to the investigation of shock instabilities for high-order schemes, ultimately aiding in mitigating this challenging problem.

Copyright of Chinese Journal of Computational Mechanics / Jisuan Lixue Xuebao is the property of Chinese Journal of Computational Mechanics Editorial Office, Dalian University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Engineering Geology / Gongcheng Dizhi Xuebao is the property of Journal of Engineering Geology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An adaptive mesh refinement–rotated lattice Boltzmann flux solver (AMR-RLBFS) is presented to simulate two and three-dimensional compressible flows with complex shock structures. In the method, the RLBFS, which has a strong shock-capturing capability and can effectively eliminate the shock instability phenomenon, is applied to solve the flow filed by reconstructing the fluxes at each cell interface adaptively with the mesoscopic lattice Boltzmann model. To locally and dynamically improve the resolution of intricate shock structures and optimize the required computational resources, a block-structured adaptive mesh refinement (AMR) technique is introduced. The validity and effectiveness of the proposed method are confirmed through a range of two and three-dimensional numerical cases, including the shock tube problem, the four-wave Riemann problem, explosion within a rectangular box, and the vorticity induced by a shock. The results obtained using the AMR-RLBFS exhibit excellent agreement with published data and demonstrate high accuracy in capturing complex shock structures. The computational efficiency of the AMR-RLBFS can be also improved significantly compared to the RLBFS on uniform grids. Furthermore, the numerical outcomes underscore the capability of the AMR-RLBFS to eliminate shock instability effects while efficiently capturing a broader spectrum of small-scale vertical structures. These findings highlight the ability of AMR-RLBFS to improve the computational efficiency and capture intricate shock structures effectively, making it a valuable tool for studying a wide range of compressible flows from aerodynamics to astrophysics. [ABSTRACT FROM AUTHOR]

The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT (matrix stability analysis tool), an open-source tool to quantitatively analyze the shock instability problem. Based on the finite-volume approach on the structured grid, MSAT can be employed to investigate the mechanism of the shock instability problem, evaluate the robustness of numerical schemes, and then help to develop robust schemes. Also, MSAT has the ability to analyze the practical simulation of supersonic or hypersonic flows, evaluate whether it will suffer from shock instabilities, and then assist in selecting appropriate numerical schemes accordingly. As a result, MSAT is a helpful tool that can investigate the shock instability problem and help to cure it.

As Computational Fluid Dynamics (CFD) progresses rapidly, there is a growing demand for precise prediction of aerodynamic heating in hypersonic flows, which remains a considerable challenge. In this paper, an accurate and robust line-hybrid method is proposed for aero-heating computations. Two HLL-type schemes are combined by utilising lines constructed in the computational grid. The high-dissipation HLL scheme is exerted at the source of shock instability to suppress disturbance propagation, while the low-dissipation HLLC scheme is applied in the wall-normal direction. The resulting HLLC-Line scheme exhibits superior robustness and accuracy simultaneously compared to conventional individual or hybrid schemes. Accurate and smoothly distributed heat profiles can be obtained by the new scheme, including those in the stagnation or interaction regions. Numerical experiments validate that the line-hybrid method is highly effective in predicting aerodynamic heating involving complex configurations. [ABSTRACT FROM AUTHOR]

A Rotated Lattice Boltzmann Flux Solver (RLBFS) with improved stability is proposed in this paper for the effective simulation of compressible flows with intense shock waves at high Mach numbers. Unlike the conventional LBFS or conventional Riemann solvers, which evaluate its mass, momentum and energy fluxes in the normal direction of each cell interface, the present RLBFS decomposes the outer normal direction of the cell interface into two perpendicular directions of the velocity difference and their counterparts. The fluxes in these two directions are respectively evaluated by the LBFS and the final fluxes at each interface are obtained by weighted summations of the two obtained fluxes. The matrix-based stability theory is applied to analyze the proposed solver by simulating the strong normal shock problem from Mach number 1.5 to 10. It reveals that the RLBFS has a smaller negative eigenvalue than the original LBFS, indicating that the RLBFS is more stable. The numerical accuracy of the RLBFS is evaluated by simulating the advection of density perturbation. The performance of the RLBFS is further demonstrated by simulating seven more compressible flows, including the transonic flow over RAE2822 airfoil, odd-even grid perturbation problem, a Mach 3 wind tunnel with a step, supersonic corner flow, double Mach reflection of a strong shock, shock-boundary layer interaction and hypersonic viscous flow around a blunt body. The obtained results agree well with the published data, showing that the RLBFS can effectively avoid the phenomenon of shock instability and thus has a potential for simulating practical compressible flows with intense shock waves at high Mach numbers. • A rotated lattice Boltzmann flux solver with improved stability is proposed. • A matrix-based stability theory is used to analyze the RLBFS and the LBFS. • Theoretical analysis show that the RLBFS is more stable than LBFS. • Numerical examples show the reliability and capability of the RLBFS. [ABSTRACT FROM AUTHOR]

An adaptive mesh refinement–rotated lattice Boltzmann flux solver (AMR-RLBFS) is presented to simulate two and three-dimensional compressible flows with complex shock structures. In the method, the RLBFS, which has a strong shock-capturing capability and can effectively eliminate the shock instability phenomenon, is applied to solve the flow filed by reconstructing the fluxes at each cell interface adaptively with the mesoscopic lattice Boltzmann model. To locally and dynamically improve the resolution of intricate shock structures and optimize the required computational resources, a block-structured adaptive mesh refinement (AMR) technique is introduced. The validity and effectiveness of the proposed method are confirmed through a range of two and three-dimensional numerical cases, including the shock tube problem, the four-wave Riemann problem, explosion within a rectangular box, and the vorticity induced by a shock. The results obtained using the AMR-RLBFS exhibit excellent agreement with published data and demonstrate high accuracy in capturing complex shock structures. The computational efficiency of the AMR-RLBFS can be also improved significantly compared to the RLBFS on uniform grids. Furthermore, the numerical outcomes underscore the capability of the AMR-RLBFS to eliminate shock instability effects while efficiently capturing a broader spectrum of small-scale vertical structures. These findings highlight the ability of AMR-RLBFS to improve the computational efficiency and capture intricate shock structures effectively, making it a valuable tool for studying a wide range of compressible flows from aerodynamics to astrophysics.

In this study, we aim to develop a robust and accurate Riemann solver to resolve strong shock waves with compact stencil information by applying the rotated hybrid concept to the RoeM scheme. The original RoeM scheme offers good robustness from the perspective of avoiding shock instability, also called the carbuncle phenomenon, in hypersonic flows by introducing Mach-number-based functions. These functions rely on multi-dimensional shock discontinuity sensing terms to capture grid-aligned shock waves, thus necessitating sharing of values of cells adjacent to each vertex. Consequently, constructing a data structure that can handle such a wide stencil is necessary, particularly for implementations on unstructured grids. Moreover, communication of this stencil information on parallel computation can increase computational costs. To overcome these complexities, the rotated hybrid concept is applied to the RoeM scheme, to replace the multi-dimensional terms while still resolving the grid-aligned shock instability. In addition, a Mach-number-based weighting function is introduced to distinguish between the flow around the boundary layer and the onset of shock instability. The damping characteristic of the proposed scheme is compared to that of the original RoeM scheme by conducting a linear perturbation analysis. The findings reveal that the proposed scheme effectively controls spurious oscillations across strong shock waves. Numerous test cases are presented to demonstrate the robustness and accuracy of the proposed approach. Notably, the proposed scheme successfully resolves thermal boundary layers and accurately predicts the heat transfer rate around stagnation points in hypersonic flows. • New Riemann solver resolves multi-dimensional shock waves compactly. • RoeM scheme enhanced with rotated hybrid concept for robustness. • Mach-number-based weighting distinguishes boundary layers and shocks. • Accurate simulation predicts heat transfer rates over blunt bodies. [ABSTRACT FROM AUTHOR]

In this paper, continuous research is undertaken to explore the underlying mechanism of numerical shock instabilities of Godunov-type schemes for strong shocks. By conducting dissipation analysis of Godunov-type schemes and a sequence of numerical experiments, we are able to clarify that the instability may be attributed to insufficient entropy production inside the numerical shock structure. As a result, a general entropy-control technique for improving the robustness of various Godunov-type schemes at strong shocks is developed. It plays a part in guaranteeing that enough entropy is produced inside the numerical shock structure. Furthermore, such a modified approach does not introduce any additional numerical dissipation on linear degenerate waves to suppress the shock instability. Numerical results that are obtained for various test cases indicate that the proposed methods have a good performance in terms of accuracy and robustness. [ABSTRACT FROM AUTHOR]

A hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability. [ABSTRACT FROM AUTHOR]

Copyright of Chinese Journal of Computational Mechanics / Jisuan Lixue Xuebao is the property of Chinese Journal of Computational Mechanics Editorial Office, Dalian University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Low dissipation shock-capturing methods, including the popular Roe scheme, will encounter the shock instability phenomenon in the computation of the multidimensional strong shock wave problems. This will seriously affect the schemes’ accurate simulation of the hypersonic flow problems. The small perturbation analysis of the Roe scheme was carried out. The results show that, all perturbations in the longitudinal direction of the shock front are damped, but the perturbations of density and shear velocity in the transverse direction are undamped. The viscosities corresponding to the entropy wave and shear wave were added to the flux transverse to the shock front to suppress instability of the Roe scheme. To prevent the improper viscosity from influencing the resolution of contact discontinuity and shear layers, 2 switching functions were defined, so that the viscosity was only added to the transverse flux in the subsonic region of the shock layer. The numerical tests show that, the modified Roe scheme not only retains the merit of high resolution of the original Roe scheme, but also has better robustness and eliminates the shock wave instability. [ABSTRACT FROM AUTHOR]

The approximate Riemann solver of Harten--Lax--van Leer (HLL) and its variant HLLC (HLL with Contact restoration) solver are widely used as flux functions of finite volume Godunov-type methods for the solution of the gas dynamic Euler equations. However, the HLLC solver suffers from two significant difficulties: an accuracy problem at low-speed flows and shock instability at high-speed flows. To remedy such drawbacks, a novel low-Mach shock-stable HLLC-type scheme called HLLC+ is developed for all speeds. The antidissipation pressure fix is introduced first to overcome the accuracy problem in low Mach number limits. Then, shear viscosity is identified and scaled into the original HLLC scheme to overcome shock instability. A new pressure-based factor function without switching coefficients is devised to prevent shear viscosity from smearing the boundary layer. The new HLLC+ scheme involves no empirical parameters and is easy to implement. Asymptotic analysis and low Mach number test cases show the excellent behaviors of HLLC+ in low Mach number limits: no global cut-off problem, damping pressure checkerboard modes, having expected ${Ma^2}$ scaling of pressure and density fluctuations, and satisfaction of divergence constraint. Furthermore, this work manifests that the accuracy problem is associated with the normal velocity jumps of the flux interface, while shock instability is related to the transverse velocity jumps. Numerical test cases across a wide range of Mach numbers demonstrate the superior performance and potentiality of HLLC+ to simulate all Mach number flows. [ABSTRACT FROM AUTHOR]

The common defects of the Roe scheme are the non-physical expansion shock and shock instability. By removing the momentum interpolation mechanism (MIM), an improved method with several advantages has been presented to suppress the shock instability. However, it cannot prevent the expansion shock and is incompatible with the traditional curing method for expansion shock. To solve the problem, the traditional curing mechanism is analyzed. Effectiveness of the traditional curing method is discussed, and several defects are identified, one of which leads to incompatibility between curing shock instability and expansion shock. Consequently, an improved Roe scheme is proposed, which is with low computational costs, concise, easy to implement, and robust. More importantly, the proposed scheme can simultaneously solve the problem of shock instability and expansion shock without additional costs. [ABSTRACT FROM AUTHOR]

In this study, we investigate the so called carbuncle phenomenon by means of numerical experiments and heuristic considerations. We identify two main sources for the carbuncle: instability of the 1d shock position and low numerical viscosity on shear waves. We also describe how higher order stabilizes the 1d shock position and, thus, reduces the carbuncle. [ABSTRACT FROM AUTHOR]

It is well known that low diffusion Riemann solvers with minimal smearing on contact and shear waves are vulnerable to shock instability problems, including the carbuncle phenomenon. In the present study, we concentrate on exploring where the instability grows out and how the dissipation inherent in Riemann solvers affects the unstable behaviors. With the help of numerical experiments and a linearized analysis method, it has been found that the shock instability is strongly related to the unstable modes of intermediate states inside the shock structure. The consistency of mass flux across the normal shock is needed for a Riemann solver to capture strong shocks stably. The famous carbuncle phenomenon is interpreted as the consequence of the inconsistency of mass flux across the normal shock for a low diffusion Riemann solver. Based on the results of numerical experiments and the linearized analysis, a robust Godunov-type scheme with a simple cure for the shock instability is suggested. With only the dissipation corresponding to shear waves introduced in the vicinity of strong shocks, the instability problem is circumvented. Numerical results of several carefully chosen strong shock wave problems are investigated to demonstrate the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

This work presents a new approach for curing the carbuncle instability. The idea underlying the approach is to introduce some dissipation in the form of right-hand sides of the Navier–Stokes equations into the basic method of solving Euler equations; in so doing, we replace the molecular viscosity coefficient by the artificial viscosity coefficient and calculate heat conductivity assuming that the Prandtl number is constant. For the artificial viscosity coefficient we have chosen a formula that is consistent with the von Neumann and Richtmyer artificial viscosity, but has its specific features (extension to multidimensional simulations, introduction of a threshold compression intensity that restricts additional dissipation to the shock layer only). The coefficients and the expression for the characteristic mesh size in this formula are chosen from a large number of Quirk-type problem computations. The new cure for the carbuncle flaw has been tested on first-order schemes (Godunov, Roe, HLLC and AUSM + schemes) as applied to one- and two-dimensional simulations on smooth structured grids. Its efficiency has been demonstrated on several well-known test problems. [ABSTRACT FROM AUTHOR]

The shock instability phenomenon is a well-known problem for hypersonic flow computation by the shock-capturing Roe scheme. The pressure checkerboard is another well-known problem for low-Mach-number flow computation. The momentum interpolation method (MIM) is necessary for low-Mach-number flows to suppress the pressure checkerboard problem, and the pressure-difference-driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high-Mach-number flows or low-Mach-number cells in numerical shock. Therefore, MIM should be retained for low-Mach-number flows and be completely removed for high-Mach-number flows and low-Mach-number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

While traditional approaches to prevent the carbuncle phenomenon in gas dynamics simulations increase the viscosity on entropy and shear waves near shocks, it was quite recently suggested to instead decrease the viscosity on the acoustic waves for low Mach numbers. The goal is to achieve what, in this paper, we call Mach number consistency: for all waves, the numerical viscosity decreases with the same order of the Mach number when the Mach number tends to zero. We take the simple approach that was used for the proof of concept together with the simple model for the increased numerical viscosity on linear waves and investigate the possibilities of combining both in an adaptive manner while locally maintaining Mach number consistency. • The concept of Mach number consistency can be generalized. • Curing inappropriate dissipation in transverse direction is Mach number consistent. • Curing inappropriate dissipation on shear waves is Mach number consistent. • A Mach number consistent adaptive blending of both approaches is introduced. • Provided simple solver proves already sufficient for most high order schemes. [ABSTRACT FROM AUTHOR]

• Numerous applications in various compressible flow field investigations. • The technique's filtering quality shows less prominent elements of a flow that are difficult to observe in the actual visualization records. • Showing the evolution of flow features as a function of location and time by sequences of streaks obtained at different positions of the flow field. • Inherent uncertainties in the measurements with this technique are highlighted. This paper discusses a digital version of the traditional streak imaging technique combined with different flow visualization methods. The basic characteristics of the technique, its advantages and its shortcomings are presented alongside a number of applications in fluid mechanics. Streak images are useful to track and quantify the propagation or development of distinct flow features as a function of time. In addition, the appearance of a new trace, or the change of grey scale (or color in the case of polychrome visualizations), can be used to graphically reveal the location and instant of the onset of certain flow phenomena. In addition, the method also has a filtering quality that allows one to show less prominent elements of a flow that may be difficult to observe in the actual visualization records. A sequence of streaks obtained at different positions of the flow field can be used to track the evolution of flow features as a function of location and time. Some features of the digital technique are highlighted via a direct comparison with results from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5 % extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high-order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high-order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock-capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high-order weighted compact nonlinear scheme is not only robust for shock-capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The construction of Euler fluxes is an important step in shock-capturing/upwind schemes. It is well known that unsuitable fluxes are responsible for many shock anomalies, such as the carbuncle phenomenon. Three kinds of flux vector splittings (FVSs) as well as three kinds of flux difference splittings (FDSs) are evaluated for the shock instability by a fifth-order weighted compact nonlinear scheme. The three FVSs are Steger–Warming splitting, van Leer splitting and kinetic flux vector splitting (KFVS). The three FDSs are Roe's splitting, advection upstream splitting method (AUSM) type splitting and Harten–Lax–van Leer (HLL) type splitting. Numerical results indicate that FVSs and high dissipative FDSs undergo a relative lower risk on the shock instability than that of low dissipative FDSs. However, none of the fluxes evaluated in the present study can entirely avoid the shock instability. Generally, the shock instability may be caused by any of the following factors: low dissipation, high Mach number, unsuitable grid distribution, large grid aspect ratio, and the relative shock-internal flow state (or position) between upstream and downstream shock waves. It comes out that the most important factor is the relative shock-internal state. If the shock-internal state is closer to the downstream state, the computation is at higher susceptibility to the shock instability. Wall-normal grid distribution has a greater influence on the shock instability than wall-azimuthal grid distribution because wall-normal grids directly impact on the shock-internal position. High shock intensity poses a high risk on the shock instability, but its influence is not as much as the shock-internal state. Large grid aspect ratio is also a source of the shock instability. Some results of a second-order scheme and a first-order scheme are also given. The comparison between the high-order scheme and the two low-order schemes indicates that high-order schemes are at a higher risk of the shock instability. Adding an entropy fix is very helpful in suppressing the shock instability for the two low-order schemes. When the high-order scheme is used, the entropy fix still works well for Roe's flux, but its effect on the Steger–Warming flux is trivial and not much clear. [ABSTRACT FROM PUBLISHER]

HLLC is one kind of schemes which resolve shock, contact, and rarefaction waves exactly and has high value of application in compressible fluid simulation. HLL scheme cannot deal with the contact and shear waves accurately, thus it has much more dissipation compared with the HLLC scheme. However, numerical tests have illustrated that HLLC scheme may suffer from shock instability phenomena in the vicinity of strong shock waves while it performs well in low speed and subsonic flow. HLL-HLLC scheme which is suitable for a larger range of flow is proposed through mathematical analyses of HLLC and HLL scheme. The new proposed scheme indicates the property of HLLC which has low order dissipation in low speed flow and can overcome the shortcoming of shock instability phenomena in high speed flow. The effectiveness and robustness of the proposed scheme free from shock instability phenomena is demonstrated through three numerical simulations: hypersonic flow on double wedge configuration, supersonic flow over backward-facing step and hypersonic flow over blunt body. [ABSTRACT FROM AUTHOR]

Due to its high resolution for discontinuities and high efficiency, the HLLEM scheme is a popular numerical method to calculate the conservative hyperbolic systems. When capturing multidimensional strong shock waves, however, it will encounter different forms of numerical oscillations, such as the odd-even decoupling and carbuncle phenomenon, which is a great hindrance to its application in hypersonic flow problems. In this paper the numerical shock instability of the HLLEM scheme is investigated from a mathematical perspective and a corresponding cure is proposed. A special linearized small perturbation analysis shows that perturbations evolved in the streamwise direction will be attenuated while the density and shear velocity perturbations evolved in the transverse direction will not be attenuated and thus induce the occurrence of instability. In order to suppress the numerical oscillation in capturing strong shock waves, a function based on the pressure ratio at the cell interface is defined to control the dissipation of the HLLEM flux. And a BVD (boundary variation diminishing) algorithm is implemented to pick the smallest density jump from all combinations among reconstructed density values to reduce the numerical dissipation, thus the contact resolving capability of the HLLEM scheme is further improved. The higher resolution for contact discontinuities and better stability for shock waves brought by the current strategies are demonstrated through solving a series of one- and two-dimensional benchmark test problems. • The shock instability of the HLLEM solver is studied in depth. • The shock instability is eliminated by the antidiffusion controlling technique with a shock indicator function. • The resolution of the HLLEM solver for contact discontinuities is improved by a BVD algorithm. • Numerical results demonstrate that the strategies adopted provide higher resolution and better shock stability. [ABSTRACT FROM AUTHOR]

Abstract: Many researchers have reported failures of the approximate Riemann solvers in the presence of strong shock. This is believed to be due to perturbation transfer in the transverse direction of shock waves. We propose a simple and clear method to prevent such problems for the Harten–Lax–van Leer contact (HLLC) scheme. By defining a sensing function in the transverse direction of strong shock, the HLLC flux is switched to the Harten–Lax–van Leer (HLL) flux in that direction locally, and the magnitude of the additional dissipation is automatically determined using the HLL scheme. We combine the HLLC and HLL schemes in a single framework using a switching function. High-order accuracy is achieved using a weighted average flux (WAF) scheme, and a method for v-shear treatment is presented. The modified HLLC scheme is named HLLC–HLL. It is tested against a steady normal shock instability problem and Quirk’s test problems, and spurious solutions in the strong shock regions are successfully controlled. [Copyright &y& Elsevier]

Abstract: We have developed a numerical simulation scheme for three-dimensional magnetohydrodynamical flow with shocks. The numerical scheme is based on the Roe-type scheme and includes additional numerical diffusion in the direction tangential to the shock front to care the carbuncle instability. The numerical viscosity is added only in the regions where the characteristics of either fast or slow wave converges, i.e., in the regions potentially dangerous to the carbuncle instability. Accordingly the numerical viscosity is as small as that of the Roe scheme except near the shock fronts. It is demonstrated from comparison with the HLLE scheme that the magnetic Reynolds number is higher in the simulations obtained with our scheme. We show application of the scheme to the magnetohydrodynamical simulations of type II supernova. It is also proved that the scheme is free from the odd–even decoupling. [Copyright &y& Elsevier]

In this paper, we propose new Euler flux functions for use in a finite-volume Euler/Navier–Stokes code, which are very simple, carbuncle-free, yet have an excellent boundary-layer-resolving capability, by combining two different Riemann solvers into one based on a rotated Riemann solver approach. We show that very economical Euler flux functions can be devised by combining the Roe solver (a full-wave solver) and the Rusanov/HLL solver (a fewer-wave solver), based on a rotated Riemann solver approach: a fewer-wave solver automatically applied in the direction normal to shocks to suppress carbuncles and a full-wave solver applied, again automatically, across shear layers to avoid an excessive amount of dissipation. The resulting flux functions can be implemented in a very simple and economical manner, in the form of the Roe solver with modified wave speeds, so that converting an existing Roe flux code into the new fluxes is an extremely simple task. They require only 7–14% extra CPU time and no problem-dependent tuning parameters. These new rotated fluxes are not only robust for shock-capturing, but also accurate for resolving shear layers. This is demonstrated by an extensive series of numerical experiments with standard finite-volume Euler and Navier–Stokes codes, including various shock instability problems and also an unstructured grid case. [Copyright &y& Elsevier]

A multidimensional dissipation technique is incorporated into Roe's flux difference splitting scheme to prevent numerical-shock instability that may occur in compressible flow solutions. The damping characteristic of the proposed technique is demonstrated through a linear perturbation analysis on the problem of a moving shock along an odd-even grid perturbation. The proposed technique is tested on well-known problems that exhibit the numerical-shock instability. The technique is then further extended to achieve higher-order accurate solution and evaluated by several benchmark test cases. [ABSTRACT FROM AUTHOR]

The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle. [Copyright &y& Elsevier]

Dissipation mechanisms of Godunov-type schemes are presented in the framework of unified representation. The causes of inaccuracy at the contact discontinuity and the dissipation mechanism in the numerical mass flux of the HLLEM scheme are examined first. A “vacuum preserving property” is defined and the prominent role of the numerical signal speed involved with the rarefaction waves in the expanding region is analyzed. Through a linear perturbation analysis on the odd–even decoupling problem, necessary conditions for designing a shock stable scheme are discussed. As a result, an improved HLLE(HLLE+) scheme is proposed and its dissipation mechanism is analyzed. The diffusivity of the Godunov-type schemes is examined by two-dimensional hypersonic viscous flow. [Copyright &y& Elsevier]

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, 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. [Copyright &y& Elsevier]

• An extra term is added to the low diffusion schemes to overcome the shock anomaly. • The revised schemes turn out to be carbuncle-free and shock-stable. • The accuracy of the original schemes in the boundary-layer is not contaminated. The appearance of shock anomaly is a major unsolved problem for some low diffusion schemes when simulating the hypersonic flow. In this paper, a simple method is proposed to enhance the robustness of the low diffusion schemes to overcome the shock anomaly. The main idea of this method is adding an appropriate extra term to the original low diffusion schemes without influencing the accuracy in aerodynamic heating prediction. This extra term is derived from the difference between the flux splitting scheme (FVS) and the advection upstream splitting method+ (AUSM+). Adding this term to three low diffusion schemes, seven typical numerical tests are conducted to examine the capability of those schemes. Numerical results show that the three new schemes turn out to be carbuncle-free and shock-stable without losing their original accuracy in prediction of aerodynamic heating, validating the feasibility and reliability of the proposed method. [ABSTRACT FROM AUTHOR]

• Derivation of a centralized flux formulation for the HLLC solver. • Dissipation control in transverse direction to shock propagation. • Smooth reduction of nonlinear signal speeds suppresses shock instability. • HLLC-LM proves stable for extreme resolutions also in three dimensions. The purpose of this paper is twofold. First, the application of high-order methods in combination with the popular HLLC Riemann solver demonstrates that the grid-aligned shock instability can strongly affect simulation results when the grid resolution is increased. Beyond the well-documented two-dimensional behavior, the problem is particularly troublesome with three-dimensional simulations. Hence, there is a need for shock-stable modifications of HLLC-type solvers for high-speed flow simulations. Second, the paper provides a stabilization of the popular HLLC flux based on a recently proposed mechanism for grid aligned-shock instabilities Fleischmann et al. (2020) [8]. The instability was found to be triggered by an inappropriate scaling of acoustic and advection dissipation for local low Mach numbers. These low Mach numbers occur during the calculation of fluxes in transverse direction of the shock propagation, where the local velocity component vanishes. A centralized formulation of the HLLC flux is provided for this purpose, which allows for a simple reduction of nonlinear signal speeds. In contrast to other shock-stable versions of the HLLC flux, the resulting HLLC-LM flux reduces the inherent numerical dissipation of the scheme. The robustness of the proposed scheme is tested for a comprehensive range of cases involving strong shock waves. Three-dimensional single- and multi-component simulations are performed with high-order methods to demonstrate that the HLLC-LM flux also copes with latest challenges of compressible high-speed computational fluid dynamics. [ABSTRACT FROM AUTHOR]