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• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the fully conservative, high-order numerical method previously developed by Johnson and Kercher (2020) [14] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. (2020) [16] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients, including a simple linear-scaling limiter, required for the discrete solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee discrete satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, advection of a low-density Gaussian wave, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that the formulation can achieve optimal high-order convergence in smooth flows. Furthermore, we find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case. Finally, mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

This paper primarily concerns the discontinuities capturing problems in nonconservative and nonconvex conservative hyperbolic systems. For the Godunov scheme of nonconservative hyperbolic systems, the numerical dissipation at discontinuous points in the simulation process is analyzed grid point by grid point through a new perspective. The numerical paths implied in the nonconservative variables represent different averaging and dissipating processes from the conservative cases. Unphysical dissipation of nonconservative variables ruins Rankine-Hugoniot relations, contributing to incorrect jumps, wrong propagation speed, and spurious fluctuations in other characteristic fields. For the discontinuities capturing problem of nonconservative hyperbolic systems, a novel numerical path preserving (NPP) method is proposed to modify the original Godunov schemes so that the dissipation of the numerical methods at discontinuities is carried out strictly following the consistent numerical path. Numerical simulations of the nonconservative systems are performed for isothermal and Euler equations. The results indicate that the NPP method can correctly capture the discontinuous structures and verify the correctness of our theoretical analysis. Additionally, the NPP method is extended to nonconvex hyperbolic conservation systems, and the Alfvénic wave (discontinuity) of the one-dimensional ideal magnetohydrodynamic (MHD) equations is simulated. It is found that the nonconvex nature of the flux causes unphysical compound wave structures while simulating the Alfvénic wave with the popular schemes nowadays, e.g., Roe of flux difference splitting method and WENO with flux vector splitting method. Modifying the original Godunov schemes with the NPP method proposed in this paper ensures correct simulation of Alfvénic discontinuity in MHD equations. It validates the effectiveness of the NPP method for nonconvex hyperbolic conservation systems. [ABSTRACT FROM AUTHOR]

In this paper, we develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique, where the numerical fluxes in the DG scheme are treated separately. However, if the numerical approximations are close to the steady state, the numerical fluxes are not independent, and it is possible to use the relationship to obtain a new penalty parameter which is zero at the steady state and the full scheme is PP. To be more precise, we first reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods. • Construct positivity-preserving and well-balanced schemes for Euler equations with gravitation. • The penalty is chosen to be zero in the Lax-Friedrichs fluxes at the steady-states. • A novel positivity-preserving technique is developed. • The scheme is straightforward to implement. [ABSTRACT FROM AUTHOR]

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein–Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (ρ = p = 0) up to t = 1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un–tractable within the Z4 formalism. Due to the above features, we consider that our new algorithm can be particularly beneficial for the numerical study of quasi normal modes of oscillations, both of black holes and of neutron stars. • Strongly hyperbolic first order Z4 formulation of the Einstein-Euler equations. • New and simple well-balanced discontinuous Galerkin schemes for numerical general relativity. • Robust conversion from conservative to primitive variables also in the presence of vacuum. • Stable long-time simulations of black holes and TOV stars in two and three space dimensions. • Head-on collision of two puncture black holes in three space dimensions using the Z4 formulation. [ABSTRACT FROM AUTHOR]

This paper presents a novel projection-based embedded discrete fracture model (pEDFM) applicable to general two-phase flow simulation in anisotropic reservoirs for the first time. The novel pEDFM is developed based on hybrid two-point flux approximation and mimetic finite difference (TPFA-MFD) methods, which retains the basic features of two-point flux approximation (TPFA) based generic pEDFM workflow and incorporates new features of mimetic finite difference (MFD) to handle full-tensor permeabilities. This is achieved by introducing the concept of effective connection, formulating additional pressure degrees of freedom for effective connections, assigning effective matrix-matrix (m-m) and matrix-fracture (m-f) connections to the corresponding sides of matrix-cell interfaces, presenting methods corresponding to different types of connections to handle low-conductivity fractures, and further deriving formulas of numerical fluxes for effective connections. The resulting global equation system is solved by a Newton-Raphson nonlinear solver with the temporary discretization of the implicit backward Euler scheme. The 2D and 2.5D numerical examples show the novel pEDFM could achieve the same level of computation accuracy and convergence as the reference discrete fracture model (DFM), while avoiding the gridding difficulty of complex fracture networks. It also significantly outperforms classical EDFM based on MFD and generic pEDFM workflow in the computational accuracy for cases with anisotropic full tensor permeabilities. The novel pEDFM, to our best knowledge, is for the first time proposed and developed for general anisotropic two-phase flow simulation and maybe the numerical simulation framework for multi-phase flow in fractured reservoirs with the best comprehensive performance so far. [ABSTRACT FROM AUTHOR]

In this paper we discuss the MUSCL-Hancock scheme for the two-dimensional conservation laws and derive the bound-preserving conditions by a parameterized convex decomposition method which is proposed in [ Journal of Computational Physics, 474:111805, 2023 ] for one-dimensional problems. The CFL number equals the half of the one-dimensional value and then ensures the speed-up compared with the classical two-stage MUSCL scheme. The bound-preserving slope limiter on the preliminary reconstruction becomes problem dependent. The maximum-minimum preserving slope limiter for the scalar problems becomes the same with its one-dimensional version. A positivity-preserving slope limiter for the Euler system is designed including a different parameter from the scalar problems and more efficient operators than the existing limiters. Numerical experiments are reported to demonstrate the robustness of the proposed scheme. • Developed bound-preserving conditions of the 2D MUSCL-Hancock scheme. • The same slope limiter and half the CFL number compared to the 1D case. • Solved important models, such as nonlinear problem, incompressible advection problem and Euler equations. • Numerically checked the sharpness and the robustness of the scheme. [ABSTRACT FROM AUTHOR]

This paper proposes a hybrid limiter in discontinuous Galerkin (DG) methods for hyperbolic conservation laws. This hybrid limiter combines the high-order DG approximation with the low-order limited solution to achieve better resolution for multiscale structures. Meanwhile, the essentially non-oscillatory property can also be retained. The indicator, based on the jump information at the cell interface, provides the non-linear weight for the hybrid limiter, which also acts as a shock detection technique to identify troubled cells in the DG approximation. The presented limiter also maintains the conservation of mass cell by cell. In addition, the hybrid limiter maintains the local data structure and can, therefore, preserve the parallel efficiency of the DG method. The hybrid limiter is very simple in design and can be easily implemented with arbitrary high-order accuracy on both structured and unstructured meshes. Benchmark examples, mainly on Euler equations, are presented to demonstrate the performance of these DG methods with associated hybrid limiters. [ABSTRACT FROM AUTHOR]

In this paper we blend the high order Compact Approximate Taylor (CAT) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 2D. The resulting scheme presents high accuracy on smooth solutions, essentially non-oscillatory behavior on irregular ones, and almost fail-safe property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behavior of the CAT-MOOD scheme. • Hig-order one-step Lax-Wendroff type scheme for 2D systems of conservation laws. • Positivity-preserving and conservative scheme. • Finite-difference scheme. • A-posteriori order adaptive scheme. • 2D numerical experiments as Euler equations with smooth and non-smooth initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, a family of stencil selection algorithms is presented for WENO schemes on unstructured meshes. The associated freedom of stencil selection for unstructured meshes, in the context of WENO schemes present a plethora of various stencil selection algorithms. The particular focus of this paper is to assess the performance of various stencil selection algorithm, investigate the parameters that dictate their robustness, accuracy and computational efficiency. Ultimately, efficient and robust stencils are pursued that can provide significant savings in computational performance, while retaining the non-oscillatory character of WENO schemes. This is achieved when making the stencil selection algorithms adaptive, based on the quality of the cells for unstructured meshes, that can in turn reduce the computational cost of WENO schemes. For assessing the performance of the developed algorithms well established test problems are employed. These include the least square approximation of polynomial functions, linear advection equation of smooth functions and solid body rotation test problem. Euler and Navier-Stokes equations test problems are also pursued such as the Shu-Osher test problem, the Double Mach Reflection, the supersonic Forward Facing step, the Kelvin-Helmholtz instability, the Taylor-Green Vortex, and the flow past a transonic circular cylinder. • Various central and directional stencils algorithms presented. • Uniqueness, Cover, Width and Compactness the most dominant parameters for directional stencils. • Algorithms suitability is dependent on the numerical framework. • Adaptive stencils reduce the computational cost of the WENO schemes. [ABSTRACT FROM AUTHOR]

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. The methods are valid on unstructured curvilinear grids using tensor-product basis functions. To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks. [ABSTRACT FROM AUTHOR]

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared. [ABSTRACT FROM AUTHOR]

This paper studies three high-order structure-preserving finite volume weighted essentially non-oscillatory (WENO) methods, which are not only well balanced (WB) for a general known hydrostatic equilibrium state but also preserve the positivity of density and pressure, for the compressible Euler equations under gravitational fields. These methods are built on a simple local scaling positivity-preserving (PP) limiter and a modified WENO-ZQ reconstruction exactly preserving the cell average value and scaling invariance. The WB properties of these three methods are achieved based on suitable numerical fluxes and approximation to the gravitational source terms. Based on some convex decomposition techniques as well as several critical properties of the admissible states and numerical flux, we carry out rigorous positivity-preserving analyses for these three WB schemes. We rigorously prove that the three WB methods, coupled with the PP limiter and a strong-stability-preserving time discretization, are always PP under suitable Courant-Friedrichs-Lewy conditions. Extensive numerical examples are provided to confirm WB and PP properties of three methods. [ABSTRACT FROM AUTHOR]

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure. • A positivity-preserving high order accurate finite difference WENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • WENO reconstruction is performed on variables from a positivity-preserving convection diffusion flux splitting. • The method is robust and suitable for solving the compressible flow with low density and low pressure. [ABSTRACT FROM AUTHOR]

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library. • Resolving extreme pressure ratio shock waves in near vacuum conditions. • Constructing entropy-stable high order schemes for the hypersonic Euler equations. • Deriving Entropy-conservative fluxes for flows modeled with the RRHO method. • Comparing real chemistry versus ideal chemistry in hypersonic flows. • Hypersonic NS solver based on the entropy-stable Euler scheme and the BR1 approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fifth order well-balanced positivity-preserving finite difference scale-invariant AWENO scheme for the compressible Euler equations with gravitational fields. By using the scale-invariant WENO (Si-WENO) operator and well-balanced modification of the interpolated conservative variables, the finite difference discretization is well-balanced with respect to the priorly known isothermal and isentropic hydrostatic states. To ensure positivity of the density and pressure throughout the whole computation, we introduce interpolation-based and flux-based positivity-preserving limiters to both the density and pressure. Meanwhile, modifications are made to the discretization of the pressure equilibrium to restore well-balancedness. We point out that by using the Si-WENO operator we can compute all ingredients in the discretization of the source term prior to the time evolution, and the well-balanced and positivity-preserving modifications are made based on these ingredients, which can improve computational efficiency. Moreover, we carefully derive the positivity-preserving CFL conditions in one and two dimensions. Finally, the accuracy, robustness, effectiveness and numerical symmetry of our approach are demonstrated by a variety of numerical examples, where the time-marching strategy is used in two-dimensional problems to avoid strong dependence on p / ρ in the CFL conditions. • The AWENO scheme preserves both well-balanced (WB) and positivity-preserving (PP) properties. • WB is achieved by reformulation of the source term, Si-WENO interpolation and hydrostatic modification. • Interpolation- and flux-based PP limiters are designed, and WB is restored simultaneously. • The theoretical CFL condition is derived. • Numerical examples illustrate the accuracy, robustness and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we build wellposed intrusive generalised Polynomial Chaos (gPC) based reduced models for uncertain photonics. We solve the reduced models with a Monte-Carlo (MC) scheme. Care is taken to highlight under which condition a reduced model (gPC based or not) is wellposed. The analysis is carried out thanks to an analogy between the construction of reduced models for uncertainty quantification and the construction of reduced models for kinetic equations. In order to enforce the aforementioned wellposedness conditions, several strategies, inspired from the hyperbolicity-preserving ones [1–8] are reviewed, adapted and analysed. The resolution of the reduced models is performed thanks to an astute combination of the Implicit Semi-analog MC (ISMC, see [9]) scheme for photonics and of MC-gPC (see [10]) for uncertainty propagation. This work demonstrates that MC-gPC can be efficiently applied to a stiff nonlinear set of partial derivative equations if the MC resolution allows a fast convergence with respect to both the time and spatial discretisations (the latter properties being allowed by ISMC). Several benchmarks are investigated in the last section, they allow illustrating important aspects of the new ISMC-gPC solver for uncertain photonics. • We build mathematically wellposed hierarchical reduced models to accurately capture uncertainties in photonic computations. • We provide a proof of the mathematical wellposedness of the reduced models we build. • The resolution is based on both the recent Implicit Semi-analog Monte-Carlo (ISMC) solver capturing the diffusion limit and the recent Monte-Carlo-generalised Polynomial Chaos (MC-gPC) methodology to capture efficiently uncertainties. • Some theoretical ingredients of the paper are also closely related to the hyperbolicity-preserving strategies used to build gPC based reduced models for Euler equations. Regarding these ingredients, the paper provides interesting comparisons, applied to an integro-differential system. [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]

• Detailed description of the Spectral Difference Raviart-Thomas (SDRT) approach on 2D hybrid grid. • Linear stability analysis based on an eigenvalues analysis. • Determination of stable SDRT schemes for a 5 th and 6 th order of accuracy. • Convergence study on regular and irregular triangular grids as well as on regular hybrid mesh. • Validation on 2D Euler and Navier-Stokes test cases on high-order- triangular and hybrid grids. In the present paper, a stable Spectral Difference formulation on triangles is defined using a flux polynomial expressed in the Raviart-Thomas basis up to the sixth-order of accuracy. Compared to the literature on the Spectral Difference approach, the present work increases the order of accuracy that the stable formulation can deal with. The proposed scheme is based on a set of flux points defined in the paper. The sets of points leading to a stable formulation are determined using a Fourier stability analysis of the linear advection equation coupled with an optimization process. The proposed Spectral Difference formulation differs from the Flux Reconstruction method on hybrid grids: the distinction between the two approaches is highlighted through the definition of the number of interior flux points. Validation starts from a convergence study using Euler equations and continues with simulations of laminar viscous flows over the NACA0012 airfoil using quadratic triangles and around a cylinder using a hybrid grid. [ABSTRACT FROM AUTHOR]

• A positivity-preserving high order accurate finite volume hybrid HWENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • The method is more efficient and more robust than the conventional HWENO methods. • The method is especially applicable for solving gas dynamics equations in low density and low pressure regime. In this paper, we construct a positivity-preserving high order accurate finite volume hybrid Hermite Weighted Essentially Non-oscillatory (HWENO) scheme for compressible Navier-Stokes equations, by incorporating a nonlinear flux and a positivity-preserving limiter. HWENO schemes have more compact stencils than WENO schemes but with higher computational cost due to the auxiliary variables. The hybrid HWENO schemes use linear reconstructions in smooth region thus are more efficient than conventional HWENO schemes. However, the hybrid HWENO is not robust for many demanding problems. The positivity-preserving hybrid HWENO scheme in this paper is not only more efficient but also much more robust than the conventional HWENO method for both compressible Euler and compressible Navier-Stokes equations, especially for solving gas dynamics equations in low density and low pressure regime. Numerical tests on low density and low pressure problems are performed to demonstrate the robustness and the efficiency of the positivity-preserving hybrid HWENO scheme. [ABSTRACT FROM AUTHOR]

Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Parés in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2 p and use (2 p + 1) -point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Parés, Russo and Zorío introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy – and thus the width of the stencils – is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests. • Extension of adaptive compact approximate Taylor methods to balance laws. • Construction of Arbitrary high order well balanced shock capturing schemes for balance laws. • Well balanced property is proven theoretically and verified numerically. • 1D and 2D numerical test cases are presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximation of the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. The proposed approximation technique is robust, formally second-order accurate in space, invariant-domain preserving, and works for every equation of state, tabulated or analytic, provided the pressure is nonnegative. An entropy surrogate functional that grows across shocks is proposed. The numerical method is verified with novel analytical solutions and then validated with several computational benchmarks seen in the literature including problems with composite waves. • Approximation technique for the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. • The method is verified with novel analytical solutions and validated with computational benchmarks from the literature. [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]

Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, numerical scale-consistent transport process, and similar domain of dependence and influence in the physical and the numerical wave propagations. Therefore, a physically soundable numerical scheme should be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number (∼1). In order to keep the compactness of the scheme and be high-order, subcell flow distributions or the equivalent degrees of freedom beyond the cell averaged flow variables are needed, such as the gradients of the flow variables inside each control volume. Under the above consideration, the direct use of Riemann solver as the evolution model doesn't provide such a mechanism. High-order dynamic evolution process should be used to update the time accurate cell-averaged flow variables and their gradients. The direct modeling of flow evolution under generalized initial condition will be developed. In order to provide reliable cell averaged flow variables and their gradients for the compact data reconstruction, the evolution process is constructed to provide possible discontinuous time-dependent flow variables across a cell interface. At the same time, the time accurate flux function at a cell interface can become a discontinuous function of time in its evolution process. Same as the spatial limiter in the conventional computational fluid dynamics methods, such as the total variation diminishing and weighted essentially non-oscillatory methods, the temporal limiter for the time accurate flux function is designed as well. The direct modeling method in this paper will update flow variables differently on both sides of a cell interface and nonlinearly limit the high-order time derivatives of the flux function in case of shock wave passing through a cell interface within a time step. The direct modeling method unifies the concepts of nonlinear limiters in spatial data reconstruction and temporal flux transport. The equivalent treatment of space and time makes the compact gas-kinetic scheme be super robust and accurate for the compressible flow simulation. At the same time, the 4th to 8th-order compact schemes can use a large CFL number (∼ 0.8) in flow computation, even for the hypersonic flow simulation. • A general principle for the development of high-order compact scheme is presented. • Same as the nonlinear limiter in space, the nonlinear limiter in time for the flux function is proposed. • According to the principle the high-order compact gas-kinetic scheme is developed. • Accurate solutions of compressible Euler and Navier-Stokes equations are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approximation strategy for time-dependent hyperbolic systems of conservation laws for the Euler system of gas dynamics that aims to represent the dynamics of strong interacting discontinuities. The goal of our method is to allow an approximation with a high-order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients, in particular across shocks. Following the Multidimensional Optimal Order Detection (MOOD) ([15,17]) approach, a candidate solution is computed at a next time level via a high-order accurate explicit scheme ([3,5]). A so-called detector determines if the candidate solution reveals any spurious oscillations or numerical issue and, if so, only the troubled cells are locally recomputed via a more dissipative scheme. This allows to design a family of "a posteriori" limited, robust and positivity preserving, as well as high accurate, non-oscillatory and effective scheme. Among the detecting criteria of the novel MOOD strategy, two different approaches from literature, based on the work of [15,17] and on [35] , are investigated. Numerical examples in 1D and 2D, on structured and unstructured meshes, are proposed to assess the effective order of accuracy for smooth flows, the non-oscillatory behaviour on shocked flows, the robustness and positivity preservation on more extreme flows. • Presentation of a novel MOOD strategy for multidimensions. • MOOD on high order Residual Distribution schemes for compressible flows. • Application of the strategy to triangle and quad meshes. [ABSTRACT FROM AUTHOR]

Three asymptotic limits exist for the Euler equations at low Mach number - purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit. In this paper, we review the low-Mach behaviour using the modified equations - the continuous Euler equations augmented with artificial diffusion terms - which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single- and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting. As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows. • Asymptotic analysis of numerical schemes at low Mach number. • Consider the convective, acoustic, and mixed convective-acoustic low Mach limits. • Derivation of required asymptotic scaling of artificial diffusion at each limit. • Use of modified equations applies to finite-volume or finite-difference schemes. • Application to Roe schemes shows excellent agreement with previous literature. [ABSTRACT FROM AUTHOR]

The third-order edge-based (EB3) scheme for the differential conservation law is widely used on node-centered control volumes. It is much efficient and easy to implement, since the second derivatives for both fluxes and solutions are not required to compute and store, and the single quadrature point at per edge is enough to preserve the third-order solution accuracy on triangular grids. In this paper, the scheme is extended to the regular cell-centered control volume in two dimensions, and by analogy, it is named FB3, where "F" denotes the face of cell-centered elements. Compatible source term discretization is carefully derived, and based on the "collapsing-cell" idea, accuracy preserving boundary scheme is developed. More importantly, within the scope of FB3 scheme, the regularity of triangular grids is specifically discussed from the analytical and numerical perspectives. Numerical experiments based on the linear convective and Euler equations are implemented, and results indicate that the third-order solution accuracy is achieved at both internal and boundary fields, as long as the control volumes consist of regular triangles. • EB3 scheme for the differential conservation law is extended from the node-centered to regular cell-centered control volume. • Compatible source term discretization is constructed according to the truncation error analysis. • Accuracy preserving boundary scheme is developed based on the "collapsing-cell" idea. • Relationship between the truncation and discretization errors is examined and verified again on the cell-centered scheme. [ABSTRACT FROM AUTHOR]

In this paper we develop a family of arbitrarily high-order non-oscillatory hybrid Discontinuous Galerkin(DG)-Finite Volume(FV) schemes for mixed-element unstructured meshes. Their key ingredient is a switch between a DG method and a FV method based on the CWENOZ scheme when invalid solutions are detected by a troubled cell indicator checking the unlimited DG solution. Therefore, the high order of accuracy offered by DG is preserved in smooth regions of the computational domain, while the robustness of FV is utilized in regions with strong gradients. The high-order CWENOZ variant used has the same spatial order of accuracy as the DG variant, while representing one of the most compact applications on unstructured meshes, therefore simplifying the implementation, reducing the computational overhead associated with large stencils of the original WENO reconstruction without sacrificing the desirable non-oscillatory properties of the schemes. We carefully investigate several parameters associated with the switching between DG and FV methods including the troubled cell indicators in a priori fashion. For the first time in the literature, we investigate the definition of the bounds for an admissible solution, the frequency by which we use the troubled cell indicators, and the evolution of the percentage of troubled cells for unsteady test problems. The 2D and 3D Euler equations are solved for well established test problems and compared with computational or experimental reference solutions. All the methods have been implemented and deployed within the UCNS3D open-source high-order unstructured Computational Fluid Dynamics (CFD) solver. The present coupling has the potential to improve the shortcomings of both FV-DG in a computational efficient manner. The improved accuracy and robustness provided is a characteristic of paramount importance for industrial-scale CFD applications, and favours the extension to other systems of governing equations. [ABSTRACT FROM AUTHOR]

• A novel numerical method to simulate cavitation induced by long-pulsed laser. • The processes of laser radiation, vaporization, and bubble/fluid dynamics are simulated simultaneously. • An embedded boundary method is proposed for laser-fluid coupling, well-posed with 2nd-order accuracy. • The level set method is combined with a phase transition law to track continuous vaporization. • Predictive capability demonstrated by simulations of different laser experiments. A computational method for simulating thermal cavitation induced by long-pulsed laser is presented. This method accounts for the absorption of laser light by a liquid, the formation of vapor bubbles due to localized heating, and the dynamics of the bubbles and the surrounding liquid. The physical model combines the Euler equations for a compressible inviscid two-phase fluid flow, a reduced form of the radiative transfer equation for laser radiation, and a local thermodynamic model of vaporization. The Euler equations are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). Following this method, numerical fluxes across phase boundaries are computed by constructing and solving one-dimensional bimaterial Riemann problems. The paper focuses on numerical methods for coupling the laser and fluid governing equations and tracking the vapor bubbles. An embedded boundary finite volume method is proposed to solve the laser radiation equation on the same mesh created for the Euler equations, which usually does not resolve the boundary and propagation directions of the laser beam. To impose boundary conditions, ghost nodes outside the laser domain are populated by mirroring and interpolation techniques. The existence and uniqueness of solution are proved for the two-dimensional case, leveraging the special geometry of the laser domain. The order of accuracy of the method is also proved, and verified using numerical tests. A method of latent heat reservoir is proposed to predict the onset of vaporization, which accounts for the accumulation and release of latent heat. A unique challenge associated with long-pulsed laser is that the dynamics of vapor bubbles is driven not only by the inertia of the bubble nuclei, but also by the continuation of vaporization. In this work, the localized level set method is employed to track the bubble surface, and a method of local correction and reinitialization is proposed to account for continuous phase transitions. Several numerical tests are presented to verify the convergence of these methods. Two realistic simulations of laser-induced cavitation are presented at the end, showing that the computational method is able to capture the key phenomena in these events, including non-spherical bubble expansion, shock waves, and the "Moses effect". [ABSTRACT FROM AUTHOR]

This paper develops a numerical method for predicting the nonlinear structural-acoustic interactions of a hyperelastic solid immersed in an unbounded compressible viscous fluid. The nonlinear acoustic wave model with perfectly matched layers adopted is proposed in an arbitrary Lagrangian-Eulerian frame to absorb acoustic waves at truncated fluid boundaries, whereas the geometric and material nonlinearities of the hyperelastic solid are characterized in a total Lagrangian form. Both the fluid and the solid are discretized by the finite element method, and both models are coupled together by the momentum equilibrium and kinematic compatibility on the common interface. The convergence and accuracy of the proposed method are assessed by comparing the computed results with existing solutions. A hyperelastic ring in a viscous fluid is analyzed to provide physical insights into the nonlinear dynamics behaviors of the coupled hyperelastic solid and acoustic fluid system. The hyperelastic ring in acoustic fluid exhibits deformation instability under a uniform distributed excitation load, producing a series of subharmonics, ultrasubharmonics, and superharmonics in the structural and acoustic responses. Nonlinear internal resonance and bifurcation phenomena in the vibration of the ring and the acoustic waves of the fluid are also discussed. • A numerical method for nonlinear structural-acoustic interaction of a hyperelastic solid immersed in an unbounded fluid is proposed. • Arbitrary Lagrangian-Eulerian technique is adopted for the nonlinear acoustic wave model in a time-dependent acoustic domain. • A loosely coupled serially staggered procedure for nonlinear structural-acoustic interaction is presented. • Internal resonance and bifurcation phenomena of a hyperelastic ring and the acoustic waves are discussed. [ABSTRACT FROM AUTHOR]

This paper presents an almost arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve the compressible Euler equations in one and two space dimensions, and consider their positivity preserving property of states density and pressure. The ALE-DG method coupling with a modified strong stability-preserving Runge-Kutta method is developed to ensure the geometric conservation law and the positivity property of the scheme with the positivity preserving limiter developed by Zhang et al. (2010) [34] and Zhang et al. (2012) [38]. For the Lax-Friedrichs, HLL, and HLLC numerical fluxes, we prove that our proposed ALE-DG method can keep the positivity property of approximations of density and pressure from the first-order ALE-DG method and the cell averages of approximations of density and pressure from high order ALE-DG method under suitable time step. Numerical examples demonstrate that the proposed positivity-preserving ALE-DG method can keep the high order accuracy, numerical stability, and positivity on moving meshes for Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations. • First implicit two-derivative deferred correction scheme for Navier-Stokes equations. • High order spatial and temporal accuracy. • Flexible matrix-free environment with simple preconditioning strategy. [ABSTRACT FROM AUTHOR]

• Robust steady-state convergence for Euler equations with strong shock waves. • Better performance of the essentially non-oscillatory (ENO) property. • Less dissipative and dispersive errors. • The simple and effective smoothness detector. For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations. [ABSTRACT FROM AUTHOR]

This paper designs and analyzes positivity-preserving well-balanced (WB) central discontinuous Galerkin (CDG) schemes for the Euler equations with gravity. A distinctive feature of these schemes is that they not only are WB for a general known stationary hydrostatic solution, but also can preserve the positivity of the fluid density and pressure. The standard CDG method does not possess this feature, while directly applying some existing WB techniques to the CDG framework may not accommodate the positivity and keep other important properties at the same time. In order to obtain the WB and positivity-preserving properties simultaneously while also maintaining the conservativeness and stability of the schemes, a novel spatial discretization is devised in the CDG framework based on suitable modifications to the numerical dissipation term and the source term approximation. The modifications are based on a crucial projection operator for the stationary hydrostatic solution, which is proposed for the first time in this work. This novel projection has the same order of accuracy as the standard L 2 -projection, can be explicitly calculated, and is easy to implement without solving any optimization problems. More importantly, it ensures that the projected stationary solution has the same cell averages on both the primal and dual meshes, which is a key to achieve the desired properties of our schemes. Based on some convex decomposition techniques, rigorous positivity-preserving analyses for the resulting WB CDG schemes are carried out. Several one- and two-dimensional numerical examples are performed to illustrate the desired properties of these schemes, including the high-order accuracy, the WB property, the robustness for simulations involving the low pressure or density, high resolution for the discontinuous solutions and the small perturbations around the equilibrium state. • New central discontinuous Galerkin schemes for Euler equations with gravitation are developed and rigorously analyzed. • Well-balance and provably positivity-preserving properties are simultaneously achieved. • The schemes are designed based on a crucial projection operator, which is proposed for the first time. • The novel projection is simple and explicitly expressible and has the same order of accuracy as standard L 2 -projection. • Extensive benchmark numerical tests demonstrate the desired properties of the proposed schemes. [ABSTRACT FROM AUTHOR]

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall ((2018) [1]). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example. [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]

• A compact fifth-order gas-kinetic scheme is proposed for compressible Euler equations. • It is based on the framework of the one-stage efficient high-order gas-kinetic scheme. • A newly-developed simple and efficient compact reconstruction technique is used. • It is more efficient than the original one-stage EHGKS with WENO reconstruction. In this paper, a compact fifth-order gas-kinetic scheme (GKS) is proposed to solve the Euler equations of compressible flows. It is based on the framework of a one-stage efficient high-order GKS (EHGKS), which can achieve arbitrary high-order accuracy in both space and time, and the newly-developed compact reconstruction technique, combining both the ideas of the simple weighted essentially non-oscillatory scheme (WENO) and Hermite WENO (HWENO). The evolution model of EHGKS provides the time-dependent variables at cell interfaces, which can directly compute the cell-average slope. Thus the necessary information for a fifth-order compact HWENO reconstruction technique is obtained, and also making it possible to improve simple WENO as a compact reconstruction by determining the quartic polynomial based on the compact stencil. The one-stage time discretization technique of EHGKS makes it possible to utilize the reconstruction only once in each computing time step, which is very different from the present compact multi-stage GKS schemes. The newly-developed compact reconstruction provides the polynomial, not only the pointwise variables but also their spatial derivatives for the interface variables evaluations directly. Several numerical examples are given to validate the accuracy and effectiveness of the proposed compact one-stage scheme. And the numerical results also demonstrate that the new scheme is more efficient than the original EHGKS method with WENO reconstruction technique. [ABSTRACT FROM AUTHOR]

This paper discusses a new class of optimized implicit-explicit Runge-Kutta methods for the numerical solution of the dispersive and non-dispersive hyperbolic systems. Optimized implicit-explicit methods are formulated for the better stability and dispersion properties. Moreover, for present methods inversion of the coefficient matrices is not necessary, which makes these methods very attractive in terms of computational cost and complexity. To validate the efficiency of the developed methods, we have solved the one- and two-dimensional dispersive rotating shallow water equations and benchmark problems from acoustics. Computed solutions are also compared with the exact and experimental results available in the literature. The present methods compete well with the existing multi-stage time-integration methods in terms of accurately resolving the physical characteristics for the chosen problems. Furthermore, the computational costs of the proposed methods are significantly lower as compared to the four-stage, fourth-order explicit Runge-Kutta (RK 4) method. • Formulation of a new class of optimized implicit-explicit Runge-Kutta methods. • Developed methods do not require inversion of coefficient matrices-reduced computational cost. • Spectral analysis is used to assess the numerical properties in the spectral plane. • Constrained optimization is performed to freeze the values of free parameters in spectral plane. • Methods are gauged for non-dispersive benchmark problems from acoustics and dispersive shallow water equations. [ABSTRACT FROM AUTHOR]

• Nonunique solutions to 2D compressible Euler equations are studied numerically. • Three parameters in the initial data are tuned to generate multiple solutions. • The compressibility of the equation affects the solution structure. In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

This paper proposes and implements the Random Choice Method (RCM) based on a direct Eulerian generalized Riemann problem (GRP) scheme for one-dimensional Euler equations in gas dynamics. It is an application of the second order accurate GRP scheme proposed by Ben-Artzi et al. (2006) [6]. Since the RCM was introduced as a numerical tool in the gas dynamics and it has the advantage of capturing discontinuities with sharp resolution, we here implement the RCM by choosing the GRP scheme as the "building block". The initial data are assumed to be piecewise linear function, and the local GRP is resolved directly and analytically at each interface. Special attention is paid to the treatment of resolving smooth wave. Numerical simulations on some typical problems show that the proposed method achieves good performance. [ABSTRACT FROM AUTHOR]

• A conservative sharp interface method is developed to simulate displacements of and collisions among particle clouds. • A hard sphere model is proposed to handle multi-body collisions among a number of particles. • Simulations are performed for the interaction between shocks and 2D/3D particles. • Results are verified against experimental data and benchmark solutions. In this paper we present a second-order accurate conservative sharp interface method capable of simulating displacements and collisions of many particles in compressible inviscid flows. We implement a cut cell algorithm to resolve moving particles of arbitrary shape on a Cartesian mesh, thereby generating unstructured body-fitted meshes near the particle surfaces and structured meshes away from the particles. A second-order finite volume method in the arbitrary Lagrange-Eulerian framework is used for the discretization of the Euler equations, so that exact conservation of mass, momentum and energy is enforced for the flow computations, even in the presence of moving particles. The boundary condition at the particle surfaces is enforced by solving a local Riemann problem, which is constructed in the direction normal to the solid surface. The movement of a particle is affected by the forces exerted by the surrounding fluid and its collisions with other particles. A hard sphere model is proposed to deal with the particle collisions; in particular, it is capable of handling multi-body collisions occurring in no-dilute particle clouds, regardless of the number of particles involved. Furthermore, the model ensures the conservation of momentum and energy during the collision process. The method is validated by comparing against benchmark solutions or experimental data available in the literature, for test cases such as supersonic flows past a stationary cylinder and lift-off of rigid cylinders after shock impact. Good agreement has been achieved qualitatively and quantitatively. The method is also used to investigate the complicated flow phenomena, e.g. transport of particle clouds in supersonic flows through a confined channel and the interaction between shocks and three-dimensional particles. [ABSTRACT FROM AUTHOR]

• An indirect ALE DG model on moving triangular meshes with mesh topological adaptability is developed. • The best features of the Lagrangian description, DG method and adaptive topology optimization strategy are incorporated. • The present scheme can handle hydrodynamics problems which involve complex deformations, large distortions and strong shock. • The new indirect ALE DG scheme is more accurate and stable than the conventional Lagrangian DG scheme. In this paper, we present a novel cell-centered indirect Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme on moving unstructured triangular meshes with mesh topological adaptability, aiming to deal with the strong distortions and large deformation flow problems. The scheme combines the explicit time marching Lagrangian DG methodology with the adaptive mesh topology optimization technique. The scheme consists of the following three steps. Firstly, we utilize the Runge-Kutta DG method to solve the compressible Euler equation in Lagrangian framework, and employ a nodal solver to obtain the nodal velocity and numerical fluxes across element boundaries. The physical variable and nodal position are updated in this step. Secondly, the adaptive mesh topology optimization technique, which includes the mesh refinement, edge collapse operation and mesh regularization, is implemented to eliminate the highly distorted elements and improve the mesh quality. Thirdly, the conservative remapping algorithm is employed, which can maintain the conservative interpolation of the Lagrangian solution onto the remeshed grid. The present indirect ALE DG scheme can ensure the high quality of the mesh by optimizing the topology connectivity, so that the present scheme can successfully simulate complex vortical flow problems for a sufficient simulation time. Due to the inherent Lagrangian nature, the present scheme can naturally track the multi-material flow interface, rather than using algorithms with interface reconstruction or diffuse interfaces. The scheme is validated with several benchmark flow problems. It is demonstrated that the present indirect ALE DG scheme with topological adaptability can accurately simulate flow problems with large fluid deformations and distortions. It can achieve remarkable improvements compared with the conventional Lagrangian DG method with fixed topological connectivity. [ABSTRACT FROM AUTHOR]

In this paper a fully Eulerian solver for the study of multiphase flows for simulating the propagation of surface gravity waves over submerged bodies is presented. We solve the incompressible Navier–Stokes equations coupled with the volume of fluid technique for the modeling of the liquid phases with the interface, an immersed body method for the solid objects and an iterative strong–coupling procedure for the fluid–structure interaction. The flow incompressibility is enforced via the solution of a Poisson equation which, owing to the density jump across the interfaces of the liquid phases, has to resort to the splitting procedure of Dodd and Ferrante [1]. The solver is validated through comparisons against classical test cases for fluid–structure interaction like migration of particles in a pressure–driven channel, multiphase flows, 'water exit' of a cylinder and a good agreement is found for all tests. Furthermore, we show the application of the solver to the case of a surface gravity wave propagating over a submerged reversed pendulum and verify that the solver can reproduce the energy exchange between the wave and the pendulum. Finally the three–dimensional spilling breaking of a wave induced by a submerged sphere is considered. [ABSTRACT FROM AUTHOR]

• Linearized/nonlinear Euler equations are coupled by a domain decomposition method. • Method captures wave propagation and interactions with shock and heated flow. • Computation is three times faster than a purely nonlinear solver for same accuracy. • Results in excellent agreement with linearized interaction analysis. In this paper, a domain decomposition technique in the finite volume framework is presented to propagate small amplitude acoustic and entropy waves in a linearized Euler region and simulate the interaction of these waves with an initially steady normal shock in a nonlinear region. An overset method is used to two-way couple the linear and nonlinear regions that overlap each other. Linearized solvers alone cannot capture this interaction due to the discontinuity encountered at shocks. On the other hand, nonlinear solvers based on second order shock-capturing schemes will result in excessive dissipation and dispersion for the small disturbances. The domain decomposition technique provides a good balance between minimizing dissipation and dispersion errors while enabling nonlinear shock-acoustic interactions. To preserve low dispersion and dissipation, a DRP scheme is used to simulate the incoming and outgoing waves in the linear region. To capture the shock wave interaction and motion, a hybrid central-upwind flux scheme is used in the nonlinear region that contains the shock. Grid sensitivity studies for an acoustic wave propagating in stationary flow were performed to compare the linear, nonlinear, and domain decomposition solvers. The nonlinear solver required ten times the mesh resolution to achieve similar accuracy as the linear solver, resulting in a forty-fold increase in computational time. For modest cell size ratios, the domain decomposition solver reduced the computational time by a factor of three compared to the nonlinear solver while achieving similar accuracy. Interaction of standing shocks with acoustic and entropy waves of amplitudes ϵ = ± 10 − 2 and ± 10 − 5 was investigated using the domain decomposition technique. The numerical results for ϵ = ± 10 − 2 compared well with the linearized interaction analysis (LIA) with less than 3% discrepancy in terms of the amplification factors. The domain decomposition technique acts as a low pass filter that averages the post-shock oscillations generated by the slow-moving shocks in the nonlinear region, resulting in the correct amplification factors in the linear region. For the smaller amplitudes of ϵ = ± 10 − 5 , the amplification factors deviated from LIA predictions by up to 70%. Numerical results suggest that the large discrepancy for the small amplitude cases is due to insufficient mesh resolution for capturing extremely slow-moving shocks. [ABSTRACT FROM AUTHOR]

High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes equations. Numerical experiments confirm the high order accuracy and robustness of the proposed strategy. • New sparse low order summation-by-parts operators. • Minimally dissipative positivity-preserving limiter for entropy stable DGSEM. • Verification using challenging test cases in unsteady viscous compressible flow. [ABSTRACT FROM AUTHOR]