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High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely — interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf. Nordström and Ruggiu (2017) [38] and Manzanero et al. (2017) [39] ). [ABSTRACT FROM AUTHOR]

A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable split-form of Burgers' equation. Moreover, overcoming limitations of the finite difference framework, stability for curvilinear grids and dense norms is obtained for SBP CPR methods by using a suitable way to compute the Jacobian. [ABSTRACT FROM AUTHOR]

Energy-conserving discretizations are widely regarded as a fundamental requirement for high-fidelity simulations of turbulent flows. The skew-symmetric splitting of the nonlinear term is a well-known approach to obtain semi-discrete conservation of energy in the inviscid limit. However, its computation is roughly twice as expensive as that of the divergence or advective forms alone. A novel time-advancement strategy that retains the conservation properties of skew-symmetric-based schemes at a reduced computational cost has been developed. This method is based on properly constructed Runge–Kutta schemes in which a different form (advective or divergence) for the convective term is adopted at each stage. A general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation. Simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew-symmetric-based techniques. [ABSTRACT FROM AUTHOR]

Energy-conserving numerical methods are widely employed within the broad area of convection-dominated systems. Semi-discrete conservation of energy is usually obtained by adopting the so-called skew-symmetric splitting of the non-linear convective term, defined as a suitable average of the divergence and advective forms. Although generally allowing global conservation of kinetic energy, it has the drawback of being roughly twice as expensive as standard divergence or advective forms alone. In this paper, a general theoretical framework has been developed to derive an efficient time-advancement strategy in the context of explicit Runge–Kutta schemes. The novel technique retains the conservation properties of skew-symmetric-based discretizations at a reduced computational cost. It is found that optimal energy conservation can be achieved by properly constructed Runge–Kutta methods in which only divergence and advective forms for the convective term are used. As a consequence, a considerable improvement in computational efficiency over existing practices is achieved. The overall procedure has proved to be able to produce new schemes with a specified order of accuracy on both solution and energy. The effectiveness of the method as well as the asymptotic behavior of the schemes is demonstrated by numerical simulation of Burgers' equation. [ABSTRACT FROM AUTHOR]

Abstract: The form of convective terms for compressible flow equations is discussed in the same way as for an incompressible one by Morinishi et al. [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys. 124 (1998) 90], and fully conservative finite difference schemes suitable for shock-free unsteady compressible flow simulations are proposed. Commutable divergence, advective, and skew-symmetric forms of convective terms are defined by including the temporal derivative term for compressible flow. These forms are analytically equivalent if the continuity is satisfied, and the skew-symmetric form is secondary conservative without the aid of the continuity, while the divergence form is primary conservative. The relations between the present and existing quasi-skew-symmetric forms are also revealed. Commutable fully discrete finite difference schemes of convection are then derived in a staggered grid system, and they are fully conservative provided that the corresponding discrete continuity is satisfied. In addition, a semi-discrete convection scheme suitable for compact finite difference is presented based on the skew-symmetric form. The conservation properties of the present schemes are demonstrated numerically in a three-dimensional periodic inviscid flow. The proposed fully discrete fully conservative second-order accurate scheme is also used to perform the DNS of compressible isotropic turbulence and the simulation of open cavity flow. [Copyright &y& Elsevier]

Abstract: A new high-order finite-volume method is presented that preserves the skew symmetry of convection for the compressible flow equations. The method is intended for Large-Eddy Simulations (LES) of compressible turbulent flows, in particular in the context of hybrid RANS–LES computations. The method is fourth-order accurate and has low numerical dissipation and dispersion. Due to the finite-volume approach, mass, momentum, and total energy are locally conserved. Furthermore, the skew-symmetry preservation implies that kinetic energy, sound-velocity, and internal energy are all locally conserved by convection as well. The method is unique in that all these properties hold on non-uniform, curvilinear, structured grids. Due to the conservation of kinetic energy, there is no spurious production or dissipation of kinetic energy stemming from the discretization of convection. This enhances the numerical stability and reduces the possible interference of numerical errors with the subgrid-scale model. By minimizing the numerical dispersion, the numerical errors are reduced by an order of magnitude compared to a standard fourth-order finite-volume method. [Copyright &y& Elsevier]

Abstract: The effect on aliasing errors of different formulations describing the cubically nonlinear convective terms within the discretized Navier–Stokes equations is examined in the presence of a non-trivial density spectrum. Fourier analysis shows that the existing skew-symmetric forms of the convective term result in reduced aliasing errors relative to the conservation form. Several formulations of the convective term, including a new formulation proposed for cubically nonlinear terms, are tested in direct numerical simulation (DNS) of decaying compressible isotropic turbulence both in chemically inert (small density fluctuations) and reactive cases (large density fluctuations) and for different degrees of resolution. In the DNS of reactive turbulent flow, the new cubic skew-symmetric form gives the most accurate results, consistent with the spectral error analysis, and at the lowest cost. In marginally resolved DNS and LES (poorly resolved by definition) the new cubic skew-symmetric form represents a robust convective formulation which minimizes both aliasing and computational cost while also allowing a reduction in the use of computationally expensive high-order dissipative filters. [Copyright &y& Elsevier]

We present a numerical formulation for the treatment of nonlinear instabilities in shock-free compressible turbulence simulations. The formulation is high order and contains no artificial dissipation. Numerical stability is enhanced through semi-discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation. The numerical implementation is achieved using a conservative skew-symmetric splitting of the nonlinear terms. The robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high Reynolds number. Results show the scheme is capable of capturing the statistical equilibrium of low Mach number compressible turbulent fluctuations at infinite Reynolds number. Comparisons with the entropy splitting technique [J. Comput. Phys. 162 (2000) 33; J. Comput. Phys. 178 (2002) 307], staggered method [J. Comput. Phys. 191(2) (2003) 392], and skew-symmetric like schemes [J. Comput. Phys. 161 (2000) 114] confirm the superiority of the current approach. We also discuss a flaw in the skew-symmetric splitting implemented in the literature. Very good results are obtained based on the proper splitting. [Copyright &y& Elsevier]

The dimensions of maximal isotropic subspaces of symmetric and skew-symmetric forms of degree d in general position are found. [ABSTRACT FROM AUTHOR]

Structure-preserving discretization of the Rosenbluth–Fokker–Planck equation is still an open question especially for unlike-particle collision. In this paper, a mass-energy-conserving isotropic Rosenbluth–Fokker–Planck scheme is introduced. The structure related to the energy conservation is skew-symmetry in mathematical sense, and the action–reaction law in physical sense. A thermal relaxation term is obtained by using integration-by-parts on a volume integral in the energy moment equation, so the discontinuous Galerkin method is selected to preserve the skew-symmetry. The discontinuous Galerkin method enables ones to introduce the nonlinear upwind flux without violating the conservation laws. Some experiments show that the conservative scheme maintains the mass-energy-conservation only with round-off errors, and analytic equilibria are reproduced only with truncation errors of its formal accuracy. [ABSTRACT FROM AUTHOR]

In this paper, an energy-consistent finite difference formulation for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied in the discrete level without explicitly solving the total energy equation. The shock waves and discontinuities in the numerical solution are stabilized by nonlinear filtering schemes. An energy-consistent discretization of the filtering schemes is also derived by introducing the viscous and resistive heating rates. The resulting energy-consistent formulation can be implemented with the various kinds of central difference, nonlinear filtering, and time integration schemes. The second- and fifth-order schemes are implemented based on the proposed formulation. The conservation properties and the robustness of the present schemes are demonstrated via one- and two-dimensional numerical tests. The proposed schemes successfully handle the most stringent problems in extremely high Mach number and low beta conditions. • An energy-consistent finite difference formulation for the compressible hydrodynamics and magnetohydrodynamics. • Characteristic-based nonlinear filtering scheme with the energy consistency. • High degree of robustness in the high Mach-number and low-beta conditions. • Capability to be implemented with wide variety of central difference, nonlinear filtering, and time integration schemes. [ABSTRACT FROM AUTHOR]