Discrete energy-conservation properties in the numerical simulation of the navier–stokes equations

Nonlinear convective terms pose the most critical issues when a numerical discretization of the Navier–Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier–Stokes equations. The invariant character of quadratic quantities such as global kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.Postprint (published version

Efficient adaptive pseudo-symplectic numerical integration techniques for Landau-Lifshitz dynamics

Numerical time integration schemes for Landau-Lifshitz magnetization dynamics are considered. Such dynamics preserves the magnetization amplitude and, in the absence of dissipation, also implies the conservation of the free energy. This property is generally lost when time discretization is performed for the numerical solution. In this work, explicit numerical schemes based on Runge-Kutta methods are introduced. The schemes are termed pseudo-symplectic in that they are accurate to order p, but preserve magnetization amplitude and free energy to order q > p. An effective strategy for adaptive time-stepping control is discussed for schemes of this class. Numerical tests against analytical solutions for the simulation of fast precessional dynamics are performed in order to point out the effectiveness of the proposed methods. This work was carried out in the frame of Program for the Support of Individual Research 2016 funded by University of Naples Parthenope.Peer ReviewedPostprint (published version

Derivation of new staggered compact schemes with application to navier-stokes equations

A method is proposed for the derivation of new classes of staggered compact derivative and interpolation operators. The algorithm has its roots in an implicit interpolation theory consistent with compact schemes and reduces to the computation of the required staggered derivatives and interpolated quantities as a combination of nodal values and collocated compact derivatives. The new approach is cost-effective, simplifies the imposition of boundary conditions, and has improved spectral resolution properties, on equal order of accuracy, with respect to classical schemes. The method is applied to incompressible Navier-Stokes equations through the implementation into a staggered flow solver with a fractional step procedure, and tested on classical benchmarks.Postprint (published version

Explicit Runge–Kutta schemes for incompressible flow with improved energy-conservation properties

The application of pseudo-symplectic Runge–Kutta methods to the incompressible Navier–Stokes equations is discussed in this work. In contrast to fully energy-conserving, implicit methods, these are explicit schemes of order p that preserve kinetic energy to order q, with q>p. Use of explicit methods with improved energy-conservation properties is appealing for convection-dominated problems, especially in case of direct and large-eddy simulation of turbulent flows. A number of pseudo-symplectic methods are constructed for application to the incompressible Navier–Stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations.Peer ReviewedPostprint (author's final draft

Approximate projection method for the incompressible Navier–Stokes equations

Postprint (published version

Energy preserving turbulent simulations at a reduced computational cost

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.Postprint (published version

An efficient time advancing strategy for energy-preserving simulations

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.Postprint (published version

Numerically stable formulations of convective terms for turbulent compressible flows

A systematic analysis of the discrete conservation properties of non-dissipative, central-difference approximations of the compressible Navier–Stokes equations is reported. A generalized splitting of the nonlinear convective terms is considered, and energy-preserving formulations are fully characterized by deriving a two-parameter family of split forms. Previously developed formulations reported in literature are shown to be particular members of this family; novel splittings are introduced and discussed as well. Furthermore, the conservation properties yielded by different choices for the energy equation (i.e. total and internal energy, entropy) are analyzed thoroughly. It is shown that additional preserved quantities can be obtained through a suitable adaptive selection of the split form within the derived family. Local conservation of primary invariants, which is a fundamental property to build high-fidelity shock-capturing methods, is also discussed in the paper. Numerical tests performed for the Taylor–Green Vortex at zero viscosity fully confirm the theoretical findings, and show that a careful choice of both the splitting and the energy formulation can provide remarkably robust and accurate results.Peer ReviewedPostprint (author's final draft

Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations

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.Comment: 9 pages, 4 figure

Numerical treatment of the energy equation in compressible flows simulations

We analyze the conservation properties of various discretizations of the system of compressible Euler equations for shock-free flows, with special focus on the treatment of the energy equation and on the induced discrete equations for other thermodynamic quantities. The analysis is conducted both theoretically and numerically and considers two important factors characterizing the various formulations, namely the choice of the energy equation and the splitting used in the discretization of the convective terms. The energy equations analyzed are total and internal energy, total enthalpy, pressure, speed of sound and entropy. In all the cases examined the discretization of the convective terms is made with locally conservative and kinetic-energy preserving schemes. Some important relations between the various formulations are highlighted and the performances of the various schemes are assessed by considering two widely used test cases. Together with some popular formulations from the literature, also new and potentially useful ones are analyzed