Entropy Stability Research Articles

Nonlinearly stable flux reconstruction high-order methods in split form

The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted for Burgers' equation verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy.

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

The algebraic flux correction (AFC) schemes presented in this work constrain a standard continuous finite element discretization of a nonlinear hyperbolic problem to satisfy relevant maximum principles and entropy stability conditions. The desired properties are enforced by applying a limiter to antidiffusive fluxes that represent the difference between the high-order baseline scheme and a property-preserving approximation of Lax–Friedrichs type. In the first step of the limiting procedure, the given target fluxes are adjusted in a way that guarantees preservation of local and/or global bounds. In the second step, additional limiting is performed, if necessary, to ensure the validity of fully discrete and/or semi-discrete entropy inequalities. The limiter-based entropy fixes considered in this work are applicable to finite element discretizations of scalar hyperbolic equations and systems alike. The underlying inequality constraints are formulated using Tadmor’s entropy stability theory.The proposed limiters impose entropy-conservative or entropy-dissipative bounds on the rate of entropy production by antidiffusive fluxes and Runge–Kutta (RK) time discretizations. Two versions of the fully discrete entropy fix are developed for this purpose. The first one incorporates temporal entropy production into the flux constraints, which makes them more restrictive and dependent on the time step. The second algorithm interprets the final stage of a high-order AFC-RK method as a constrained antidiffusive correction of an implicit low-order scheme (algebraic Lax–Friedrichs in space + backward Euler in time). In this case, iterative flux correction is required, but the inequality constraints are less restrictive and limiting can be performed using algorithms developed for the semi-discrete problem. To motivate the use of limiter-based entropy fixes, we prove a finite element version of the Lax–Wendroff theorem and perform numerical studies for standard test problems. In our numerical experiments, entropy-dissipative schemes converge to correct weak solutions of scalar conservation laws, of the Euler equations, and of the shallow water equations.

Different unilateral force control strategies between athletes and non-athletes

This study investigated continuous visuomotor tracking capabilities between athletes and non-athlete controls using isometric force control paradigm. Nine female athletes and nine female age-matched controls performed unilateral hand-grip force control tasks with their dominant and non-dominant hands at 10% and 40% of maximal voluntary contraction (MVC), respectively. Three conventional outcome measures on force control capabilities included mean force, force accuracy, and force variability, and we additionally calculated two nonlinear dynamics variables including force regularity using sample entropy and force stability using maximal Lyapunov exponent. Finally, we performed correlation analyses to determine the relationship between nonlinear dynamics variables and conventional measures for each group. The findings indicated that force control capabilities as indicated by three conventional measures were not significantly different between athlete and non-athlete control groups. However, the athletes revealed less force regularity and greater force stability across hand conditions and targeted force levels than those in non-athlete controls. The correlation analyses found that increased force regularity (i.e., less sample entropy values) at 10% of MVC and decreased force regularity (i.e., greater sample entropy values) at 40% of MVC were significantly related to improved force accuracy and variability for the athlete group, and these patterns were not observed in the non-athlete control group. These findings suggested that the athletes may use different adaptive force control strategies as indicated by nonlinear dynamics tools.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

Development and analysis of entropy stable no-slip wall boundary conditions for the Eulerian model for viscous and heat conducting compressible flows

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Phys A Stat Mech Appl 506:350–375, 2018). The spatial discretization is based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier–Stokes models. The numerical results obtained with the two models are compared, and similarities and differences are then highlighted. However, the differences are very small and probably smaller than what the current experimental technology allows to measure.

Entropy Stability of Bicompact Schemes in Gas Dynamics Problems

Entropy Stability of Bicompact Schemes in Gas Dynamics Problems

Entropy Stable Discontinuous Galerkin Methods for Nonlinear Conservation Laws on Networks and Multi-Dimensional Domains

We present a high-order entropy stable discontinuous Galerkin method for nonlinear conservation laws on both multi-dimensional domains and on networks constructed from one-dimensional domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment.

Improved variational mode decomposition method for vibration signal processing of flood discharge structure

It is crucial for flood discharge structure vibration safety evaluations to filter low-frequency noise, separate dense-frequency components and obtain high-frequency component accurately from vibration signals. Variational mode decomposition, a novel signal adaptive decomposition method, effectively processes flood discharge structures. However, the mode number and quadratic penalty item uncertainty in variational mode decomposition directly affects the vibration signal decomposition. Therefore, an improved variational mode decomposition method for vibration signal processing is proposed in this study. The proposed method adaptively determines the mode number based on singular entropy and frequency stability to completely separate the structural vibration components (including dense-frequency components and high-frequency components) and noise components from the vibration signal. Next, an objective quadratic penalty item function based on sample entropy and mutual information is proposed to quantify the mode mixing between the structural vibration components. Finally, a particle swarm optimisation algorithm based on beetle antenna search is proposed to optimise the quadratic penalty item, which overcomes the shortcomings of traditional algorithms and suppresses the mode mixing between the structural vibration components. The validity and feasibility of the proposed method was verified by the simulation signal and was applied to a sluice prototype project. The results showed that the method effectively filtered noise, greatly improved the vibration response signal-to-noise ratio and obtained the structural vibration component time history signal, which provides a foundation for flood discharge structure vibration safety evaluation and health monitoring.

Kinetic Functions for Nonclassical Shocks, Entropy Stability, and Discrete Summation by Parts

We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions for scalar conservation laws in one space dimension, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.

Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations

In this article, we propose high order discontinuous Galerkin entropy stable schemes for ten-moment Gaussian closure equations, based on the suitable quadrature rules (see [1]). The key components of the proposed schemes are the use of an entropy conservative numerical flux [2] in each cell and an appropriate entropy stable numerical flux at the cell edges. These fluxes are then used in the entropy stable DG framework of [1] to obtain entropy stability of the semi-discrete schemes. We also extend these schemes to a source term that models plasma laser interaction. For the time discretization, we use strong stability preserving methods. The proposed schemes are then tested on several test cases to demonstrate stability, accuracy and robustness.

A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations

The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable Discontinuous Galerkin Spectral Element Method (DGSEM) based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order summation-by-parts (SBP) operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the Legendre-Gauss-Lobatto (LGL) nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. With tensor products and careful design of the low order scheme on curved elements, we are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.

Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws

This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems.

Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws

In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable/entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems.

High Thermoelectric Performance and Defect Energetics of Multipocketed Full Heusler Compounds

Thermoelectrics are used in energy harvesting technology for power generation and refrigeration, but await breakthroughs with a higher figure of merit ($Z\phantom{\rule{0}{0ex}}T$), especially at room-to-cryogenic temperatures. Via explicit treatment of electron-phonon scattering, this study shows that multipocketed full Heusler compounds Sr${}_{2}$BiAu and Sr${}_{2}$SbAu could feature notably high theoretical $Z\phantom{\rule{0}{0ex}}T$. Stability and defects analysis also suggest that these compounds may be synthesizable and favorably $n$-type. A successful experimental realization of these compounds could pave new grounds in bulk thermoelectrics.

Thermodynamics of rotating regular black holes

This paper presents a study on the thermodynamics of rotating Bardeen as well as Hayward regular black hole and their anti-de Sitter versions. We analyze the temperature, entropy, heat capacity and thermodynamical stability for the considered black hole geometries. The Hawking temperature is calculated using the surface gravity as well as tunneling probability method leading to the same result. We conclude that the considered rotating as well as rotating anti-de Sitter regular black holes are less hot and thermodynamically more stable than ordinary Kerr and Kerr-anti-de Sitter solutions, respectively.

Entropy-stable p-nonconforming discretizations with the summation-by-parts property for the compressible Navier–Stokes equations

The entropy-conservative/stable, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernández et al. (2019) is extended from the compressible Euler equations to the compressible Navier–Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to construct the interface coupling terms, thereby localizing the discrete geometric conservation law constraints to each individual element. The resulting scheme is entropy conservative/stable, element-wise conservative, and freestream preserving. Viscous interface dissipation operators that retain the entropy stability of the base scheme are developed. The accuracy and stability of the resulting numerical scheme are shown to be comparable to those of the original conforming scheme in Carpenter et al. (2014) and Parsani et al. (2016), i.e., this scheme achieves ∼p+1/2 convergence on geometrically high-order distorted element grids; this is demonstrated in the context of the viscous shock problem, the Taylor–Green vortex problem at a Reynolds number of Re=1,600, and a subsonic turbulent flow past a sphere at Re=2,000.

Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L∞-boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition.

Exploring the dispersion measure of the Milky Way halo

Abstract Fast radio bursts offer the opportunity to place new constraints on the mass and density profile of hot and ionized gas in galactic haloes. We test here the X-ray emission and dispersion measure predicted by different gas profiles for the halo of the Milky Way. We examine a range of models, including entropy stability conditions and external pressure continuity. We find that incorporating constraints from X-ray observations leads to favouring dispersion measures on the lower end of the range given by these models. We show that the dispersion measure of the Milky Way halo could be less than 10 cm−3 pc in the most extreme model we consider, which is based on constraints from O vii absorption lines. However, the models allowed by the soft X-ray constraints span more than an order of magnitude in dispersion measures. Additional information on the distribution of gas in the Milky Way halo could be obtained from the signature of a dipole in the dispersion measure of fast radio bursts across the sky, but this will be a small effect for most cases.

Fermion tunneling, instability, and first law of Rindler modified Schwarzschild black hole as a thermodynamic system

This study investigated quantum tunneling of spin-half particles through the event horizon of a Rindler modified Schwarzschild black hole (RMSBH) in the presence of quintessence matter. We analyzed the thermodynamics of RMSBH in Kiselev coordinates. In particular, we explore the RMSBH’s entropy and thermal stability. We showed that RMSBH serves as an unstable system, causing fluctuations. The integral formulation of the first law of RMSBH in the absence of a cosmological constant was also represented. By using the first law, we have studied the Ruppeiner geometry for Rindler acceleration and pressure with fixed ensembles.

Thermodynamic consistency of cell-centered lagrangian schemes

It is well known that gas dynamics equations may develop singularities,i.e., shock waves, at a finite time. The related discontinuous solutions are thus sought under a weak form which corresponds to the integral form of the underlying conservation laws of mass, momentum and total energy. Weak solutions being not uniquely defined, the physically relevant solution is singled out by means of an extra admissibility criterion termed entropy condition. In other words, this means that the solutions of gas dynamics equations have to be consistent with the second law of thermodynamics: for smooth flows entropy has to be conserved, whereas for non smooth flows, such as shock waves, it has to be dissipated ensuring the conversion of kinetic energy into internal energy.Bearing this in mind, the purpose of the present work is to address the thermodynamic consistency of cell-centered Finite Volume discretizations dedicated to the numerical simulation of Lagrangian hydrodynamics. Firstly, we describe explicitly a general procedure to construct affordable entropy conservative numerical fluxes, extending Tadmor’s work to the Lagrangian framework. Secondly, the entropy stability of these fluxes enhanced by proper dissipation operators is investigated. Then, a multi-dimensional extension of this work is explored. Finally, these theoretical studies are assessed by various numerical experiments.