3D Compressible Euler Equations Research Articles

A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling

AbstractNonhydrostatic atmospheric models often use semi‐implicit temporal discretisations in order to negate the time‐step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the nonlinear solver is typically truncated to a fixed number of iterations, often using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third‐order, four stage Rosenbrock–Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank–Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock–Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly‐stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow‐water equations and the 3D compressible Euler equations at both planetary and nonhydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank–Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock–Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at nonhydrostatic scales.

Entropy and energy conservation for thermal atmospheric dynamics using mixed compatible finite elements

Atmospheric systems incorporating thermal dynamics must be stable with respect to both energy and entropy. While energy conservation can be enforced via the preservation of the skew-symmetric structure of the Hamiltonian form of the equations of motion, entropy conservation is typically derived as an additional invariant of the Hamiltonian system, and satisfied via the exact preservation of the chain rule. This is particularly challenging since the function spaces used to represent the thermodynamic variables in compatible finite element discretisations are typically discontinuous at element boundaries. In the present work we negate this problem by constructing our equations of motion via a novel formulation which allow for the necessary cancellations required to simultaneously conserve entropy and energy without the chain rule. We show that such formulations allow for stable simulation of turbulent dynamics for both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.

Long time existence of smooth solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

Long time existence of smooth solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping

<abstract><p>We are concerned with the space-time decay rate of high-order spatial derivatives of solutions for 3D compressible Euler equations with damping. For any integer $ \ell\geq3 $, Kim (2022) showed the space-time decay rate of the $ k(0\leq k\leq \ell-2) $th-order spatial derivative of the solution. By making full use of the structure of the system, and employing different weighted energy methods for $ 0\leq k \leq \ell-2, k = \ell-1, k = \ell $, it is shown that the space-time decay rate of the $ (\ell-1) $th-order and $ \ell $th-order spatial derivative of the strong solution in weighted Lebesgue space $ L_\sigma^2 $ are $ t^{-\frac{3}{4}-\frac{\ell-1}{2}+\frac{\sigma}{2}} $ and $ t^{-\frac{3}{4}-\frac{\ell}{2}+\frac{\sigma}{2}} $ respectively, which are totally new as compared to that of Kim (2022) <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

Implicit summation by parts operators for finite difference approximations of first and second derivatives

Implicit finite difference approximations are derived for both the first and second derivatives. The boundary closures are based on the banded-norm summation-by-parts framework and the boundary conditions are imposed using a weak (penalty) enforcement. Up to 8th order global convergence is achieved. The finite difference approximations lead to implicit ODE systems. Spectral resolution characteristics are achieved by proper tuning of the internal difference stencils. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D and the 2D compressible Euler equations.

Space-time decay rate for 3D compressible Euler equations with damping

Space-time decay rate for 3D compressible Euler equations with damping

Local well-posedness for the motion of a compressible gravity water wave with vorticity

In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in Coutand-Shkoller [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delicate elliptic estimates, nor the higher regularity on the flow-map or initial vorticity. Instead, we adapt the Alinhac's good unknowns to the estimates of full spatial derivatives.

Absence of singularities in solutions for the compressible Euler equations with source terms in

The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$ , $d\geq 1$ , which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$ -norm of $\nabla u$ is exponential decay.

Vanishing dissipation limit to the planar rarefaction wave for the three-dimensional compressible Navier-Stokes-Fourier equations

We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. First, we prove that the 3D compressible Navier-Stokes-Fourier equations admit a family of smooth solutions that converge to the planar rarefaction wave solution of the 3D compressible Euler equations with arbitrary strength. Second, we obtain a uniform convergence rate in terms of the viscosity and heat-conductivity coefficients. Due to the 3D setting the approach for the two-dimensional case could not be applied directly. Instead, the analysis of the 3D case is carried out in the original non-scaled variables, and consequently the dissipation terms are more singular. Novel ideas and techniques are developed to establish the uniform estimates. More accurate a priori assumptions with respect to the dissipation coefficients are crucially needed for the stability analysis, and some new observations on the cancellations of the physical structures for the flux terms are essentially used to justify the 3D limit. Moreover, we find that the decay rate with respect to the dissipation coefficients is determined by the nonlinear flux terms in the original variables for the 3D limit.

Rough sound waves in 3D compressible Euler flow with vorticity

We prove a series of intimately related results tied to the regularity and geometry of solutions to the 3D compressible Euler equations. The results concern “general” solutions, which can have nontrivial vorticity and entropy. Our geo-analytic framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric “(sound) wave-part” coupled to a “transport-div-curl-part” (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the \(H^{2^+}(\mathbb {R}^3)\)-norm of the wave-part of the initial data and various Sobolev and Hölder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: Lindblad (Math Res Lett 5(5):605–622, 1998) showed that shock singularities can instantly form if one only assumes a bound for the \(H^2(\mathbb {R}^3)\)-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy affect the Ricci curvature of the acoustical metric and therefore, via Raychaudhuri’s equation, influence the evolution of the geometry of acoustic null hypersurfaces, i.e., sound cones; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and (iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability.

Local existence with low regularity for the 2D compressible Euler equations

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

Spectral steady-state solutions to the 2D compressible Euler equations for cross-mountain flows

Spectral steady-state solutions to the 2D compressible Euler equations for cross-mountain flows

The DGDD method for reduced-order modeling of conservation laws

The discontinuous Galerkin domain decomposition (DGDD) method couples subdomains of high-fidelity polynomial approximation to regions of low-dimensional resolution for the numerical solution of systems of conservation laws. In the low-fidelity regions, the solution is approximated by empirical modes constructed by Proper Orthogonal Decomposition and a reduced-order model is used to predict the solution. The high-dimensional model instead solves the system of conservation laws only in regions where the solution is not amenable to a low-dimensional representation. The coupling between the high-dimensional and the reduced-order models is then performed in a straightforward manner through numerical fluxes at discrete cell boundaries. We show results from application of the proposed method to parametric problems governed by the quasi-1D and 2D compressible Euler equations. In particular, we investigate the prediction of unsteady flows in a converging-diverging nozzle and over a NACA0012 airfoil in presence of shocks. The results demonstrate the stability and the accuracy of the proposed method and the significant reduction of the computational cost with respect to the high-dimensional model.

The 2D compressible Euler equations in bounded impermeable domains with corners

The 2D compressible Euler equations in bounded impermeable domains with corners

Global existence and convergence rates of solutions for the compressible Euler equations with damping

The Cauchy problem for the 3D compressible Euler equations with damping is considered. Existence of global-in-time smooth solutions is established under the condition that the initial data is small perturbations of some given constant state in the framework of Sobolev space $ H^3(\mathbb{R}^{3}) $ only, but we don't need the bound of $ L^1 $ norm. Moreover, the optimal $ L^{2} $-$ L^{2} $ convergence rates are also obtained for the solution. Our proof is based on the benefit of the low frequency and high frequency decomposition, here, we just need spectral analysis of the low frequency part of the Green function to the linearized system, so that we succeed to avoid some complicate analysis.

On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases

For 2D compressible full Euler equations of Chaplygin gases, when the initial axisymmetric perturbation of a rest state is small, we prove that the smooth solution exists globally. Compared with the previous references, there are two different key points in this paper: both the vorticity and the variable entropy are simultaneously considered, moreover, the usual assumption on the compact support of initial perturbation is removed. Due to the appearances of the variable entropy and vorticity, the related perturbation of solution will have no decay in time, which leads to an essential difficulty in establishing the global energy estimate. Thanks to introducing a nonlinear ODE which arises from the vorticity and entropy, and considering the difference between the solutions of the resulting ODE and the full Euler equations, we can distinguish the fast decay part and non-decay part of solution to Euler equations. Based on this, by introducing some suitable weighted energies together with a class of weighted \begin{document}$ L^\infty $\end{document} - \begin{document}$ L^\infty $\end{document} estimates for the solutions of 2D wave equations, we can eventually obtain the global energy estimates and further complete the proof on the global existence of smooth solution to 2D full Euler equations.

On the breakdown of 2D compressible Eulerian flows in bounded impermeable regions with corners

We consider smooth solutions of the 2D compressible Euler equations with suitable external forces in impermeable domains with corners. If the corner angles are small enough, we obtain results which have the following corollary: under a mild condition on the equation of state and for suitable external forces, the solutions can be continued in time (with no loss of smoothness) as long as there is no accumulation of vorticity or velocity divergence or pressure gradient or entropy gradient. For some special (e.g. barotropic) flows, this formulation can be simplified. Our results in the small corner angles case correspond to similar results obtained by Chemin in all space (of any dimension), which contain in particular a version for compressible flows of the Beale-Kato-Majda and Ponce incompressible flows criteria. For larger corner angles, we give examples where our assumptions are satisfied but continuation in time is only possible with a loss of smoothness.

A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere

A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a non-hydrostatic gravity wave as driven by a temperature perturbation.

A New Formulation of the 3D Compressible Euler Equations with Dynamic Entropy: Remarkable Null Structures and Regularity Properties

We derive a new formulation of the 3D compressible Euler equations exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work here is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity (defined to be vorticity divided by density), transport equations for the entropy and its gradient, and some additional transport–divergence–curl-type equations involving special combinations of the derivatives of the solution variables. The good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the “wave part” of the system. In a forthcoming application, we will use the new formulation to give a sharp, constructive proof of finite-time shock formation, tied to the intersection of acoustic “wave characteristics,” for solutions with nontrivial vorticity and entropy at the singularity. In the present article, we derive the new formulation and provide an overview of the central role that it plays in the proof of shock formation. Although the equations are significantly more complicated than they are in the barotropic case, they enjoy many of same remarkable features, including: (i) all derivative-quadratic inhomogeneous terms are null forms relative to the acoustical metric, which is the Lorentzian metric driving the propagation of sound waves and (ii) the transport–divergence–curl-type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity, assuming that the initial data enjoy the same gain in regularity. This represents a gain of one derivative compared to standard estimates. This gain of a derivative, which seems to be new for the entropy, is essential for closing the energy estimates in our forthcoming proof of shock formation, since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations.

Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

For 2D compressible isentropic Euler equations of polytropic gases, S. Alinhac establishes that the axisymmetric smooth solution blows up. In the present paper, for 2D compressible isentropic Euler equations of Chaplygin gases, we show that the small perturbed smooth solution exists globally when the rotationally invariant initial data are a perturbation of size ε>0 of a rest state. Near the light cone, 2D Euler equations of Chaplygin gases can be transformed into a second order quasilinear wave equation, which satisfies both the null conditions. This will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone. However, away from the light cone, the hydrodynamical waves have no decay in time and strongly affect the related acoustical waves. By introducing a nonlinear ODE, we can distinguish the fast decay part and non-decay part so that the global energy estimates are derived.