Euler Equations Of Gas Dynamics Research Articles

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

In this work, we enhance the fifth-order Weighted Essentially Non-Oscillatory (WENO) shock-capturing scheme by integrating deep learning techniques. We improve the established WENO algorithm by training a compact neural network to dynamically adjust the smoothness indicators within the WENO scheme. This modification boosts the accuracy of the numerical results, particularly in proximity to abrupt shocks. Notably, our approach eliminates the need for additional post-processing steps, distinguishing it from previous deep learning-based methods. We substantiate the superiority of our new approach through the examination of multiple examples from the literature concerning the two-dimensional Euler equations of gas dynamics. Through a thorough investigation of these test problems, encompassing various shocks and rarefaction waves, our novel technique consistently outperforms the traditional fifth-order WENO scheme. This superiority is especially evident in cases where numerical solutions exhibit excessive diffusion or overshoot around shocks.

Interactions between elementary waves and weak discontinuity in two-layer blood flow through artery

Blood flow through arterial circulation can be characterized by fluid transport in flexible tubes and modeled mathematically using the conservation of mass and momentum. A one-dimensional model for two-layer blood flow with different blood velocities and the same constant density in each layer derived from the Euler equations of gas dynamics by taking the vertical average across each layer. This work presents interactions of elementary waves with a weak discontinuity for the quasilinear 3 × 3 system of conservation laws governing the two-layer blood flow in arteries. Exploiting elementary waves as a single-parameter curve, we study the Riemann solution uniquely and consequently establish the condition on initial data for the existence of a solution to the Riemann problem. Furthermore, we discuss the evolution of weak discontinuity waves and subsequently derive their amplitudes; in what follows, we investigate the interactions of weak discontinuity with contact discontinuity and shocks. Finally, a series of numerical tests have been performed to understand the impact of shock strength and the initial data on the amplitudes of reflected and transmitted waves and the jumps in shock acceleration.

ON THE INTEGRAL CONVERGENCE OF NUMERICAL SCHEMES CALCULATING GAS-DYNAMIC SHOCK WAVES

A comparative experimental accuracy study of shock-capturing schemes such as RBM(Rusanov-Burstein-Mirin), CWA(Compact high order Weak Approximation) and A-WENO(Alternative Weighted Essentially Non-Oscillatory) schemes is carried out by numerically solving a Cauchy problem with smooth periodic initial data for the Euler equations of gas dynamics. It is shown that in the presence of shock waves, RBM and CWA schemes(in the construction of which nonlinear flux correction is not used) have a higher order of integral convergence, which provides significantly higher accuracy to these schemes (compared to A-WENO scheme) in the areas of shock waves influence, despite noticeable non-physical oscillations at their fronts. This makes it possible to use RBM and CWA schemes as basic ones when constructing combined schemes that monotonically localize shock wave fronts and at the same time maintain higher order accuracy in their influence areas.

The Role of Pickup Ions in the Interaction of the Solar Wind with the Local Interstellar Medium. I. Importance of Kinetic Processes at the Heliospheric Termination Shock

The role of pickup ions (PUIs) in the solar wind interaction with the local interstellar medium is investigated with 3D, multifluid simulations. The flow of the mixture of all charged particles is described by the ideal MHD equations, with the source terms responsible for charge exchange between ions and neutral atoms. The thermodynamically distinct populations of neutrals are governed by individual sets of gas dynamics Euler equations. PUIs are treated as a separate, comoving fluid. Because the anisotropic behavior of PUIs at the heliospheric termination shocks is not described by the standard conservation laws (a.k.a. the Rankine–Hugoniot relations), we derived boundary conditions for them, which are obtained from the dedicated kinetic simulations of collisionless shocks. It is demonstrated that this approach to treating PUIs makes the computation results more consistent with observational data. In particular, the PUI pressure in the inner heliosheath (IHS) becomes higher by ∼40%–50% in the new model, as compared with the solutions where no special boundary conditions are applied. Hotter PUIs eventually lead to charge-exchange-driven cooling of the IHS plasma, which reduces the IHS width by ∼15% (∼8–10 au) in the upwind direction, and even more in the other directions. The density of secondary neutral atoms born in the IHS decreases by ∼30%, while their temperature increases by ∼60%. Simulation results are validated with New Horizons data at distances between 11 and 47 au.

The compressible Euler equations in a physical vacuum: A comprehensive Eulerian approach

This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problem is limited to Lagrangian coordinates, in high-regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard-style well-posedness theory for this problem in low-regularity Sobolev spaces. In particular, we give new proofs for existence, uniqueness, and continuous dependence on the data with sharp, scale-invariant energy estimates, and a continuation criterion.

On the Accuracy of Shock-Capturing Schemes Calculating Gas-Dynamic Shock Waves

A comparative experimental accuracy study of three shock-capturing schemes (the second-order CABARET, third-order Rusanov, and fifth-order in space third-order in time A-WENO schemes) is carried out by numerically solving a Cauchy problem with smooth periodic initial data for the Euler equations of gas dynamics. In the studied example, the solution breaks down and shock waves emerge. It is shown that the CABARET and A-WENO schemes, which are constructed using nonlinear limiters as a stabilization mechanism, have approximately the same accuracy in the areas of shock wave influence, while the nonmonotone Rusanov scheme has significantly higher accuracy in these areas despite producing noticeable nonphysical oscillations in the immediate vicinities of shock waves. At the same time, the combined scheme obtained based on the Rusanov and CABARET schemes localizes shock wave fronts, which are captured in a non-oscillatory manner, and preserves higher accuracy in the areas of the shock influence.

A fast dynamic smooth adaptive meshing scheme with applications to compressible flow

We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, det⁡∇ψ(x,t)=G∘ψ(x,t). The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map ψ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method [21] that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2D and 3D mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers [22,18], while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2D gas dynamics. Specifically, we implement the C-method [64,65] and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.

Symmetry group analysis of several coupled fractional partial differential equations

Under investigations were several coupled time–space fractional integrable equations which are called the (1+1)-dimensional KdV-type equations, the (2+1)-dimensional Burgers equations and the (3+1)-dimensional unsteady Euler equations of gas dynamics. In this article, these above considered time–space fractional models through the symmetry analysis approach were explored. As the most basic results, symmetries of these researched equations, were obtained. With the help of the multiple-parameters Erdélyi-Kober fractional operators, they can be reduced into lower dimensional systems. We noticed that this reduction process was somewhat different from the integer case. Besides, we also obtained the explicit power series solution and analyze the convergence to it. By considering the new fractional conservation theorem, conserved quantity of time and space of these time–space fractional systems, were also obtained.

Numerical approximation of living-man steady state solutions for blood flow in arteries using a well-balanced discontinuous Galerkin scheme

This study developed a well-balanced discontinuous Galerkin method for the one-dimensional blood flow model capable of preserving zero and non-zero velocity steady state solutions. The strategy used was to incorporate the source term into the flux function to rewrite the law of balance in a conservative form, in terms of global equilibrium variables that remained constant in space and time under steady states. Subsequently, the conservative variables were recalculated using the equilibrium variables. Recently, this concept was applied to shallow water and the Euler equations of gas dynamics. Numerical tests verified the well-balanced property of the designed scheme and its ability to capture small steady-state perturbations on relatively coarse meshes accurately.

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

To improve the shock-capturing capability of the third-order WENO scheme and enhance its computational efficiency, in this paper, we design a new WENO scheme independent of the local smoothing factor, WENO-SIF. The weight function of the WENO-SIF scheme is the segmentation function of the sub-stencil, which is guaranteed to achieve the desired accuracy at higher order critical points. WENO-SIF does not need to compute the smoothing factor during the computation, which effectively reduces the computational consumption. The present WENO-SIF is compared with WENO-JS and other WENO schemes for numerical experiments at one- and two-dimensional benchmark problems with a suitable choice of $$\lambda =0.13$$ . The results demonstrate that the WENO scheme can further improve the resolution of WENO-JS, achieve optimal accuracy at high-order critical points, and significantly reduce the computational consumption.

Numerical investigation of Mach number consistent Roe solvers for the Euler equations of gas dynamics

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.

A robust common-weights WENO scheme based on the flux vector splitting for Euler equations

This paper proposes a common-weights weighted essentially non-oscillatory (Co-WENO) scheme for solving the Euler equations of gas dynamics. Different from the usual component-wise weighting methods, common-weights means that, on one global stencil, a set of weights is commonly shared by the split flux vector of Euler equations in one spatial dimension. The common-weights WENO scheme has two significant advantages. First, since only one set of weights is calculated and used for the split flux vector, the method has an improved computational efficiency. Second, for a stencil (or each cell on the stencil), the Co-WENO scheme keeps the same contribution on each component numerical flux in a hyperbolic system of equations. How to calculate the weights is one of the vital issues in developing this kind of Co-WENO schemes. In this paper, based on the flux vector split method, the product of density, pressure, and the split flux of energy equation(Γ±=ρpfE±) is proposed to calculate the common weights. This is based on the following considerations: (1) the density jumps at shocks and contact discontinuities; (2) the split energy flux contains the term of the third power of the velocity (for example, u3) and makes the resulting scheme has upwind characteristic; (3) the pressure always jumps at shocks, and it can help improve the stability in high speed flows, in which the kinetic energy is much larger than the internal energy. Numerical experiments also show that the proposed common-weights WENO scheme has good robustness and low numerical dissipation, and it can help suppress phase errors.

ENO-ET: a reconstruction scheme based on extended ENO stencil and truncated highest-order term

High-order numerical methods for hyperbolic balance laws must circumvent Godunov’s theorem, for which non-linear spatial reconstruction is a successful procedure; this allows the design of non-linear semi-discrete and fully discrete high-order finite volume methods. The Essentially-Non-Oscillatory (ENO) method pioneered by Harten and collaborators in 1987 was a fundamental breakthrough in providing the bases for constructing schemes of both high-order of accuracy in smooth regions and essentially non-oscillatory at discontinuities. Several variations of ENO have since appeared in the literature, notably the Weighted Essentially-Non-Oscillatory (WENO) method of Jiang and Shu. In the present article, after revisiting the main elements of reconstruction methods, including their analysis, we propose a new, conservative ENO-type reconstruction procedure, called ENO-ET, that preserves the main advantages of the classical ENO and successfully overcomes its well-known weaknesses. The newly proposed reconstruction scheme is systematically assessed and compared with the best existing reconstruction schemes. We do so in two ways: (i) the methods are purely regarded as local, non-linear interpolation objects and (ii) the reconstruction methods are implemented in the context of high-order fully-discrete ADER numerical schemes for solving hyperbolic balance laws. The resulting schemes are of arbitrary order of accuracy in both space and time. In this paper the methods are implemented up to seventh order of accuracy and tested on carefully chosen numerical examples. Tests solved include the linear advection equation, the Euler equations of gas dynamics and the blood flow equations. Particular features of interest here are: (a) attaining theoretically expected convergence rates for smooth solutions arising from demanding initial conditions, (b) compute essentially non-oscillatory profiles for discontinuous solutions and (c) attain successfully steady-steady state solutions by time marching. As compared to existing methods, the newly proposed ENO-ET scheme performs very well for all these problems.

Local characteristic decomposition based central-upwind scheme

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics to illustrate the performance on a variety of examples. The obtained numerical results clearly demonstrate that the proposed new schemes outperform the original central-upwind schemes.

Improvements of the fifth-order WENO-JS-type scheme with normalized smoothing factor for gas dynamic Euler equations

In this paper, we developed a novel high-resolution fifth-order WENO-JS-type scheme that can achieve the optimal accuracy order at or near the arbitrary order critical points. To enhance the accuracy of the weight function of WENO-JS, we normalized the local smoothness factor to obtain a series of normalized variables. By introducing a mapping function that can increase the accuracy of the normalization variables, we developed a new normalized WENO-JS-type scheme called WENO-NSI. The WENO-NSI scheme can improve the shock capture ability by selecting the apposite parameter λ in the mapping function. Numerical experiments of one- and two-dimensional gas dynamic Euler equations for the present WENO-NSI scheme are compared with those of the WENO-JS/Z/M/D/IM and TENO-Z schemes. The results show that the present scheme can achieve theoretical accuracy at or near the arbitrary order critical points, and it has a better shock capturing ability than other schemes.

14-moment maximum-entropy modeling of collisionless ions for Hall thruster discharges

Ions in Hall effect thrusters are often characterized by a low collisionality. In the presence of acceleration fields and azimuthal electric field waves, this results in strong deviations from thermodynamic equilibrium, introducing kinetic effects. This work investigates the application of the 14-moment maximum-entropy model to this problem. This method consists in a set of 14 partial differential equations (PDEs) for the density, momentum, pressure tensor components, heat flux vector, and fourth-order moment associated with the particle velocity distribution function. The model is applied to the study of collisionless ion dynamics in a Hall thruster-like configuration, and its accuracy is assessed against different models, including the Vlasov kinetic equation. Three test cases are considered: a purely axial acceleration problem, the problem of ion-wave trapping, and finally the evolution of ions in the axial-azimuthal plane. Most of this work considers ions only, and the coupling with electrons is removed by prescribing reasonable values of the electric field. This allows us to obtain a direct comparison among different ion models. However, the possibility to run self-consistent plasma simulations is also briefly discussed, considering quasi-neutral or multi-fluid models. The maximum-entropy system appears to be a robust and accurate option for the considered test cases. The accuracy is improved over the simpler pressureless gas model (cold ions) and the Euler equations for gas dynamics, while the computational cost shows to remain much lower than direct kinetic simulations.

Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.

High Order Semi-implicit WENO Schemes for All-Mach Full Euler System of Gas Dynamics

In this paper, we propose high order semi-implicit schemes for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. The schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes. Besides, they can well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

Three-level order-adaptive weighted essentially non-oscillatory schemes

Classical fifth-order weighted essentially non-oscillatory (WENO) schemes are based on reconstructions from three consecutive second-order polynomials. They give a third-order accurate scheme when two consecutive polynomials are smooth and one is non-smooth. This situation means losing one order of accuracy if compared with a fourth-order scheme. In classical WENO schemes, weighting functions that determine the weights for the polynomials and that eventually define the reconstruction are directly related to a smoothness indicator. This property may lead to a non-zero residual of a very small value in the Taylor expansion. To handle this situation a weighting procedure that does not directly rely on the smoothness indicator is presented. Once the smooth polynomials are determined, the smoothness indicator has no role in the further steps, which are taken care of by a new switch function. Contrary to other approaches this switch function does not involve any conditional statements. A tuning parameter is also included so that the resulting order-adaptive property can be adjusted to specific requirements. The performance of the resulting schemes for cell-average and point-value reconstructions is studied. In addition, the effect of the choice of numerical flux or Riemann solver on these reconstruction strategies is analysed. Numerical tests for the Euler equations of gas dynamics in one and two space dimensions are presented. Improvements in resolution and the stability are found when compared with the conventional WENO techniques.

Enhanced fifth order WENO shock-capturing schemes with deep learning

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method. Moreover, the formal order of accuracy of the resulting scheme can be proven.We demonstrate our findings with the inviscid Burgers’ equation, the Buckley–Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks. Finally, the straight-forward extension of the method to two-dimensional problems is included and illustrated using the 2D Burgers’ equation.