Nonlinear Euler Equations Research Articles

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped $p$-system

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta–Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows us to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform, which we avoid here, developing the “physical space version” of the hyperbolic hypocoercivity approach introduced in Beauchard and Zuazua [Arch. Ration. Mech. Anal. 199 (2011), 177–227], to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows us to recover the natural heat-like time decay of solutions under sharp rank conditions, without employing Fourier analysis or L^{1} assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped p -system.

Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics

The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.

A High-Order Immersed Moving Boundary Method Using Ghost Points and Characteristics for Acoustics

Realistic acoustic problems often involve interactions between arbitrary-shaped moving objects at the interface with a fluid domain. The present work aims at developing numerical methods to take into account moving boundaries for acoustic problems, with a high order of convergence. The reference problem considered in this work is the solution of the non-linear Euler equations by a finite-difference time-domain method, describing the formation of one-dimensional waves created by a moving piston. The most suitable class of methods to solve this type of problem in the case of complex moving boundaries is the ghost-point method for sharp interface with 4th order reconstruction. 1 In order to accurately formulate the immersed boundaries with ghost points, the boundary conditions are formulated in terms of characteristic waves of the non-linear equations. The proposed approach is also extended to represent moving boundaries with a prescribed acoustic impedance. This is achieved by coupling a time-domain acoustic impedance model to the immersed boundary technique. The method is also applied to a two-dimensional example, where an acoustic pulse is reflected by a moving wall.

Data-Driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. In this work, we propose a technique for highly accelerated prediction of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA). Specifically, we create and compare ERA reduced-order models (ROMs) based on the training data generated by solving the linearized and nonlinear Euler equations, and we use them to predict the aeroacoustic response of an airfoil in a purely predictive setting subject to different types of gust loading. Activating each input channel separately in the full-order model (FOM) to generate the Markov sequence for training makes it computationally challenging to use the ERA in aeroacoustics applications. We address this bottleneck by first proposing a multifidelity gappy POD method to reduce the computation cost on the FOM and ROM levels by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of the ERA for accurate online acoustic response prediction, and they reduce the offline computation cost of ROMs.

Utilizing time-reversibility for shock capturing in nonlinear hyperbolic conservation laws

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.

ДЕЯКІ ЗАДАЧІ ПРО РУХ ТВЕРДОГО ТІЛА У СЕРЕДОВИЩІ З ОПОРОМ

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

A comparison of smoothed particle hydrodynamics simulation with exact results from a nonlinear water wave model

The aim of this paper is to verify the velocity profile and the pressure variation inside the fluid domain over one wavelength obtained from a numerically simulated Smoothed Particle Hydrodynamics model with some exact qualitative results (i.e., increasing/decreasing trend or constant value of a flow field) from a fully nonlinear Euler equation for water wave model. A numerical wave flume has been modeled and a regular wave train is created by the horizontal displacement of a wave paddle on one side of the flume. A passive beach is used to dissipate the energy of the wave on the other side. The extracted numerical results are compared with some recently available exact results from a nonlinear steady water wave model based on the Euler equations for irrotational flow. The flow properties under wave crests, wave troughs, and along the distance from the wave crest to the wave trough over one wavelength are investigated. The horizontal and vertical velocity components and the pressure in the fluid domain agree well with the analytical results.

Efficient Flutter Prediction Using Reduced-Order Modeling

Typical aerodynamic shape optimization and multidisciplinary optimization algorithms omit high-fidelity flutter predictions due to the associated computational costs. This paper presents a model order reduction framework as a step toward flutter-constrained aircraft optimization. The Euler equations linearized about a steady-state solution of the nonlinear Euler equations are used as the governing unsteady flow equations. Using a proper orthogonal decomposition approach, a reduced basis is constructed onto which the governing equations are projected. The result is a linear reduced-order model (ROM) with significantly fewer degrees of freedom capable of rapidly approximating aerodynamic forces. This ROM is coupled to a linear structural model to create a single monolithic aeroelastic system. The eigenvalues of the resulting system are analyzed for various flow conditions to determine the onset of flutter in the system. To ensure the stability of the ROM, the use of a stabilizing inner product is demonstrated. The flutter boundaries obtained for both a two-degree-of-freedom airfoil structure and the AGARD 445.6 wing model show good agreement with the full-order model and with the literature.

Kinetic Modeling of Multiple Scattering of Acoustic Waves in Randomly Heterogeneous Flows

We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak random fluctuations of the mean particle velocity and speed of sound. We more particularly address the regime where the acoustic wavelengths are comparable to the correlation lengths of the weak inhomogeneities--the so-called weak coupling limit. The analysis is carried on starting from the linearized Euler equations and the convected wave equation with variable density and speed of sound, which can be derived from the nonlinear Euler equations. We use a multi-scale expansion of the Wigner distribution of a velocity potential associated to the waves to derive a radiative transfer equation describing the evolution of the angularly resolved wave action in space/time phase space. The latter experiences convection, refraction and scattering when it propagates through the heterogeneous ambient flow, although the overall wave action is conserved. The convection and refraction phenomena are accounted for by the convective part of the transport equation and depend on the smooth variations of the ambient quantities. The scattering phenomenon is accounted for by the collisional part of the transport equation and depends on the cross-power spectral densities of the fluctuations of the ambient quantities at the wavelength scales. The refraction, phase shift, spectral broadening, and multiple scattering effects of the high-frequency regimes described in various previous publications are thus encompassed by the proposed model. The overall derivation is based on the interpretation of spatial-temporal Wigner transforms in terms of semiclassical operators in their standard quantization.

Numerical investigation of wave steepening and shock coalescence near a cold Mach 3 jet.

Wave steepening and shock coalescence due to nonlinear propagation effects are investigated for a cold Mach 3 jet. The jet flow and near pressure fields are computed using large-eddy simulation. The near acoustic field is propagated to the far field by solving the linearized or the weakly nonlinear Euler equations. Near the angle of peak levels, the skewness factors of the pressure fluctuations for linear and nonlinear propagations display positive values that are almost identical. Thus, the positive asymmetry of the fluctuations originates during the wave generation process and is not due to nonlinear propagation effects. Compressions in the signals are much steeper for a nonlinear than for a linear propagation, highlighting the crucial role of nonlinear distortions in the formation of steepened waves. The power transfers due to nonlinear propagation are examined for specific frequencies by considering the spatial distribution of the Morfey-Howell indicator in the near and far acoustic fields. They are in good agreement with the direct measurements performed by comparing the spectra for nonlinear and linear propagations. This shows the suitability of the Morfey-Howell indicator to characterize nonlinear distortions for supersonic jets.

A new variation for the relativistic Euler equations

The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

Solving the problem of moving a solid body around a fixed point in a resisting medium

In this paper, we consider the problem of motion of an axisymmetric solid around a fixed point in a resistive medium, described by nonlinear dynamic Euler equations. An analytical solution to the problem when the moments of external resistance forces are proportional to the corresponding projections of the angular velocity of the body is obtained by partial discretization of nonlinear differential equations based on the theory of generalized functions. Curves of angular velocity change are constructed and graphs of comparison of these curves with previously known exact solutions are given.

Decoupling of hyperbolic conservation laws using an improved eigenvector matrix

The resolution of Harten’s non-oscillatory explicit second-order accurate total variation diminishing scheme was relatively high for computations of nonlinear wave equations. However, this scheme shows false dissipation for sharp discontinuities, such as shocks and contacts, when applied to 1D Euler equation. The selection of the entropy fixing parameter, flux limiter, and appropriate eigenvector matrix is crucial to control the amount of such dissipation. This study focuses on the formulation of eigenvector matrix, which is used for the decoupling of a highly coupled nonlinear 1D Euler equation into a nonlinear wave equation. The selection of the eigenvector matrix is crucial to limit the unwanted false dissipation, which is inherent to the numerical schemes derived for nonlinear wave equations. A generalized form of the scaling factor for the eigenvector matrix is then derived, which is satisfied by the scaling factors used by Harten, Hofmann, and Yee. LAX, SOD, and inverse shock test cases of the shock tube problem are solved to examine and analyze the corresponding physical aspects. The findings show that multiplying the scaling factor by itself or by a constant does not affect the results. However, the variables inside the scaling factor, as well as powers of these variables play a vital role in controlling false dissipation. These scaling factors can effectively capture physics in one case, but might not exhibit satisfactory performance for other problems. In addition, Yee’s scaling factor can effectively capture shock and contact discontinuities, especially for the inverse shock test case.

Image restoration from noisy incomplete frequency data by alternative iteration scheme

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

A new very high-order upwind directional multi-layer compact (DMLC) scheme for multi-dimensional flows

In the authors’ previous paper (Bai and Zhong, J. Comput. Phys. 2019), a very high-order upwind multi-layer compact (MLC) scheme designed to solve smooth multi-scale flow problems with complex physics is presented. In the MLC scheme, the auxiliary equations for the first derivatives are introduced. Accordingly, the first derivatives are evolved simultaneously with the function values and used as additional degrees of freedom in the finite difference approximation. The upwind MLC scheme is subsequently derived on a centered stencil, with an adjustable parameter to introduce small dissipation for stability. It was shown in [1] that the scheme achieved spectral-like resolution within a compact stencil, and it produced very high-order accuracy in flow simulations. However, the MLC scheme shows weak numerical instabilities for a small range of wavenumbers when it is applied to multi-dimensional flows [2], which are mainly triggered by the inconsistency between its one- and two-dimensional formulations. The instability may lead to divergence in long-time multi-dimensional simulations. Moreover, the cross-derivative approximation in the MLC scheme requires an ad-hoc selection of supporting grid points. Also, the cross-derivative approximation is relatively expensive for very high-order cases. In this paper, a new directional multi-layer compact (DMLC) scheme is presented in order to address the remaining problems of the MLC scheme and obtain better performance for multi-dimensional flow simulations. The main idea of the new DMLC scheme is to introduce the auxiliary equation for the cross derivative in multi-dimensional cases, consequently, the spatial discretization can be fulfilled along each dimension independently. With this directional discretization technique, the one-dimensional formulation can be applied to all spatial derivatives in a multi-dimensional governing equation. Therefore, the DMLC scheme overcomes inconsistency between one- and two-dimensional formulations of the original MLC scheme; and it also avoids the ad-hoc cross-derivative approximations. The two-dimensional Fourier analysis demonstrates that all modes of the DMLC scheme are stable for wavenumbers in [0, 2π], and it has better spectral resolution and smaller anisotropic error than the MLC scheme. The stability analysis through matrix method indicates that stable boundary closure schemes are much easier to be obtained in the DMLC scheme. The numerical tests in the linear advection equation and the nonlinear Euler equations validate that the DMLC scheme are more accurate and require less CPU time than the MLC scheme on the same mesh. In particular, the long-time simulation results reveal that the DMLC scheme is always stable for both periodic and non-periodic boundary conditions; while the MLC scheme may diverge in some cases. Overall, the new DMLC scheme shows comprehensive improvements from the original MLC scheme with uniform stability, higher accuracy and spectral resolution, and better computational efficiency for multi-dimensional flows. The improvement is even more significant in very high-order cases which makes the DMLC scheme appropriate for solving complex flow problems.

Numerical simulation of upwinding schemes applied to complex fluid dynamics equations

In this work, the nonlinear Euler equations of gas dynamics are employed for a comparative study of the frequently used flux limiters, namely: the consecrated ARORA-ROE, van Albada, CUBISTA and ADBQUICKEST; and two rational functions TVD-CBC-based TOPUS and SDPUS-C1 schemes. The comparison is mainly done in terms of their abilities in resolving discontinuities, reproducing complex flow phenomena and accuracy. It is concluded that both TOPUS and SDPUS-C1 can be considered as alternative tools for computer simulations of fluid flow. As application, the corresponding SDPUS-C1 scheme is employed to simulate 2D/3D incompressible flows with moving free surfaces.

On the Forced Euler and Navier–Stokes Equations: Linear Damping and Modified Scattering

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier–Stokes equations close to Couette flow on $$\mathbb {T}\times I$$ T×I. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $$H^{s}, s>-1$$ Hs,s>-1. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier–Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of Deng and Masmoudi (Long time instability of the Couette flow in low Gevrey spaces, 2018. arXiv:1803.01246).

Adjoint-based optimization of sound reinforcement including non-uniform flow.

The determination of optimal geometric arrangements and electronic drives of loudspeaker arrays in sound reinforcement applications is an ill-posed inverse problem. This paper introduces an innovative method to determine complex driving functions, also considering complex environmental conditions. As an alternative to common frequency domain methods, the authors present an adjoint-based approach in the time domain: Acoustic sources are optimized in order to generate a given target sound field. Instead of the Helmholtz equation, the full non-linear Euler equations are considered. This enables an easier treatment of non-uniform flow and boundary conditions. As proof of concept, a circular and a linear monopole array are examined. For the latter, the environmental conditions include wind and thermal stratification. For all examples, the method is able to provide appropriate driving functions.

Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Olsson and Oliger (Energy and maximum norm estimates for nonlinear conservation laws, 1994), Gerritsen and Olsson (J Comput Phys 129:245–262, 1996) and Yee et al. (J Comput Phys 162:33–81, 2000) is revisited. The objective of this paper is to prove for the first time that the entropy split methods based on physical entropies are entropy stable methods for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. Standard high order spatial central differencing as well as high order central spatial dispersion relation preserving spatial differencing is part of the entropy stable methodology framework. The proof is to replace the spatial derivatives by SBP difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving methods (Acta Numer 12:451–512, 2003) using mathematical entropies and more recently the methods in Winters and Gassner (J Comput Phys 304:72–108, 2016), do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed (Roanocha in Generalized summation-by-parts operators and variable coefficients, 2018. arXiv:1705.10541v2 [math.NA]). Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy and Grubber splitting.