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We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu \cite{Hmidi-Mateu} for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point-vortex limit. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem., 37 pages, 3 figures

In this work we present a numerical study of surface water waves over variable topographies for the linear Euler equations based on a conformal mapping and Fourier transform. We show that in the shallow-water limit the Jacobian of the conformal mapping brings all the topographic effects from the bottom to the free surface. Implementation of the numerical method is illustrated by a MATLAB program. The numerical results are validated by comparing them with exact solutions when the bottom topography is flat, and with theoretical results for an uneven topography.

This paper proposes a new accelerating technique for simulating low speed flows, termed as pSeudo High Speed method (SHS), which uses governing equations and numerical methods of compressible flows. SHS method has advantages of simple formula, easy manipulation, and only need to modify flux of Euler equations. It can directly employ the existing well-developed schemes of hyperbolic conservation laws. To verify the technique, several numerical experiments are performed, such as: flow past airfoils and flow past a cylinder. Analysis of SHS method and comparisons with some precondition methods are made numerically. All tests show that SHS method can greatly improve the efficiency of compressible method simulating low speed flow fields, which exhibits in accelerating the convergence rate and increasing the accuracy of the numerical results.

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Nonlinear steady rotational waves in a low-pressure region are investigated. The problem is formulated in a simplified canonical domain through the use of a conformal mapping, which flattens the free surface. Steady waves are computed numerically using a Newton’s method and classified into three types. Besides, our results indicate that there is a region in which steady waves do not exist. The thickness of this region is compared with the one predicted by the weakly nonlinear, weakly dispersive regime.

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α = 1 , γ = 2 ).

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature.

We consider a 2 × 2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x → ± ∞ , reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.

From the formal expansion of the solutions of Euler-Voigt equations in R + 2 with no-slip boundary conditions, the boundary layer equations of Euler-Voigt equations to Euler equations are obtained. In case of the analytic data, one obtains the local existence and uniqueness of the solutions for the boundary layer equations by abstract Cauchy-Kovalevskaya theorem.

A design method for three-dimensional shape transition scramjet nozzles is introduced. The method employs streamline tracing to capture the properties of an axisymmetric thrust nozzle in a complex ...

We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.

The 3-D point cloud segmentation has played an important role in spatial structure analysis. Nowadays, segmentation methods either use a primitive-based strategy to fit points in predefined geometric shapes or group points based on their attributes (e.g., spatial distance). However, the required segmentation results, e.g., primitive level or object level, depend on the application. Therefore, this letter develops a semiautomatic method to extract contours for the users' desired segmentation. First, we initialize a 3-D closed curve for the target. Second, we calculate the internal and external force based on the proposed vector flow to deform the curve. The deformation equation is solved based on the Euler equation and calculated iteratively. Finally, the curve is converged as object contours. After one removes contours, those disjoint points are grouped as the users' desired instances. Experiments are conducted on various point clouds to demonstrate the effectiveness in terms of accuracy and consistency. Our quantitative evaluation outperformed selected primitive- and object-based methods, which presents a new viewpoint to the point cloud processing.

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.

In this paper, we consider the inverse problem for the estimation of a point-wise approximation error occurring at the discretization of the system of partial differential equations. We analyse the...

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

We consider the ultra-relativistic Euler equations for an ideal gas, which are described in terms of the pressure p and the spatial part u ∈ R 3 of the dimensionless four-velocity. We also numerically investigate the ultra-relativistic Euler equations using the modified Rusanov scheme compared with classical Rusanov scheme. Our scheme consists of predictor and corrector stage. The predictor stage contains a parameter of control ( α i + 1 2 n ) is responsible for the numerical diffusion of this scheme. In order to control this parameter, we use a strategy depending on limiter theory and using Riemann invariants. The corrector stage recovers the balance conservation equation. The scheme can compute the numerical flux corresponding to the real state of solution without relying on Riemann problem solvers. The numerical results show the high resolution of the proposed finite volume scheme (modified Rusanov) and confirm its capability to provide accurate simulations for the ultra-relativistic Euler equations under regimes with strong shocks and rarefactions.

This chapter introduces the theory and applications of calculus of variation. We introduce the Euler equation, which is vital in solving the functional problem. We examine the solution methods in one-dimensional and in both higher dimensions and functions of multiple variables. This theory plays a vital part in the derivation of the variational-based data assimilation theory.

The hydrodynamical model of the collective behavior of animals consists of the Euler equation with additional non-local forcing terms representing the repulsive and attractive forces among individuals. This paper deals with the system endowed with an additional white-noise forcing and an artificial viscous term. We provide a proof of the existence of a dissipative martingale solution -- a cornerstone for a subsequent analysis of the system with stochastic forcing.

In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with $$k=3$$ and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Robust numerical methods for CFD applications, such as WENO schemes, quickly evolved in the past few decades. Together with the Inverse Lax–Wendroff (ILW) procedure, WENO ideas were also applied in the boundary treatment. Those methods are known for their high-resolution property, i.e., good representation of nonlinear phenomena, which is an important property in solving challenging engineering problems. In light of that, the objective of this work is to present a review of well-established high-resolution numerical methods to solve the Euler equations and adapt the Navier–Stokes viscous terms discretization and boundary treatment. To test the modifications, we employed the positivity-preserving Lax–Friedrichs splitting, multi-resolution WENO scheme, third-order strong stability preserving Runge–Kutta time discretization, and ILW boundary treatment. The first problems were simple flows with analytical solutions for accuracy tests. We also tested the accuracy with nontrivial phenomena in the vortex flow. Oblique shock and complicated flow structures were captured in the Rayleigh–Taylor instability and flow past a cylinder. We showed the discretization and boundary treatment can handle non-constant viscosity, are high-order, high-resolution, and behave similarly to the well-established numerical methods. Furthermore, the methods discussed here can preserve symmetry and no approximations regarding the boundary layer were made. Therefore, the discretization and boundary treatment can be considered when solving direct numerical simulations.

In this paper, the flow field characteristics of Oblique Detonation Waves (ODWs) induced by a finite wedge under argon dilution are studied by solving the Euler equations with a detailed chemical model of hydrogen and air. First, the effects of the expansion waves, argon concentration, geometric parameters, and Mach number on the ODW are discussed. The results show that the changes of these parameters may make the oblique detonation not be initiated. Then, the ODW initiation criterion of the finite wedge is summarized, as the characteristic length of the induction zone LC and the characteristic length of the oblique wedge LW meet the condition LC/LW

This paper is devoted to the robust optimization of nacelle liners (acoustic treatments). The full computational aeroacoustic model is based on the convected Helmholtz equation in presence of a non...

We investigate the empirical implications of the investment-based model of asset pricing for the Hansen-Jagannathan and Kozak-Nagel-Santosh discount factors in the linear span of equity returns. We find that the stochastic discount factors satisfying the Euler equation for equity returns cannot satisfy the Euler equation for investment returns because returns on corporate investment covary inversely with the sources of equity risk relative to returns on equity. As a result, the model fails to replicate the level of the risk premium. Our results suggest that joint restrictions on the optimality of investment and consumption pose stringent conditions for candidate production models.

We introduce a unified deep learning method that solves dynamic economic models by casting them into nonlinear regression equations. We derive such equations for three fundamental objects of economic dynamics – lifetime reward functions, Bellman equations and Euler equations. We estimate the decision functions on simulated data using a stochastic gradient descent method. We introduce an all-in-one integration operator that facilitates approximation of high-dimensional integrals. We use neural networks to perform model reduction and to handle multicollinearity. Our deep learning method is tractable in large-scale problems, e.g., Krusell and Smith (1998). We provide a TensorFlow code that accommodates a variety of applications.

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].

Detonation reflections over cylindrical convex-straight coupled surfaces were numerically analyzed using the density-based compressible–reactive solver DCRFoam developed on the framework of OpenFOAM-V7. The two-dimensional reactive Euler equations were adopted to calculate the reflection dynamics, considering a detailed chemical mechanism. Effects of the global wedge angle (ranging from 20° to 50°) and the radius of the former convex section (ranging from 25 mm to 125 mm) on the detonation reflections were discussed. It was founded that the original regular reflection could transform into the Mach reflection on the former cylindrical convex section for coupled surfaces with global wedge angles smaller than 50°. The peak pressures of the reflection points in the configurations of the regular reflection were much larger than those in the Mach reflection, and the increase in the wedge angle led to the increase of the average peak pressure of the reflection points in the Mach reflection on the following straight section. For a coupled surface with a global wedge angle of 50°, no Mach reflection configurations were established on the former cylindrical convex section. The existence of the convex section was proven to delay the establishment of the Mach reflection on the following straight section. For coupled surfaces with the same global wedge angles, the transition angles from the regular reflection to the Mach reflection increased with the increase in the radius.

A two-dimensional version of the conservative entropy-stable discontinuous Galerkin method is proposed for Euler equations in variables: density, momentum density, and pressure. For the equation describing the dynamics of the mean pressure in a finite element, an approximation is constructed that is conservative in the total energy. A special slope limiter ensures that the entropy inequality and a two-dimensional analog of the monotonicity conditions for the numerical solution are satisfied. The developed method is tested on some model gas-dynamic problems.

Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.

An alternative cell-vertex entropy-stable finite volume method for the system of Euler equations is presented. It is derived from the residual distribution method, using the signals from each triangular element as a foundation for controlling entropy. Each signal has an entropy-conserved and entropy-stable components. From a median-dual area perspective, these signals can be interpreted as a line integral of the fluxes along the control volume boundaries of the cell-vertex approach. The new method includes a first-order version which is positive together with a linearity-preserving second order approach. A second order limited version is also presented using entropy as the guiding principle for limiting. Results herein demonstrate that the new method is more accurate and robust relative to the current cell-vertex entropy-stable finite volume method.

The paper deals with the calculation of the internal singularities of the Schwarz function corresponding to the boundary of a planar vortex patch during its self-induced motion in an inviscid, isochoric fluid. The vortex boundary is approximated by a simple, time-dependent map onto the unit circle, whose coefficients are obtained by fitting to the boundary computed in a contour dynamics numerical simulation of the motion. At any given time, the branch points of the Schwarz function are calculated, and from them, the generally curved shape of the internal branch cut, together with the jump of the Schwarz function across it. The knowledge of the internal singularities enables the calculation of the Schwarz function at any point inside the vortex, so that it is possible to check the validity of the map during the motion by comparing left and right hand sides of the evolution equation of the Schwarz function. Our procedure yields explicit functional forms of the analytic continuations of the velocity and its conjugate on the vortex boundary. It also opens a new way to understand the relation between the time evolution of the shape of a vortex patch during its motion, and the corresponding changes in the singular set of its Schwarz function.

A variation model of the gradient anisotropic elasticity theory is constructed. Its distinctive feature is the fact that the density of potential energy, along with symmetric fourth- and sixth-rank stiffness tensors, also contains a nonsymmetric fifth-rank stiffness tensor. Accordingly, the stresses in it also depend on the curvatures and the couple stresses depend on distortions. The Euler equations are three fourth-order equilibrium equations. The spectrum of boundary-value problems is determined by six pairs of alternative boundary conditions at each nonsingular surface point. At each special point of the surface (belonging to surface edges), in the general case, additional conditions arise for continuity of the displacement vector and the meniscus force vector when crossing the surface through the edge. In order to reduce the number of the physical parameters requiring experimental determination, particular types of the fifth- and sixth-rank stiffness tensors are postulated. Along with the classical tensor of anisotropic moduli, it is proposed to introduce a first-rank stiffness tensor (a vector with a length dimension), with the help of which the fifth-rank tensor is constructed from the classical fourthrank tensor by means of tensor multiplication. The sixth-rank tensor is constructed as the tensor product of the classical fourth-rank tensor and two first-rank tensors.

Kinetic or Boltzmann schemes are interesting alternatives to the macroscopic numerical methods for solving the hyperbolic conservation laws of gas dynamics. They utilize the particle-based description instead of the wave propagation models. While the continuous particle velocity based upwind schemes were developed in the earlier decades, the discrete velocity Boltzmann schemes introduced in the last decade are found to be simpler and are easier to handle. In this work, we introduce a novel way of introducing discrete velocities which correspond to the physical wave speeds and formulate a discrete velocity Boltzmann scheme for solving Euler equations.

In this paper, we numerically study interactions of a short-pulsed plane acoustic wave with complex rigid objects by solving the linearized Euler equations. For the study, our numerical approach implements an immersed boundary method to satisfy the no-penetration condition on the surface of rigid objects, together with a fully-explicit staggered-grid finite-difference time-domain method having perfectly matched layers. First, we validate our approach for acoustic wave scattering by a circular cylinder, a well-known benchmark problem. Subsequently, we extend our simulations to two representative problems, namely interactions with a circular cylinder and a plano-concave lens which are of paramount interest in terms of acoustic force and acoustic focusing, respectively. Our simulations allow us to better understand the main mechanism of propagation, reflection, and scattering of acoustic waves as a result of interaction with rigid objects. In addition, the effects of pulse width on the interactions are closely investigated.

The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations. The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts (SBP) difference boundary closure of (Gerritsen and Olsson in J Comput Phys 129: 245–262, 1996; Olsson and Oliger in RIACS Tech Rep 94.01, 1994; Yee et al. in J Comp Phys 162: 33–81, 2000). Sjogreen and Yee (J Sci Comput https://doi.org/10.1007/s10915-019-01013-1) recently proved that the entropy split method is entropy conservative and stable. Standard high-order spatial central differencing as well as high order central spatial dispersion relation preserving (DRP) spatial differencing is part of the entropy stable split methodology framework. The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives. Due to the construction, this conservative numerical flux requires higher operations count and is less stable than the original semi-conservative split method. However, the Tadmor entropy conservative (EC) method (Tadmor in Acta Numerica 12: 451–512, 2003) of the same order requires more operations count than the new construction. Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative, a modified nonlinear filter approach of (Yee et al. in J Comput Phys 150: 199–238, 1999, J Comp Phys 162: 3381, 2000; Yee and Sjogreen in J Comput Phys 225: 910934, 2007, High Order Filter Methods for Wide Range of Compressible flow Speeds. Proceedings of the ICOSAHOM09, June 22–26, Trondheim, Norway, 2009) is proposed in conjunction with the entropy split method as the base method for problems containing shock waves. Long-time integration of 2D and 3D test cases is included to show the comparison of these new approaches.

This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N2, C2, and B2, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.

A new strategy for high-fidelity unsteady mesh adaptation dealing with fluid–structure interaction (FSI) problems is presented using a partitioned approach. The Euler equations are solved by an edge-based Finite-Volume solver whereas the linear elasticity equations are solved by the finite-element method using the Lagrange $${\mathbb {P}}_1$$ elements. The coupling between both codes is realized by imposing suitable boundary conditions on conforming meshes even at the fluid–structure interface. Small displacements of the structure are assumed and so the mesh is not deformed. The unsteady mesh adaptation process is based on a unique cavity operator which can handle non-manifold geometry, the fluid–structure interface in this work. The computation of a well-documented two-dimensional test case is finally carried out to perform validation of this new strategy as well as a three-dimensional test case to demonstrate our ability to treat complex three-dimensional test cases.

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $$\mathbb {R}^d$$ with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $$|x|^\beta $$ with $$\beta

This paper discusses four different approaches that can be followed to derive the equations of motion for a fixed and symmetrical spinning top. Starting from the usual Euler equations in the body-fixed system, after manipulation it is shown that identical equations are derived for the space-fixe system as well. All the three Cartesian components of the angular momentum vector are calculated for both the body- and the space-systems and they are formulated so that they can be used for further numerical analysis. In addition to the classical set, the Euler equations are also easily derived using a rotating system originated at the pivot but not spinning. Moreover, Lagrange equations are derived and the latter are proven to be equivalent with the Euler equations. The best way among these four methods for teaching students is probably the instructor’s preference. Moreover, using commercial software, an adequately accurate numerical solution is derived. Not only the position of the spinning top is calculated but also the support forces at the pivot are predicted

The compressible Euler equations in R N with Coriolis force is the fundamental mathematical model for the atmospheric dynamics. We use the vector and matrix technique to the N -dimensional Euler equations, to construct the following Cartesian vector form solutions u = b ( t ) + A ( t ) x . One advantage of this approach is that the Euler equations with Coriolis force can be solved both theoretically and algebraically, which can be accomplished by constructing appropriate matrices A ( t ) , and vector b ( t ) . We can obtain the new exact solutions using the required matrices A ( t ) and vector b ( t ) . As exact solutions for the compressible Euler equations are rare, the new exact solutions are valuable for the verification of the corresponding computational methods.

The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.

Since proposed, the genuinely multidimensional Riemann solvers have been attracting more attentions because it considers the waves normal and parallel to the cell interfaces at the same time. However, all these methods are built upon the hyperbolic property of the compressible Euler equations. Therefore, they can not be applied to solve the Euler equations in steady subsonic cases which are with the elliptic property. In this study, a novel multidimensional Riemann solver called MULE-AUSMPWM (Multidimensional E-AUSMPW Modified) is proposed. It is built upon the Zha–Bilgen splitting procedure and adopts the Balsara's multidimensional wave model as the ME-AUSMPW scheme. In addition, it improves the ME-AUSMPW scheme's accuracy at subsonic speeds by avoiding simulating the convective fluxes in fully upwind manners. Extensive numerical tests, such as the two dimensional Euler problem, the one dimensional moving contact discontinuity case, the two-dimensional double Mach reflection case, the two-dimensional Riemann problem, and the turbulent flow past a 2d NACA0012 airfoil case, are conducted. Results illustrate that the MULE-AUSMPWM scheme proposed is capable of simulating compressible complex flows and improving the existing multidimensional Riemann solvers' accuracy at subsonic speeds remarkably.

In this paper, we are concerned with the Riemann problem for the generalized pressureless Euler equations with a composite source term, which covers the important Eulerian droplet model associated ...

We consider convergence to the incompressible Navier–Stokes equations/Euler equations for compressible viscous flows on periodic thin domain Ω δ = T a 2 × δ T with general initial data. Compared with Caggio et al (2020 Nonlinearity 33 840–863), where similar problems are considered in the whole space, we have to deal with the persistence of acoustic waves. We prove that the weak solutions in the 3D domain converge weakly/strongly to the solutions of the 2D incompressible Navier–Stokes equations/Euler equations when the Mach number ɛ goes to 0 as well as the thickness δ/the viscosity goes to 0.

Evaluating a new survey of German consumers, we test whether individual consumption spending decisions are formed according to a Euler equation model. We find that consumers are more likely to increase current spending if they plan to increase spending in the future and if they expect higher inflation. In the subsample of financially literate households, we find an additional negative effect of nominal interest rate expectations. The effects of macroeconomic expectations become stronger if consumers observed news on monetary policy or financial markets. These news effects are particularly pronounced for consumers who save and those with low inflation forecast accuracy.

Within the incompressible three-dimensional Euler equations, we study the pancake-like high vorticity regions, which arise during the onset of developed hydrodynamic turbulence. We show that these regions have an internal fine structure consisting of three vortex layers. Such a layered structure, together with the power law of self-similar evolution of the pancake, prevents development of the Kelvin-Helmholtz instability., Comment: 4 pages, 2 figures

The Euler equation gives a set of an infinite number of relations between the values of prime numbers and their numbers. These relations are implemented by conditionally convergent series. An analytic function implementing these relations crosses a value of 1 an infinite number of times. The distance between the first and second points of this type turns out to be anomalously large.