Constant Initial Data Research Articles

A systematic analysis of three-dimensional Riemann problems for verification of compressible-flow solvers

Numerical simulation is a well-established way of analyzing compressible flows. Due to high computational demands of solvers for such flow problems, their verification is typically limited to two-dimensional (2D) cases. However, 2D simulations suppress fundamental three-dimensional (3D) aspects of the flow evolution in (compressible) turbulent flows or shock-bubble and shock-drop interactions. With increase of computational power, 3D simulations become more feasible for routine analyses. The verification of 3D simulation frameworks is often limited to transformations of lower dimensional test cases in the 3D space. There is a lack of strictly 3D reference test cases for gas dynamics. In this work, we present a set of genuine 3D Riemann problems in order to validate and verify numerical solvers for compressible flows. The problems are designed such that each octant’s constant initial data connects two neighboring states by an elementary wave only. The problem design is inspired by well-established 2D Riemann problems most prominently posted by Lax and Liu (1998). In contrast to the twenty published 2D cases, more than 300 distinct combinations can be found in 3D. We provide example solutions for the particularly interesting ones of these case combinations and show how the cases help to expose shortcomings of numerical solvers. We provide reference data from computations with an open-source compressible multiresolution flow solver. For the reference solutions, we employ the Harten-Lax-van Leer contact (HLLC) Riemann solver and a weighted essentially non-oscillatory (WENO) reconstruction stencil of fifth order. The reference solutions use an effective resolution of one billion cells. We additionally make the full compute pipeline of this work publicly available, so interested researchers may reproduce and extend the current work.

On kinematics of one‐dimensional radially symmetric shocks in non‐ideal reacting gases

In this manuscript, the advancement of one‐dimensional shock waves propagating through a non‐ideal reacting gas is examined. Considering the shock wave as a propagating discontinuity surface, a pair of evolution equations describing the strength of the shock and the first‐order discontinuity for variable initial data are obtained. The truncation approximations are used to obtain the exponent of shock velocity for the case of strong shock with constant initial data and compared with those exponents obtained from Guderley's scheme and Chisnell–Chester–Whitham (CCW) rule. The effects of non‐ideal and reaction parameters on the shock evolution in the case of arbitrary shock strength are examined and compared with those obtained from the CCW rule. Furthermore, the global behaviors of shock strength and induced discontinuity for different values of non‐ideal and reaction parameters are analyzed.

Emissions minimization on road networks via Generic Second Order Models

<abstract><p>In this paper we consider the problem of estimating emissions due to vehicular traffic on complex networks, and minimizing their effect by regulating traffic at junctions. For the traffic evolution, we consider a Generic Second Order Model, which encompasses the majority of two-equations (i.e., second-order) models available in the literature, and extend it to road networks with merge and diverge junctions. The dynamics on the whole network is determined by selecting a solution to the Riemann Problems at junctions, i.e., the Cauchy problems with constant initial data on each incident road. The latter are solved by assuming the maximization of the flow and assigning a traffic distribution coefficient for outgoing roads of diverges, and a priority rule for incoming roads of merges. A general emission model is considered and its parameters are tuned to the $ {\mathrm{NO_{x}}} $ emission rate. The minimization of emissions is then formulated in terms of the traffic distribution and priority parameters, taking into account travel times. A comparison is provided between roundabouts with optimized parameters and traffic lights, which correspond to time-varying traffic priorities. Our approach can be adapted to manage traffic in complex networks in order to reduce emissions while keeping travel time at acceptable levels.</p></abstract>

δ − shocks and vacuum states in the Riemann problem for isothermal van der Waals dusty gas under the flux approximation

In this paper, we analyze the Riemann problem for concentration and cavitation phenomena to the isothermal Euler equations for van der Waals gas with dust particles in the presence of flux approximation. By the Riemann problem, we mean that it is an initial value problem for the system of conservation laws supplemented by constant discontinuous initial data involving one breaking. The existence of δ−shock and vacuum state in the Riemann problem for the system under consideration is examined. Further, it is shown that as the flux approximation and pressure vanish, the Riemann solution containing two shock waves to the isothermal Euler equation for van der Waals gas with dust particles converges to δ−shock solution and the Riemann solution containing two rarefaction waves tends to the vacuum state solution of the transport equations. Numerical simulations presenting the formation of δ−shocks and vacuum states for different flux approximations are also shown.

Riemann problems for the 2-D pressureless gas dynamics with external forces

Abstract This paper is concerned with the pressureless gas dynamics with a space-dependent gravity or a time-dependent friction in two dimensions. The basic Riemann problems with two pieces of constant initial data are considered. By the characteristic analysis, the Riemann problems are constructively solved by two kinds of solutions: vacuum solution and delta-shock solution, which describe the phenomena of cavitation and concentration, respectively. Especially, the generalized Rankine–Hugoniot relations for delta-shock waves are clarified and applied to the Riemann problems.

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder stability with respect to perturbations are proved. Moreover, a particular example with semilinear equation, constant initial data, and standard quadratic tracking type objective functional is constructed that has at least two different locally optimal solutions. By the perturbation analysis, the existence of a problem with non-constant initial data is shown that also has at least two different locally optimal solutions.

Approximate solutions of the Riemann problem for a two-phase flow of immiscible liquids based on the Buckley–Leverett model

The article proposes an approximate method based on the "vanishing viscosity" method, which ensures the smoothness of the solution without taking into account the capillary pressure. We will consider the vanishing viscosity solution to the Riemann problem and to the boundary Riemann problem. It is not a weak solution, unless the system is conservative. One can prove that it is a viscosity solution actually meaning the extension of the semigroup of the vanishing viscosity solution to piecewise constant initial and boundary data. It is known that without taking into account the capillary pressure, the Buckley–Leverett model is the main one. Typically, from a computational point of view, approximate models are required for time slicing when creating computational algorithms. Analysis of the flow of a mixture of two immiscible liquids, the viscosity of which depends on pressure, leads to a further extension of the classical Buckley–Leverett model. Some two-phase flow models based on the expansion of Darcy’s law include the effect of capillary pressure. This is motivated by the fact that some fluids, e.g., crude oil, have a pressure-dependent viscosity and are noticeably sensitive to pressure fluctuations. Results confirm the insignificant influence of cross-coupling terms compared to the classical Darcy approach.

Application of the G.A. Geniev, N.S. Chausova method for stability of shallow shells investigation

Introduction. The work is devoted to the stability of shallow shells investigation, based on the G.A. Geniev, N.S. Chausov methodology. An analysis is given of the works of the authors working on the issues of determining the stress-strain state of this structure type. Current trends and shortcomings of the methods used are indicated.
 
 Materials and methods. The analysis of stability is based on the G.A. Geniev, N.S. Chausov fundamental work. It defines the first and second kind stability lost and substantiates the use of an equations system to describe the stress-strain state of a structure. The equations system for shallow shells with constant and variable thickness and shape of the middle surface is solved using the Bubnov – Galerkin method. The approximating functions of stresses and displacements make it possible to vary the type of structure support.
 
 Results. Implementation of the G.A. Geniev and N.S. Chausov methodology with V.Z. Vlasov approximating functions made it possible to investigation the influence of various parameters on the critical load. The geometric nonlinearity of the work of structures was taken into account. Specific values of the design parameters are given, which increase the second kind stability with constant initial data.
 
 Conclusions. The analysis of the shallow shells stability made it possible to reveal the regularities in the change in the value of the critical load when varying various geometric characteristics. The presented results can be used in the design of real structures. At the same time, it is possible to set the tasks of optimizing such structures with restrictions on the value of their volume (weight) or minimizing it due to the interconnected change in geometric characteristics while maintaining the bearing capacity.

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator H=−Δ+|x|2 in Rd and its fractional powers Hβ, β>0 in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform Vgf (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of Vgf, that is in terms of membership to modulation spaces Mp,q, 0<p,q≤∞. We prove the complete range of fixed-time estimates for the semigroup e−tHβ when acting on Mp,q, for every 0<p,q≤∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing.As an application, we establish global well-posedness for the nonlinear heat equation for Hβ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e−ct as the solution of the corresponding linear equation, where c=dβ is the bottom of the spectrum of Hβ. Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to M∞,1).

Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial-boundary-value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.

Spherically Symmetric Solutions of the Full Compressible Euler Equations in $$R^N$$RN

This paper focuses on the multidimensional spherically symmetric full compressible Euler equations and constructs rigorously a family of global self-similar bounded weak solutions for all positive time to its initial value problem with constant initial data. The main approach is to reduce the full compressible Euler equations to an autonomous system of ordinary differential equations under the spherically symmetric and self-similar assumptions. We establish the detailed structures of solutions as well as their existence by analyzing carefully the properties of the integral curves of the autonomous ODE system.

Spatial asymptotic of the stochastic heat equation with compactly supported initial data

We investigate the growth of the tallest peaks of random field solutions to the parabolic Anderson models over concentric balls as the radii approach infinity. The noise is white in time and correlated in space. The spatial correlation function is either bounded or non-negative satisfying Dalang’s condition. The initial data are Borel measures with compact supports, in particular, include Dirac masses. The results obtained are related to those of Conus et al. (Ann Probab 41(3B):2225–2260, 2013) and Chen (Ann Probab 44(2):1535–1598, 2016) where constant initial data are considered.

On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero.

Geometric rigidity of constant heat flow

Let $\Omega$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $\Omega$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $\Omega$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $\Omega$ has the constant flow property if and only if it is an {\it isoparametric tube}, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.

Global Well-Posedness of the Two-Dimensional Exterior Navier–Stokes Equations for Non-decaying Data

We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers.

Analytical solution to 1D nonlinear elastodynamics with general constitutive laws

The equations of 1D elastodynamics write as a 2×2 hyperbolic system of conservation laws. The solution to the Riemann problem (i.e. piecewise constant initial data) is addressed, both in the case of convex and nonconvex constitutive laws. In the convex case, the solution can include shock waves or rarefaction waves. In the nonconvex case, compound waves must also be considered. In both convex and nonconvex cases, a restriction on the initial velocity jump is required to ensure the existence of the solution. Admissibility regions are determined, predicting the occurrence of shocks, rarefactions and compound waves. Lastly, analytical solutions are completely detailed for various constitutive laws (hyperbola, tanh and polynomial), and reference test cases are proposed. A link to a Matlab script is also provided, allowing a direct implementation.

Wave phenomena of the Toda lattice with steplike initial data

We give a survey of the long-time asymptotics for the Toda lattice with steplike constant initial data using the nonlinear steepest descent analysis and its extension based on a suitably chosen g-function. Analytic formulas for the leading term of the asymptotic solutions of the Toda shock and rarefaction problems (including the case of overlapping background spectra) are given and complemented by numerical simulations. We provide an explicit formula for the modulated solution in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.

A simple HLLC-type Riemann solver for compressible non-equilibrium two-phase flows

A simple, robust and accurate HLLC-type Riemann solver for two-phase 7-equation type models is built. It involves 4 waves per phase, i.e. the three conventional right- and left-facing and contact waves, augmented by an extra “interfacial” wave. Inspired by the Discrete Equations Method (Abgrall and Saurel, 2003), this wave speed (uI) is assumed function only of the piecewise constant initial data. Therefore it is computed easily from these initial states. The same is done for the interfacial pressure PI. Interfacial variables uI and PI are thus local constants in the Riemann problem. Thanks to this property there is no difficulty to express the non-conservative system of partial differential equations in local conservative form. With the conventional HLLC wave speed estimates and the extra interfacial speed uI, the four-waves Riemann problem for each phase is solved following the same strategy as in Toro et al. (1994) for the Euler equations. As uI and PI are functions only of the Riemann problem initial data, the two-phase Riemann problem consists in two independent Riemann problems with 4 waves only. Moreover, it is shown that these solvers are entropy producing. The method is easy to code and very robust. Its accuracy is validated against exact solutions as well as experimental data.

On an inverse problem for scalar conservation laws

We study in what sense one can determine the flux functions k = k(x) and f = f(u), k piecewise constant, in the scalar hyperbolic conservation law ut + (k(x)f(u))x = 0 by observing the solution u(t, ·) of the Cauchy problem with suitable piecewise constant initial data u|t = 0 = uo.