An isogeometric discontinuous Galerkin method for Euler equations

An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non‐uniform rational B‐splines. This leads to the solution inherently shares the same function space as the non‐uniform rational B‐splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.     

Relaxation approximation of the Euler equations

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.     

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.     

An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion W. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of W. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of W. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations. AMS subject classification (2000)  60H15, 60H35, 65C30     

Regularity criteria for the Euler equations with nondecaying data

The local-in-time existence and uniqueness of strong solutions to the Euler equations in the whole space with nondecaying and certainly regular initial velocity are concerned. It is obtained that the spatial regularity of solutions coincides with that of initial velocity under the suitable setting of external forcing terms. Regularity criteria focusing into the vorticity are also discussed due to the similar arguments of Beale-Kato-Majda.     

An adaptive domain decomposition procedure for Boltzmann and Euler equations

In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure lead to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.     

Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinderC=×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points sinceC can be thought of as a stretched manifold with conical points att=– andt=. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.     

Isospectral theory of Euler equations

Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.     

Entropy solutions of the compressible Euler equations

We consider the three-dimensional Euler equations of gas dynamics on a bounded periodic domain and a bounded time interval. We prove that Lax–Friedrichs scheme can be used to produce a sequence of solutions with ever finer resolution for any appropriately bounded (but not necessarily small) initial data. Furthermore, with some technical assumptions, e.g. that the density remains strictly positive in the sequence of solutions at hand, a subsequence converges to an entropy solution. We provide numerical evidence for these results by computing a sensitive Kelvin–Helmholtz problem.     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

Blowup for the Euler and Euler-Poisson equations with repulsive forces

In this paper, we study the blowup of the N-dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions (ρ,V), with compact support in [0,R], where R>0 is a positive constant and in the sense which ρ(t,r)=0 and V(t,r)=0 for r≥R, under the initial condition

(1)     

Enthalpy damping for the steady Euler equations

For inviscid steady flow problems where the enthalpy is constant at steady state, it has been proposed by Jameson, Schmidt, and Turkel to use the difference between the local enthalpy and the steady state enthalpy as a driving term to accelerate convergence of iterative schemes. This idea is analyzed here, both on the level of the partial differential equation and on the level of a particular finite difference scheme. It is shown that for the two-dimensional unsteady Euler equations, a hyperbolic system with eigenvalues on the imaginary axis, there is no enthalpy damping strategy which can move all the eigenvalues into the open left half plane. For the numerical scheme, however, the analysis shows and examples verify that enthalpy damping can be effective in accelerating convergence to steady state.     

Weak solution of incompressible Euler equations with decreasing energy

Weak solution of incompressible Euler equations are L²-vector fields, satisfying integral relations, which express the mass and momentum balance. They are believed to describe the turbulent fluid motion at high Reynolds numbers. We justify this conjecture by constructing a weak solution with decreasing kinetic energy. The construction is based on Generalized Flows, introduced by Y. Brenier.     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

Precise regularity results for the Euler equations

It has already been proved, under various assumptions, that no singularity can appear in an initially regular perfect fluid flow, if the L^∞ norm of the velocity's curl does not blow up. Here that result is proved for flows in smooth bounded domains of (d?2) when the regularity is expressed in terms of Besov (or Triebel-Lizorkin) spaces.     

Non-metric two-component Euler equations on the circle

A geometric approach to the study of natural two-component generalizations of the periodic Hunter–Saxton is presented. We give rigorous evidence of the fact that these systems can be realized as geodesic equations with respect to symmetric linear connections on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle ${{\text{\sc Diff}}(\mathbb{S})}$ with the space of smooth functions on the circle. An immediate consequence of this approach is a well-posedness result of the corresponding Cauchy problems in the smooth category.