When faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order $p=r=s$ and stage order $q=p,p-1$ up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.

We construct a new type of planar Euler flows with localized vorticity. Let $\kappa _i\not =0$ , $i=1,\ldots , m$ , be m arbitrarily fixed constants. For any given nondegenerate critical point $\mathbf {x}_0=(x_{0,1},\ldots ,x_{0,m})$ of the Kirchhoff–Routh function defined on $\Omega ^m$ corresponding to $(\kappa _1,\ldots , \kappa _m)$ , we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and concentrate on curves perturbed from small circles centered near $x_{0,i}$ , $i=1,\ldots ,m$ . The proof is accomplished via the implicit function theorem with suitable choice of function spaces.

We propose a model in which financial sophistication improves portfolio returns and therefore the incentive to substitute consumption intertemporally. The model delivers an Euler equation in which consumption growth is positively correlated with financial sophistication. We test the model's prediction using panel data on consumption and financial sophistication drawn from the Italian Survey of Household Income and Wealth. We find that consumption growth is positively correlated with financial sophistication, as predicted by the model. We also provide estimates of the intertemporal elasticity of substitution in the range between 0.4 and 0.6.

We compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: A first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.

In this paper, we study the phenomena of thermo-chemical imbalance in a reactive mono-dimensional flow composed of a mixture of (79% nitrogen N2 and 21% oxygen O2). We are interested in modeling the physicochemical process that may be encountered in hypersonic flows, as vibrational excitation, dissociation and ionization, also the formation of chemical species to higher temperatures behind a detached strong shock. We put a special emphasis on vibrational relaxation model of CVD coupling. At these high temperatures, collisions electrons-atoms become very effective, taking in account the radiation that requires knowledge and modeling of all physicochemical processes (collisional and radiative). The mathematical model of flows at the atmospheric reentry is governed by Euler stationary equations coupled with the chemical kinetics and the radiative transfer equations. Our computational code is based on the finite differences method that used to discretize and resolve the obtained numerical model, where an appropriate mesh is selected in the relaxation zone in order to determine the flow parameters at each grid position

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form where and are any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

We consider the Euler equation for an incompressible fluid on a three dimensional torus,and the construction of its solution as a power series in time. We point out some generalfacts on this subject, from convergence issues for the power series to the role ofsymmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas andWu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose avery simple Fourier polynomial as an initial datum for the Euler equation and analyzed thepower series in time for the solution, determining the first 35 terms by computer algebra.Their calculations suggested for the series a finite convergence radiusτ3 in the H3 Sobolev space, with0.32 < τ3 < 0.35; they regarded this as an indicationthat the solution of the Euler equation blows up. We have repeated the calculations of E.Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238,using again computer algebra; the order has been increased from 35 to 52, using thesymmetries of the initial datum to speed up computations. As forτ3, our results agree with the original computations of E.Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238(yielding in fact to conjecture that 0.32 < τ3 < 0.33).Moreover, our analysis supports the following conclusions: (a) The finiteness ofτ3 is not at all an indication of a possible blow-up. (b)There is a strong indication that the solution of the Euler equation does not blow up at atime close to τ3. In fact, the solution is likely to exist, atleast, up to a time θ3 > 0.47. (c) There is a weakindication, based on Padé analysis, that the solution might blow up at a later time.

An anisotropic solution adaptive method based on unstructured quadrilateral meshes for inviscid compressible flows is proposed. The data structure, the directional refinement and coarsening, including the method for initializing the refined new cells, for the anisotropic adaptive method are described. It provides efficient high resolution of flow features, which are aligned with the original quadrilateral mesh structures. Five different cases are provided to show that it could be used to resolve the anisotropic flow features and be applied to model the complex geometry as well as to keep a relative high order of accuracy on an efficient anisotropic mesh.

Nous prouvons que pour toute solution u du problème de Kelvin–Helmholtz des nappes de tourbillons pour l'équation d'Euler bi-dimensionnelle, définie localement en temps, la courbe de saut de u et la densité de tourbillon sont analytiques (sous une hypothèse de régularité Holderienne de la courbe de saut). Nous donnons également un résultat de régularité partielle de la trace de u sur t=0 lorsque u est définie sur un demi-interval [O,T[.