Spacetime discontinuous Galerkin methods for convection-diffusion equations

We have developed two new methods for solving convection-diffusion systems, with particular focus on the compressible Navier-Stokes equations. Our methods are extensions of a spacetime discontinuous Galerkin method for solving systems of hyperbolic conservation laws [3]. Following the original scheme, we use entropy variables as degrees of freedom and entropy stable numerical fluxes for the nonlinear convection term. We examine two different approaches for incorporating the diffusion term: the interior penalty method and the local discontinuous Galerkin approach. For both extensions, we can show an entropy stability result for convection-diffusion systems. Although our schemes are designed for systems, we focus on scalar convectiondiffusion equations in this contribution. This allows us to highlight our main ideas behind the stability proofs, which are the same for scalar equations and systems, in a simplified setting.     

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

On the entropy conditions for some flux limited diffusion equations

In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.     

Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D

Summary. Systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms. To solve these weakly coupled systems numerically a class of explicit and implicit upwind finite volume methods on unstructured grids is presented. Provided an unique entropy solution of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing discretization parameter to this entropy solution. These results are applied to examples from combustion theory and hydrology where the existence of entropy solutions can be shown. The proofs rely on an extension of a result due to DiPerna concerning measure valued solutions to the case of weakly coupled hyperbolic systems. Received April 29, 1997     

重建极性连续统理论的基本定律和原理(Ⅸ)——热力学

对现有的微极连续统场论的基本定律进行了再研究,并指出了它们的不完整性．建立起新的微极连续统热静力学和热动力学的第一和第二基本定律．从这些定律可以很自然地和同时推导出热静力学的所有平衡方程和熵不等式以及热动力学的所有均衡方程和熵率不等式．随时对这里得到的新结果与现有微极连续力学专著和教科书中的相应结果进行了比较．着重指出的是,为什么从现有的微极连续统热动力学基本定律不能推导出局部能量均衡方程和局部熵不等式问题已经得到阐明．     

Thermodynamic and statistical principles of the theory of elastoviscoplastic deformation and strengthening of materials

We propose a thermodynamic method and a statistical one for constructing the constitutive equations of elastoviscoplastic deformation and strengthening of materials. The thermodynamic method is based on the energy conservation law as well as the equations of entropy balance and entropy generation in the presence of self-equilibrated internal microstresses, which are characterized by coupled strengthening parameters. The general constitutive equations consist of the relations between thermodynamic flows and forces, which follow from nonnegativity of entropy generation and satisfy the generalized Onsager principle, as well as the thermoelasticity relations and the expression for entropy, which follow from the energy conservation law. The specific constitutive equations are obtained on the basis of representation of the energy dissipation rate as a sum of two constituents that describe translational and isotropic strengthening and are approximated by power and hyperbolic sine laws. Starting from the stochastic microstructural concepts, we construct the constitutive equations of elastoviscoplastic deformation and strengthening on the basis of the linear model of thermoelasticity and the nonlinear Maxwell model for spherical and deviatoric components of microstresses and microstrains, respectively. The solution of the problem of the effective properties and stress-strain state of a three-component material is constructed with the use of the combined Voigt–Reuss scheme and leads to constitutive equations coinciding, as to their form, with similar equations constructed by the thermodynamic method.     

The Riemann problem for general systems of conservation laws

An extended entropy condition (E) has previously been proposed, by which we have been able to prove uniqueness and existence theorems for the Riemann problem for general 2-conservation laws. In this paper we consider the Riemann problem for general n-conservation laws. We first show how the shock are related to the characteristic speeds. A uniqueness theorem is proved subject to condition (E), which is equivalent to Lax's shock inequalities when the system is “genuinely nonlinear.” These general observations are then applied to the equations of gas dynamics without the convexity condition P_vv(v, s) > 0. Using condition (E), we prove the uniqueness theorem for the Riemann problem of the gas dynamics equations. This answers a question of Bethe. Next, we establish the relation between the shock speed σ and the entropy S along any shock curve. That the entropy S increases across any shock, first proved by Weyl for the convex case, is established for the nonconvex case by a different method. Wendroff also considered the gas dynamics equations without convexity conditions and constructed a solution to the Riemann problem. Notice that his solution does satisfy our condition (E).     

An entropy satisfying MUSCL scheme for systems of conservation laws

Summary. For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme is a nonlinear version of the classical Van Leer's MUSCL scheme. Strong convergence of this second order, entropy satisfying scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory. Received March 28, 1995 / Revised version received June 17, 1995     

Shock waves in the nonisentropic gas flow

In this paper we propose an extended entropy condition for general systems of hyperbolic conservation laws with several space variables. This entropy condition generalizes the well-known condition (E) of Volpert for a single conservation law with several space variables and reduces to the entropy condition proposed earlier by the author for systems with one space variable. The Riemann problem for general nonisentropic gas equations has a unique solution for initial data with arbitrarily large jumps. The occurrence of a vacuum region is observed. The projections of shock curves on the pressure-velocity plane are analyzed so as to study the interaction of weak shocks. Our results differ markedly from those of previous works in that we do not assume the equation of state to be polytropic. In fact our assumptions on the equation of state allow the pressure to be a nonconvex function of specific volume.The Riemann problem for this general system of gas equations was also treated by B. Wendroff when the initial data are near constant.     

A new nonlinear functional for general scalar conservation laws

The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the L¹ stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658-667] and it can be viewed as a better attempt for the generalized entropy functional for general equations.     

重建极性连续统理论的基本定律和原理(Ⅱ)——微态连续统理论和偶应力理论

重建微态连续统理论和偶应力理论的动量和动量矩均衡定律以及能量守恒定律,并由这些定律自然地推导出相应的局部和非局部均衡方程。这些结果可由耦合型微极连续统理论过渡和归结而得到。把推导出的结果和传统的质量和微惯性守恒定律以及熵不等式结合在一起就构成微态连续统理论和偶应力理论的基本均衡定律和方程体系。还弄清了以前的各种连续统理论的不完整性层次。最后,给出了几种特殊情形。     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

水动力-热动力学的极值定律

本文对水动力学和更普通性的连续介体动力学中以连续方程与运动方程所表达的现有诸经典守恒定律以外,提出另一最大能量消散率定律.这一定律的推论就是应用水力学中培纶格-波丝最小储存能学说. 凡在运动中消散了的机械能皆转化成为热能,储存在物体里.能量之消散当一定时刻一定温度都使产熵增加.所以,从最大能量消散率可引出热力学第二定律的一个新概念,即机械运动的产熵率也总是一个可能的最大值. 文中建议的这个连续介体极值定律,可从变分原理推导出来,重订热力学第二定律则可藉微观分析加以证明.两者合成水动力-热动力学极值定律     

Entropy stable schemes for initial-boundary-value conservation laws

We consider initial-boundary-value problems for systems of conservation laws and design entropy stable finite difference schemes to approximate them. The schemes are shown to be entropy stable for a large class of systems that are equipped with a symmetric splitting, derived from the entropy formulation. Numerical examples for the Euler equations of gas dynamics are presented to illustrate the robust performance of the proposed method.     

Al2O3-47 nm and Al2O3-36 nm characterizations of nonlinear differential equations for biomedical applications: Magnetized peristaltic transport

Current research models the Al₂O₃ 47nm and Al₂O₃ 36nm nanoparticles transportation through peristalsis with entropy optimization. Conservation laws for mass, momentum and energy are used to model the present flow situation. These equations elaborates the magnetohydrodynamics, Hall, thermal radiation, Joule heating, heat generation and absorption. Convective heat transfer impacts are studied at channel walls. Entropy is modeled in view of thermodynamics second law. Two different expressions for effective viscosity are accounted. Simplification of the modeled equations is done through lubrication assumptions. Solution for momentum equation is obtained analytically and for numerically for temperature equation. Built-in shooting procedure is utilized to obtain the desired numerical results. Later on these obtained results are used to sketch and discussed the flow quantities of interest for the influential parameters accounted in the problem.     

Discrete approximations of BV solutions to doubly nonlinear degenerate parabolic equations

Summary. In this paper we present and analyse certain discrete approximations of solutions to scalar, doubly nonlinear degenerate, parabolic problems of the form under the very general structural condition . To mention only a few examples: the heat equation, the porous medium equation, the two-phase flow equation, hyperbolic conservation laws and equations arising from the theory of non-Newtonian fluids are all special cases of (P). Since the diffusion terms a(s) and b(s) are allowed to degenerate on intervals, shock waves will in general appear in the solutions of (P). Furthermore, weak solutions are not uniquely determined by their data. For these reasons we work within the framework of weak solutions that are of bounded variation (in space and time) and, in addition, satisfy an entropy condition. The well-posedness of the Cauchy problem (P) in this class of so-called BV entropy weak solutions follows from a work of Yin [18]. The discrete approximations are shown to converge to the unique BV entropy weak solution of (P). Received November 10, 1998 / Revised version received June 10, 1999 / Published online June 8, 2000     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

An energy-entropy-consistent time stepping scheme for finite thermo-viscoelasticity

This paper presents a time integrator, which is based on a time discrete spatially weak finite element formulation, but fulfills the same balance laws as the underlying (five) differential equations. Namely, in addition to the balances of linear and angular momentum as well as entropy, also the balances of total energy and LYAPUNOV function are fulfilled. The spatially weak formulation is obtained by integration by parts. Where the resulting virtual stress power term is well-known, the virtual entropy production by conduction of heat is less known. The time discretisation is based on the midpoint rule and non-standard time discrete differential operators. This time integrator is a further development of the TC integrator of I. ROMERO. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Une nouvelle méthode de relaxation pour les équations de Navier–Stokes compressibles

We consider the compressible Navier–Stokes equations for gas flows endowed with general pressure and temperature laws as long as they are compatible with the existence of an entropy and Gibbs relations. We extend the relaxation method introduced for the Euler equations by Coquel and Perthame. Keeping the same “sub-characteristic” conditions for the hyperbolic fluxes and using a consistent splitting of the diffusive fluxes based on a global temperature, we prove the stability of the relaxation system via the sign of the production of a suitable entropy. A first order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method. To cite this article: E. Bongiovanni et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).