Equilibrium schemes for scalar conservation laws with stiff sources

We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

     

Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam

We introduce a set of conserved quantities of energy‐type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well‐posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd.     

Additive Invariant Functionals for Dynamical Systems

We consider the problem of defining completely a class of additive conservation laws for the generalized Liouville equation whose characteristics are given by an arbitrary system of first-order ordinary differential equations. We first show that if the conservation law, a time-invariant functional, is additive on functions having disjoint compact support in phase space, then it is represented by an integral over phase space of a kernel which is a function of the solution to the Liouville equation. Then we use the fact that in classical mechanics phase space is usually a direct product of physical space and velocity space (Newtonian systems). We prove that for such systems the aforementioned representation of the invariant functionals will hold for conservation laws which are additive only in physical space; i.e., additivity in physical space automatically implies additivity in the whole phase space. We extend the results to include non-degenerate Hamiltonian systems, and, more generally, to include both conservative and dissipative dynamical systems. Some applications of the results are discussed.     

一类非线性Schr dinger方程的高精度守恒数值格式

1引言本文讨论下面非线性Schr(?)dinger方程(NLS)方程的初边值问题:i(?)u/(?)t (?)~2u/(?)x~2 2|u~2|u=0,(1) u(x_l,t)=u(x_r,t)=0,t>0,(2) u(x,0)=u_0(x),x_l≤x≤x_r,(3)其中u(x,t)是复值函数,u_0(x)为已知的复值函数,i~2=-1.该问题有着如下的电荷与能量守恒关系:     

Digital Studies and Perspectives for N-Body Modelling in Physics

In 1977, Richard Feynman, in an invited lecture at the national APS meeting, stated unequivocally that quantum mechanical principles placed few important restrictions on how a stable atom can be held in place, and, indeed, a stable atom on the whole has a comparatively definite position fixed by its comparatively massive nucleus. In this spirit, N-body problems are studied using molecules and classical molecular potentials and also using collections of molecules, called particles, with related potentials derived through conservation of mass and energy. Detailed applications include primary vortex flow and turbulent flow for both water vapor and air, soliton collision, and the motion of a top on a smooth surface. Other applications, like microdrop collision, stress of a slotted copper plate, contact angle of adhesion, cellular self-reorganiztion, the bounce of an elastic ball, and elastic snap-through are mentioned and referenced appropriately. Lastly, numerical methodology is developed which preserves the physics of special relativity and is applied to simulate a relativistic oscillator.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅷ)—PRINCIPLE OF TOTAL WORK AND ENERGY

IntroductionRecently ,wehaverestudiedtheconventionalconservationlawofenergy ,whichhaswidelybeenusedandisusinginliteraturesofclassicalcontinuummechanicsathomeandabroad ,andpointedoutthatitistheoreticallyincompleteeveninthetraditionalframework .Thepresentsituationoftheconservationlawsofenergyforpolarcontinuaisthesameasthatforclassicalcontinuummechanics.Inordertochangethestatusquo ,itseemstobequitenecessarythatnewprinciplesoftotalworkandenergyshouldbeproposedtotaketheplaceoftheexistingconservatio…     

The incompatibility between local hidden variable theories and the fundamental conservation laws

I discuss in detail the result that the Bell’s inequalities derived in the context of local hidden variable theories for discrete quantized observables can be satisfied only if a fundamental conservation law is violated on the average. This result shows that such theories are physically nonviable, and makes the demarcating criteria of the Bell’s inequalities redundant. I show that a unique correlation function can be derived from the validity of the conservation law alone and this coincides with the quantum mechanical correlation function. Thus, any theory with a different correlation function, like any local hidden variable theory, is incompatible with the fundamental conservation laws and space-time symmetries. The results are discussed in the context of two-particle singlet and triplet states, GHZ states, and two-particle double slit interferometry. Some observations on quantum entropy, entanglement, and nonlocality are also discussed.     

Lax representations for triplets of two-dimensional scalar fields of the chiral type

We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 189–205, August, 2006.     

二次灾难

二次灾难是指由于地球被撞击引起的地震、海啸、大气扬尘、震裂冰川而导致海平面升高等间接的灾难效应.基于能量转化和守恒概念,建立简洁的力学模型,根据牛顿力学的动力学理论及有关文献,研究直径1 000 m的铁镍小行星撞击南极点对地球造成的直接影响和二次灾难.同时,还给出了一套评估灾难的方法.通过具体的计算发现,由撞击带来的直接效应是有限的,在短时间内会迅速衰减,但由此引发的二次效应却是十分显著的.