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This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown., Comment: 13 pages

The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and one-half" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schr\"odinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do they require monotonicity of the equilibrium., Comment: 49 pages

Let $$L_0=\suml_{j=1}^nM_j^0D_j+M_0^0,\,\,\,\,D_j=\frac{1}{i}\frac{\pa}{\paxj}, \quad x\in\Rn,$$ be a constant coefficient first-order partial differential system, where the matrices $M_j^0$ are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the nonhomegeneous case it is assumed that the operator is isotropic . The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: \begin{itemize} \item Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular the Dirac and Maxwell systems are explicitly treated. \item The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation. \end{itemize}

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called "weak Poincar\'e inequality" (WPI), originally introduced by Liggett [Ann. Probab., 1991]. Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup., Comment: 15 pages, improved results including precise constants in the inequalities (including Nash), expanded class of examples

This paper deals with the convergence of the second-order GRP (Generalized Riemann Problem) numerical scheme to the entropy solution for scalar conservation laws with strictly convex fluxes. The approximate profiles at each time step are linear in each cell, with possible jump discontinuities (of functional values and slopes) across cell boundaries. The basic observation is that the discrete values produced by the scheme are exact averages of an approximate conservation law , which enables the use of properties of such solutions in the proof. In particular, the “total-variation" of the scheme can be controlled, using analytic properties. In practice, the GRP code allows “sawteeth" profiles (i.e., the piecewise linear approximation is not monotone even if the sequences of averages is such). The “reconstruction" procedure considered here also allows the formation of “sawteeth" profiles, with an hypothesis of “Godunov Compatibility", which limits the slopes in cases of non-monotone profiles. The scheme is proved to converge to a weak solution of the conservation law. In the case of a monotone initial profile it is shown (under a further hypothesis on the slopes) that the limit solution is indeed the entropy solution. The constructed solution satisfies the “finite propagation speed", so that no rarefaction shocks can appear in intervals such that the initial function is monotone in their domain of dependence. However, the characterization of the limit solution as the unique entropy solution, for general initial data, is still an open problem.

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.

A direct Eulerian generalized Riemann problem (GRP) scheme is derived for compressible fluid flows. Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem (GRP) directly for the Eulerian formulation. The crucial auxiliary Lagrangian scheme in the original GRP scheme is not necessary in the present framework. The delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique.

A pure-streamfunction formulation is introduced for the numerical simulation of the two-dimensional incompressible Navier-Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacian of the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function relations need to be invoked. A compact numerical scheme, which interpolates streamfunction values as well as its first order derivatives, is presented and analyzed. A number of numerical experiments are presented, including driven and double driven cavities, where the Reynolds numbers are sufficiently large, leading to symmetry breaking of asymptotic solutions.

The high-resolution generalized Riemann problem (GRP) conservation laws scheme for compressible flows combined with Strang-type operator splitting is applied to computing an initial value problem having a discontinuous initial data. Imperfect representation of the initial data on the Cartesian grid, where the smooth curve of discontinuity is approximated by a jagged line, gives rise to spurious waves when using high-resolution integration with operator splitting. The nature of these waves is clarified by comparison to a one-dimensional model. We demonstrate that it is not the operator splitting that gives rise to these waves, but rather the better quality of the hyperbolic (one-dimensional) solver, which is not degraded by the operator splitting. It is expected that this property of retaining sharp features of initial data will also be produced by other second-order conservation laws schemes.

Motivated by the study of the fourth-order nonlinear Schrodinger equations introduced by V. Karpman [4], we give dispersion estimates for the linear group associated to i ∂ t +Δ 2 ±Δ .

Long-time behavior and regularity are studied for solutions of the Stark equationut=i(−Δ−x1+V(x))u,u(0,x)∈L2(Rn). It is shown that for a class of short-range potentialsV(x) the gain of local smoothness and the decay as |t|→∞ are close to those of the corresponding Schrödinger equationut=i(−Δ+V(x))u.

Two index formulas for operators defined by infinite band matrices are proved. These results may be interpreted as a generalization of a classical theorem of M. G. Krein on orthogonal polynomials. The proofs are based on dichotomy and nonstationary inertia theory.

A review of about a decade of development of the generalized Riemann problem (GRP) scheme is presented. The method is briefly outlined, followed by various numerical and physical extensions. The range of versatile applications of the GRP method is illustrated through numerous examples.

International audience; The convergence of a fourth-order compact scheme to the one-dimensional biharmonic problem is established in the case of general Dirichlet boundary conditions. The compact scheme invokes value of the unknown function as well as Pade approximations of its first-order derivative. Using the Pade approximation allows us to approximate the first-order derivative within fourth-order accuracy. However, although the truncation error of the discrete biharmonic scheme is of fourth-order at interior point, the truncation error drops to first-order at near-boundary points. Nonetheless, we prove that the scheme retains its fourth-order (optimal) accuracy. This is done by a careful inspection of the matrix elements of the discrete biharmonic operator. A number of numerical examples corroborate this effect. We also present a study of the eigenvalue problem u xxxx = νu. We compute and display the eigenvalues and the eigenfunctions related to the continuous and the discrete problems. By the positivity of the eigenvalues, one can deduce the stability of of the related time-dependent problem u t = −u xxxx. In addition, we study the eigenvalue problem u xxxx = νu xx. This is related to the stability of the linear time-dependent equation u xxt = νu xxxx. Its continuous and discrete eigenvalues and eigenfunction (or eigenvectors) are computed and displayed graphically.

The problem of expanding a field over the set of waveguide modes is well known. Nevertheless, one may find small differences in the way this concept is used. Some employ a superposition of waveguide modes that include all field components, and a mode is considered as a single entity which propagates undisturbed along the structure. Others prefer to expand only the transverse-field components, whereas the longitudinal ones are derived from Maxwell's equations. It is shown that the latter is correct, at least for the important class of 2-dimensional structures. The two approaches coincide, however, if the structure is the waveguide for which the set of modes was calculated. The formal mathematical proof is restricted to a nonlossy medium. It is shown that the modes with real propagation coefficients squared suffice to construct a complete set. Depending on the boundary conditions in the longitudinal z direction, some or many of these modes may or may not be needed.

We consider the 1/2-dimensional relativistic Vlasov-Maxwell system that describes the time-evolution of a plasma. We find a relatively simple criterion for spectral instability of a wide class of equilibria. This class includes non-homogeneous equilibria that need not satisfy any additional symmetry properties (as was the case in previous results), nor should they be monotone in the particle energy. The criterion is given in terms of the spectral properties of two Schr\"{o}dinger operators that arise naturally from Maxwell's equations. The spectral analysis of these operators is quite delicate, and some general functional analytic tools are developed to treat them. These tools can be applied to similar systems in higher dimensions, as long as their domain is finite or periodic., Comment: 19 pages

A smooth spectral theory is presented in an abstract Hilbert space framework. The main assumption (of smoothness) is the Holder continuity of the derivative of the spectral measure (density of states). A Limiting Absorption Principle (LAP) is derived on the basis of continuity properties of Cauchytype integrals. This abstract theory is then extended to include short-range perturbations and sums of tensor products. Applications to partial differential operators are presented. In the context of partial differential operators the spectral derivative is closely related to trace operators on compact (for elliptic operators) or non-compact manifolds.

Consider the Schrodinger equation {fx25-1}. The following estimates are proved: (A) IfV≡0 then for any 0≤α 1/2, {fx25-3} (B) If |V(x)|≤C(1+|x|2)−1−δ, δ>0, then (if 0 is neither an eigenvalue nor a resonance of −Δ+V), {fx25-4}.

Let u = u(x, t) be a solution to the IVP for the Schrodinger equation iu 1 = (−Δ + V(x))u ≡ Hu , u(x, 0) = u 0 (x) ϵ P ac L 2 (R n ) (Pac is the projection on the absolutely continuous subspace of H). Assume that for some e > 0 the multiplication operator (1+|x|) 1+i V(x):H 1-e (R n ) → L 2 (R n ) is bounded. Then u(x, t) = u1(x, t) + u2(x, t) where, for every s > 1 2 , ∫ R ∫ R n (1 + |x| 2 ) −s |(1+H) 1 4 u 1 (x,t)| 2 dxdt ⩽ C ‖u 0 ‖ L 2 (R n ) 2 , and for every integer j, sup‖(I + H) j u 2 (·, t)‖ L 2 (R n ) ⩽ C j ‖ U 0 ‖ L 2 (R n ) tϵR

L ∞ estimates are derived for the oscillatory integral ∫ + 0 ∞ e − i ( x λ + (1/ m ) t λ m ) a (λ) d λ, where 2 ≤ m ∈ R and ( x , t ) ∈ R × R + . The amplitude a (λ) can be oscillatory, e.g., a (λ) = e it p (λ) with p (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a (λ) = (1 + λ) k with 0 ≤ k ≤ 1 2 ( m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrodinger or Airy type, and of associated semilinear equations.

A GRP-scheme is introduced for the numerical integration of the Euler system of equations of compressible reactive flow in a duct of variable cross section, subject to an external potential. The GRP (generalized Riemann problem) scheme is based on an analytic solution of the GRP at jump discontinuities. It is a second-order scheme generalizing the first-order Godunov scheme, having the property of high resolution of shocks and other discontinuities. Some numerical examples are considered, including an infinite spherical reflected shock, a spherical blast wave and gas collapse under an external potential.

The flagellar filament, the bacterial organelle of motility, is the smallest rotary propeller known. It consists of 1), a basal body (part of which is the proton driven rotary motor), 2), a hook (universal joint—allowing for off-axial transmission of rotary motion), and 3), a filament (propeller—a long, rigid, supercoiled helical assembly allowing for the conversion of rotary motion into linear thrust). Helically perturbed (so-called “complex”) filaments have a coarse surface composed of deep grooves and ridges following the three-start helical lines. These surface structures, reminiscent of a turbine or Archimedean screw, originate from symmetry reduction along the six-start helical lines due to dimerization of the flagellin monomers from which the filament self assembles. Using high-resolution electron microscopy and helical image reconstruction methods, we calculated three-dimensional density maps of the complex filament of Rhizobium lupini H13-3 and determined its surface pattern and boundaries. The helical symmetry of the filament allows viewing it as a stack of identical slices spaced axially and rotated by constant increments. Here we use the closed outlines of these slices to explore, in two dimensions, the hydrodynamic effect of the turbine-like boundaries of the flagellar filament. In particular, we try to determine if, and under what conditions, transitions from laminar to turbulent flow (or perturbations of the laminar flow) may occur on or near the surface of the bacterial propeller. To address these questions, we apply the boundary element method in a manner allowing the handling of convoluted boundaries. We tested the method on several simple, well-characterized cylindrical structures before applying it to real, highly convoluted biological surfaces and to simplified mechanical analogs. Our results indicate that under extreme structural and functional conditions, and at low Reynolds numbers, a deviation from laminar flow might occur on the flagellar surface. These transitions, and the conditions enabling them, may affect flagellar polymorphism and the formation and dispersion of flagellar bundles—factors important in the chemotactic response.

This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, time-dependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic ( E 1 ) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov's scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10]. The GRP for Quasi-1-D, Compressible, Inviscid Flow In Section 4.2 we studied the Euler equations (4.45) governing the quasi-1-D flow in a duct of variable cross section. We emphasized in particular the role of the Riemann problem (“shock tube problem”), namely, the IVP subject to initial data (4.100). As we shall see in this chapter, the solution to the Riemann problem is a basic ingredient in the numerical resolution of the flow.

This chapter presents the existence, uniqueness, and regularity theory of solutions to the Navier–Stokes equations when they are formulated in vorticity form. It also discusses the large-time asymptotic behavior of solutions for sufficiently small initial data. In fact, the three-dimensional case has hardly been studied and therefore, it concentrates on the two-dimensional case. From the point of view of hydrodynamic phenomena an interesting case is the evolution of vorticity and its associated velocity field when it is initially given by isolated vortices, vortex filaments, or sheets. In the zero viscosity limit (i.e., v = 0, leading to the Euler equations) the circulation is preserved (Kelvin's theorem) and the use of vorticity in numerical methods has become very popular. In vortex methods even smooth initial data are replaced by a distribution of singular vortical objects.

It is the purpose of this paper to present some recent results concerning properties of solutions to time dependent Schrodinger-type equations of the form, $$ \begin{array}{*{20}c} {\frac{1}{i}\frac{{du}}{{dt}} = (H_0 + V)u,} \\ {u(0) = u_0 \in L^2 (\mathbb{R}^n ).} \\ \end{array} $$ (0.1)

The GRP (Generalized Riemann Problem) method was introduced and developed in [1–7, 9–10], following the pioneering work of van Leer [13].

Necessary and Sufficient Conditions for an input-output operator of a linear time variant finite dimensional system to be a strict contraction area suggested. This is a generalization of the well known bounded real lemma. Both the definite and indefinite case is considered. The results are presented in state space form.

Orthogonalization with invertible squares is studied in the special case of modules over the C*-algebra of block diagonal operators in l r 2 . For orthogonalizations with invertible squares of the system I, S,..., S m , where S is a unilateral shift, are obtained results which describe their invertibility, Fredholm and index properties.

In this paper, we study connections between Fredholm properties of block weighted shifts in Hilbert spaces, Stein inequalities for nonstationary systems, and dichotomies for such systems.

A second-order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented. Two related versions are proposed, one Lagrangian and the second direct Eulerian. They both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity, assuming initial nonvanishing slopes on both sides. While this solution is an extension of the solution to the classical Riemann problem, the resulting schemes are second-order extensions of Godunov's methods. In both cases, they are very simple to implement in computer codes. Several numerical examples are shown, where the only additional mechanism is a simple monotonicity algorithm.

Let H0=−Δ+V0( x1), H=H0+V( x) be self‐adjoint in L2(Rn). V0 depends only on one coordinate and tends monotonically to ∓∞ as x1→±∞. V is a real H0‐compact potential, short range with respect to V0. In particular, the cases V0( x1)=−(sgn x1)‖x1‖α, 0

Let L 0 = − Δ + V ( x 1 ) , L = L 0 + V p ( x ) {L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x) be selfadjoint in L 2 ( R n ) {L^2}({R^n}) . Here V , V p V,{V_p} are real functions, V ( x 1 ) V({x_1}) depends only on the first coordinate. Existence of the wave-operators W ± ( L , L 0 ) = s - lim t → ± ∞ exp ⁡ ( i t L ) exp ⁡ ( − i t L 0 ) {W_ \pm }\,(L,{L_0}) = s \text {-} {\lim _{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0}) is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of − d 2 / d t 2 + V ( t ) -{d^2}/d{t^2} + V(t) in L 2 ( R ) {L^2}(R) . Its absolute continuity is proved as well as a suitable eigenfunction expansion. V V is a "Stark-like" potential. In particular, the cases V ( x 1 ) = ( − sgn ⁡ x 1 ) | x 1 | α , 0 > α ⩽ 2 V({x_1}) = ( - \operatorname {sgn}{x_1})|{x_1}\,{|^\alpha },0 > \alpha \leqslant 2 , are included. V p {V_p} may be taken as the sum of an L 2 {L^2} -function and a function satisfying growth conditions in the + x 1 + {x_1} direction. V p ( x ) = | x | − 1 {V_p}(x) = |x|^{ - 1} is included.

The “Generalized Riemann Problem” (GRP) method is applied to 1-D compressible flows with material interfaces and variable cross section. The resulting scheme is second-order and uses a “mixed-type” grid, where cell boundaries can be either Lagrangian or Eulerian. In fact, using the analytic resolution of discontinuities at cell boundaries, provided by the GRP solution, one can extend the scheme presented here to include any adaptive mesh. Two numerical examples are studied: a planar shock-tube and exploding helium sphere. It is shown that discontinuities are sharply resolved while there are no oscillations in the smooth part of the flow. In particular, wave interactions, including formation of new shocks and reflection from the center of symmetry, are automatically taken care of.

Let H = −Δ + VE(¦x¦)+ V(x) be a Schrödinger operator in Rn. Here VE(¦x¦) is an “exploding” radially symmetric potential which is at least C2 monotone nonincreasing and O(r2) as r → ∞. V is a general potential which is short range with respect to VE. In particular, VE  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = limr → ∞ VE(r) and R(z) = (H − z)−1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset N⊆(Λ, ∞), in a suitable operator topology B(L, L∗). And L ⊆ L2(Rn) is a weighted L2-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing.

A generalized Riemann problem is introduced for the equations of reactive non-viscous compressible flow in one space dimension. Initial data are assumed to be linearly distributed on both sides of a jump discontinuity. The resolution of the singularity is studied and the first-order variation (in time) of flow variables is given in exact form.