Search

Let u(@?x, t) be a weak solution of the Euler equation, governing the inviscid polytropic gas dynamics; in addition, u(@?x, t) is assumed to respect the usual entropy conditions connected with the conservative Euler equations. We show that such entropy solutions of the gas dynamics equations satisfy a minimum entropy principle, namely, that the spatial minimum of their specific entropy, Ess inf"@?"xS(u(@?x, t)), is an increasing function of time. This principle equally applies to discrete approximations of the Euler equations such as the Godunov-type and Lax-Friedrichs schemes. Our derivation of this minimum principle makes use of the fact that there is a family of generalized entropy functions connected with the conservative Euler equations.

Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservative equation. In particular, entropy satisfying convergence follows for E schemes—those containing more numerical viscosity than Godunov’s scheme.

We consider finite-difference, pseudospectral and Fourier—Galerkin methods for the approximate solution of time-dependent problems. The paper provides a unified framework for the stability analysis of all three discrete methods. In particular, the problem of stability for highly accurate stencils is studied in some detail.

A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Convenient conditions for nonlinear difference schemes to be total variation diminishing (TVD) are derived. It is shown that such schemes share the TVD property, provided their numerical fluxes meet a certain positivity condition at local extreme values but can be arbitrary otherwise. Local TVD conditions are invariant under different incremental representations of the nonlinear schemes, and thus provide a simplified generalization of the global TVD conditions established by previous work.

New convenient stability criteria are provided in this paper for a large class of finite-difference approximations to initial-boundary value problems associated with the hyperbolic system u t = A u x + B u + f {{\mathbf {u}}_t} = A{{\mathbf {u}}_x} + B{\mathbf {u}} + {\mathbf {f}} in the quarter plane x ⩾ 0 x \geqslant 0 , t ⩾ 0 t \geqslant 0 . Using the new criteria, stability is easily established for numerous combinations of well-known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature.

Discrete approximations to hyperbolic systems of conservation laws are studied. We quantify the amount of numerical viscosity present in such schemes, and relate it to their entropy stability by means of comparison. To this end, conservative schemes which are also entropy conservative are constructed. These entropy conservative schemes enjoy second-order accuracy; moreover, they can be interpreted as piecewise linear finite element methods, and hence can be formulated on various mesh configurations. We then show that conservative schemes are entropy stable, if and—for three-point schemes—only if they contain more viscosity than that present in the above-mentioned entropy conservative ones.

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.

Using a simple symmetrizability criterion, we show that symmetric systems of conservation laws are equipped with a one-parameter family of entropy functions.

The Kuramoto–Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting a unique smooth solution, continuously dependent on its initial data. As an attractive alternative to standard energy methods, existence and stability are derived in this case, by “patching” in the large short time solutions without “loss of derivatives”.

We study some properties of the numerical radius of matrices with non-negative entries, and explicit ways to compute it. We also characterize positive matrices with equal spectral and numerical radii, i.e., positive spectral matrices.

The convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed. Numerical tests indicate that the convergence may (and in fact in some cases must) fail, with or without post-processing of the numerical solution. Instead, a new kind of spectrally accurate vanishing viscosity is introduced to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments, it is shown that this spectral viscosity prevents oscillations, and convergence to the unique entropy solution follows.

It is shown that when differencing analytic functions using the pseudospectral Fourier or Chebyshev methods, the error committed decays to zero at an exponential rate.

The stability of finite difference approximations to initial boundary hyperbolic systems is studied. As is well known, a proper specification of boundary conditions for such systems is essential for their solutions to be well defined. A discrete analogue of the above is proved - if the numerical boundary conditions are consistent with an inflow part of the problem, they render the overall computation unstable. An example of the inviscid gasdynamics equations is considered. Previously announced in STAR as N81-33874

We show that any spectrally dominant vector norm on the vector space of matrices which is invariant under isometries dominates the numerical radius r (·). Thus, the celebrated Lax-Wendroff stability condition, r (·)⩽1, defines a maximal isometrically invariant stable set.

The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.

Easily checkable sufficient stability criteria are obtained for explicit dissipative approximations to mixed initial-boundary value problems associated with the system u t = A u x {u_t} = A{u_x} in the quarter plane x ⩾ 0 x \geqslant 0 , t ⩾ 0 t \geqslant 0 . The criteria are given entirely in terms of the boundary conditions for the outflow unknowns. The results imply that certain well-known boundary conditions, when used in combination with any (stable) dissipative scheme, always maintain stability.

Convenient stability criteria are obtained for difference approximations to initial-boundary value problems associated with the hyperbolic system u t = A u x + B u + f {{\mathbf {u}}_t} = A{{\mathbf {u}}_x} + B{\mathbf {u}} + {\mathbf {f}} in the quarter plane x ⩾ 0 x \geqslant 0 , t ⩾ 0 t \geqslant 0 . The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outflow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well-known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized.

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations.

We study complex-valued symmetric matrices. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. Consequently, we provide a sharp criterion for identifying those symmetric matrices whose spectral norm does not exceed one: such strongly stable matrices are usually sought in connection with convergent difference approximations to partial differential equations. As an example, we apply the derived criterion to conclude the strong stability of a Lax-Wendroff scheme.

Hyperbolic systems of conservation laws augumented with an entropy inequality are studied. It is shown that such systems can be written in a (quasilinear) skew-selfadjoint form. Centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered. Employing the above formulation in bounded regions under the nonsmooth regime as well, a local entropy decay estimate is also concluded. Examples of the shallow-water and the full gasdynamics equations are explicitly treated.

The Kreiss matrix theorem asserts that a family of N × N matrices is L 2 -stable if and only if either a resolvent condition (R) or a Hermitian norm condition (H) is satisfied. We give a direct, considerably shorter proof of the power-boundedness of an N × N matrix satisfying (R), sharpening former results by showing that power- boundedness depends, at most, linearly on the dimension N . We also show that L 2 -stability is characterized by an H-condition employing a general H-numerical radius instead of the usual H-norm, thus generalizing a sufficient stability criterion, due to Lax and Wendroff.

We study the question of entropy stability for discrete approximations to hyperbolic systems of conservation laws. We quantify the amount of numerical viscosity present in such schemes, and relate it to their entropy stability by means of comparison. To this end, two main ingredients are used: the entropy variables and the construction of certain entropy conservative schemes in terms of piecewise-linear finite element approximations. We then show that conservative schemes are entropy stable, if they contain more numerical viscosity than the above mentioned entropy conservative ones.

We consider finite difference approximations to the mixed initial boundary value problems of the hyperbolic system v(t) = au(x) in the quarter plan x 0, t 0. The finite difference approximations may be either dissipative or non-dissipative. Easily checkable sufficient stability criteria are obtained entirely in terms of the boundary conditions for the out flow variables. It is shown how to extend the results for the two dimensional case when the boundary conditions in the y direction allow Fourier transformations. Examples are given. (Author)

Let f(x) be a bounded 2π-periodic function whose Fourier coefficients are given by $${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(k) = {{\frac{1}{2}}_{\pi }}\int\limits_{{ - \pi }}^{\pi } {f(y){{e}^{{ - ik \bullet y}}}dy, - \infty < k < \infty .} $$ (1.1) It is well-known that whenever f is a smooth function, then its spectral approximation — consisting of the partial sums $$ {{S}_{N}}f(x) \equiv {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}}_{N}}(x) = \sum\limits_{{|k| \leqslant N}} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(k){{e}^{{ik\bullet x}}},} $$ (1.2) converges pointwise to f(x). A typical error estimate in this case, asserts that for any x in the domain we have $$ |{\text{f(x)}}\;{\text{ - }}\;{{\hat{f}}_{{\text{N}}}}({\text{x}})|\; \leqslant \;{{{\text{C}}}_{{\text{S}}}}|\,|\;{\text{f}}\;|\,{{|}_{{({\text{S}})}}}\bullet {{{\text{N}}}^{{ - {\text{S + 1}}}}},\quad {\text{s}}\;{\text{ > }}\;{\text{1}}{\text{.}} $$ (1.3) Here and below, CS stands for (possibly different) generic constant bounds, and ||f||(S) denotes the largest maximum norm of f and its first s derivatives, the maximum taken over the whole domain.

We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.

We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, { fk 5 ak 1 ibk}k51 N , we form the generalized conjugate partial sum S N s @ f #~x! 5 ¥ k51 N sS ND~aksin kx 2 bkcos kx!. The classical conjugate partial sum, SN [ f ]( x), corresponds to s [ 1 and it is known that 2p log N SN@ f #~x! converges to the jump function [ f ]( x) :5 f ( x1) 2 f ( x2); thus, 2p log N SN@ f #~x! tends to “concentrate” near the edges of f. The convergence, however, is at the unacceptably slow rate of order 2(1/log N ). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called “concentration factors,” sk,N 5 sS ND . Our main result shows that an arbitrary C[0, 1] nondecreasing s(z) satisfying *1/N 1 s~x! x dxOi N3 ` 2p leads

We extend the ideas introduced in (33) for hierarchical multiscale decompositions of images. Viewed as a function f ∈ L2(­), a given image is hierarchically decomposed into the sum or product of simpler "atoms" uk, where uk extracts more refined information from the previous scale uki1. To this end, the uk's are obtained as dyadically scaled minimizers of standard functionals arising in image analysis. Thus, starting with vi1 := f and letting vk denote the residual at a given dyadic scale, ¸k ∼ 2k, the recursive step (uk,vk) = arginf QT (vki1,¸k) leads to the desired hierarchical decomposition, f ∼ P T uk; here T is a blurring operator. We characterize such QT -minimizers (by duality) and expand our previous energy estimates of the data f in terms of k ukk . Numerical results illustrate applications of the new hierarchical multiscale decomposition for blurry images, images with additive and multiplicative noise and image segmentation.

We adopt a non-oscillatory central scheme, first presented in the con- text of Hyperbolic conservation laws in (28) followed by (15), to the framework of the incompressible Euler equations in their vorticity formulation. The embedded duality in these equations, enables us to toggle between their two equivalent rep- resentations - the conservative Hyperbolic-like form vs. the convective form. We are therefore able to apply local methods, to problems with a global nature. This results in a new stable and convergent method which enjoys high-resolution with- out the formation of spurious oscillations. These desirable properties are clearly visible in the numerical simulations we present.

20 pages; International audience; Compactness of families of solutions --- or of approximate solutions --- is a feature that distinguishes certain classes of nonlinear hyperbolic equations from the case of linear hyperbolic equations, in space dimension one. This paper shows that some classical compactness results in the context of hyperbolic conservation laws, such as the Lax compactness theorem for the entropy solution semigroup associated with a nonlinear scalar conservation laws with convex flux, or the Tartar-DiPerna compensated compactness method, can be turned into quantitative compactness estimates --- in terms of epsilon-entropy, for instance --- or even nonlinear regularization estimates. This regularizing effect caused by the nonlinearity is discussed in detail on two examples: a) the case of a scalar conservation law with convex flux, and b) the case of isentropic gas dynamics, in space dimension one.

International audience; This paper is a brief presentation of those Mean Field Games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e. with diffusion) is also presented both in the stationary and dynamic case. The variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on Augmented Lagrangian Algorithms, and some new simulations are shown at the end of the paper.

International audience; This paper deals with typical questions arising in the analysis of numerical approximations for scalar conservation laws with a source term. We focus our attention on semi-discrete finite volume schemes, in the general case of a nonuniform spatial mesh. To define appropriate discretizations of the source term, we introduce the formalism peculiar to the Upwind Interface Source method and we establish conditions on the numerical functions so that the discrete solver preserves the steady state solutions. Then we formulate a rigorous definition of consistency, adapted to the class of well-balanced schemes, for which we are able to prove a Lax-Wendroff type convergence theorem. Some examples of numerical methods are discussed, in order to validate the arguments we propose.

International audience; We present numerical simulations of the Saint-Venant system for shallow waters, including small friction and viscosity, motivated by the interest in recovering the results of experimental studies on the free-surface flows over an obstacle. We use the kinetic scheme "with reflections" formulated in [B. Perthame, C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), no. 4, 201-231], appropriately extended to obtain second order accuracy according to the theory developed in [Th. Katsaounis, C. Simeoni, First and second order error estimates for the upwind source at interface method, Math. Comp. 74 (2005), no. 249, 103-122]