adaptive multiresolution approach for two dimensional pdes a multiresolution adaptive approach for the solution of two dimensional partial differential equations pdes is presented this methodology is a multidimensional extension of that presented in a previous work comput methods appl mech engrg 191 2002 3909 the method proposed is unconditionally stable by incorporating convection differencing schemes with the tvd property and the grid is dynamically adapted so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed the two desired properties of any pde solver stability and accuracy are therefore retained numerical results for four test problems are presented which serve to demonstrate the robustness and cost effectiveness of the method

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

a smoothness indicator for adaptive algorithms for hyperbolic systems the formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed in this paper a new smoothness indicator si is used to identify rough solution regions and is implemented in locally adaptive algorithms the si is based on the weak local truncation error of the approximate solution it was recently reported in s karni and a kurganov local error analysis for approximate solutions of hyperbolic conservation laws where error analysis and convergence properties were established the present paper is concerned with its implementation in scheme adaption and mesh adaption algorithms the si provides a general framework for adaption and is not restricted to a particular discretization scheme the implementation in this paper uses the central upwind scheme of a kurganov s noelle and g petrova siam j sci comput 23 707 2001 the extension of the si to two space dimensions is given numerical results in one and two space dimensions demonstrate the robustness of the proposed si and its potential in reducing computational costs and improving the resolution of the solution

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

linear stochastic models of nonlinear dynamical systems we investigate in this work the validity of linear stochastic models for nonlinear dynamical systems we exploit as our basic tool a previously proposed rayleigh ritz approximation for the effective action of nonlinear dynamical systems started from random initial conditions the present paper discusses only the case where the probability density function ansatz employed in the variational calculation is markovian i e is determined completely by the present values of the moment averages in this case we show that the rayleigh ritz effective action of the complete set of moment functions that are employed in the closure has a quadratic part which is always formally an onsager machlup action thus subject to satisfaction of the requisite realizability conditions on the noise covariance a linear langevin model will exist which reproduces exactly the joint two time correlations of the moment functions we compare our method with the closely related formalism of principal oscillation patterns pop which in the approach of penland is a method to derive such a linear langevin model empirically from time series data for the moment functions the predictive capability of the pop analysis compared with the rayleigh ritz result is limited to the regime of small fluctuations around the most probable future pattern finally we shall discuss a thermodynamics of statistical moments which should hold for all dynamical systems with stable invariant probability measures and which follows within the rayleigh ritz formalism

the common randomness capacity of a network of discrete memoryless channels we generalize our previous results on generating common randomness at two terminals to a situation where any finite number of agents interconnected by an arbitrary network of independent point to point discrete memoryless channels wish to generate common randomness by interactive communication over the network our main result is an exact characterization of the common randomness capacity of such a network i e the maximum number of bits of randomness that all the agents can agree on per step of communication as a by product we also obtain a purely combinatorial result viz a characterization of the incidence vectors of the spanning arborescences rooted at a specified vertex in a digraph and having exactly one edge exiting the root as precisely the extreme points of a certain unbounded convex polyhedron described by a system of linear inequalities

nonoscillatory high order accurate self similar maximum principle satisfying shock capturing schemes i this is the first paper in a series in which a class of nonoscillatory high order accurate self similar local maximum principle satisfying in scalar conservation law shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed in this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented using scalar one dimensional initial value problems as a model for this model the schemes are made to satisfy a local maximum principle and a nonoscillatory property the method uses a simple centered stencil with quadratic reconstruction followed by two modifications imposed as needed the first enforces a local maximum principle the second guarantees that no new extrema develop the schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size i e there are no grid size dependent limits involving free parameters as in e g math comp 49 1987 pp 105 121 math comp

application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes we consider the numerical solution of the shallow water equations on unstructured grids we focus on flows over wet areas the extension to the case of dry bed will be reported elsewhere the shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding in this case with the total energy our aim is to show the application of a particular class of conservative residual distribution rd schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties we give a review of conservative rd schemes showing relations between different approaches previously published and recall l stability and accuracy criteria characterizing the schemes in particular the accuracy of the rd method in presence of source terms is analyzed and conditions to construct rth order discretizations on irregular triangular grids are proved it is shown that the rd approach gives a natural way of obtaining high order discretizations which moreover preserves exactly the steady lake at rest solution independently on mesh topology nature of the variation of the bottom and polynomial order of interpolation used for the unknowns we also consider more general analytical solutions which are less investigated from the numerical view point on irregular grids linearity preserving rd schemes yield a truly second order approximation we also sketch a strategy to achieve discretizations which preserve exactly some of these solutions numerical results on steady and time dependent problems involving smooth and non smooth variations of the bottom topology show very promising features of the approach

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

identification via channels the authors main finding is that any object among doubly exponentially many objects can be identified in blocklength n with arbitrarily small error probability via a discrete memoryless channel dmc if randomization can be used for the encoding procedure a novel doubly exponential coding theorem is presented which determines the optimal r that is the identification capacity of the dmc as a function of its transmission probability matrix this identification capacity is a well known quantity namely shannon s transmission capacity for the dmc

identification in the presence of feedback a discovery of new capacity formulas a study is made of the identification problem in the presence of a noiseless feedback channel and the second order capacity c sub f resp c sub f for deterministic resp randomized encoding strategies is determined several important phenomena are encountered 1 although feedback does not increase the transmission capacity of a discrete memoryless channel dmc it does increase the second order identification capacity 2 noise increases c sub f 3 the structure of the new capacity formulas is simpler than c e shannon s 1948 familiar formula this has the effect that proofs of converses become easier than in the authors previous work

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

multiresolution based on weighted averages of the hat function ii nonlinear reconstruction techniques in this paper we describe and analyze a class of nonlinear mr schemes for the multiresolution setting which corresponds to discretization by local averages with respect to the hat function these schemes are based on the essentially non oscillatory eno interpolatory procedure described in a harten b engquist s osher and s chakravarthy j comput phys 71 1987 pp 231 302 we show that by allowing the approximation to fit the local nature of the data one can improve the compression capabilities of the multiresolution algorithms the question of stability for nonlinear data dependent reconstruction techniques is also addressed

matrix exponential based closures for the turbulent subgrid scale stress tensor two approaches for closing the turbulence subgrid scale stress tensor in terms of matrix exponentials are introduced and compared the first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials this formal solution of the subgrid scale stress transport equation is shown to be useful to explore special cases such as the response to constant velocity gradient but neglecting pressure strain correlations and diffusion effects the second approach is based on an eulerian lagrangian change of variables combined with the assumption of isotropy for the conditionally averaged lagrangian velocity gradient tensor and with the recent fluid deformation approximation it is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose short time expansions of the matrix exponentials are shown to provide an eddy viscosity term and particular quadratic terms and thus allow a reinterpretation of traditional eddy viscosity and nonlinear stress closures the basic feasibility of the matrix exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence the matrix exponential closure employs the drastic approximation of entirely omitting the pressure strain correlation and other nonlinear scrambling terms but unlike eddy viscosity closures the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term and using physically motivated assumptions about lagrangian decorrelation and upstream isotropy

error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method in this paper we derive a priori and a posteriori error estimates for cell centered finite volume approximations of nonlinear conservation laws on polygonal bounded domains numerical experiments show the applicability of the a posteriori result for the derivation of local adaptive solution strategies

identification plus transmission over channels with perfect feedback we determine the region of all identification and transmission rate pairs achievable over a discrete memoryless channel dmc with perfect and instantaneous feedback for both randomized and deterministic encoding as a by product we also have a new proof of kemperman s 1973 strong converse to shannon s coding theorem for dmc s with feedback

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

nonconservative hybrid shock capturing schemes we examine some efficient numerical approximations for hyperbolic systems of conservation laws the approximations are constructed by hybridizing simple accurate centered difference schemes for use in smooth regions with sophisticated shock capturing schemes for use only in narrow zones near shocks and other singularities the switching strategies we consider are very flexible in allowing one to choose schemes independently for different regions of the flow the resulting hybrid schemes need not be conservative but numerical examples in one dimension demonstrate that if the switching strategy is cautious e g if switching is prohibited too close to shocks then high accuracy can be achieved for both shock speeds and smooth regions for one switching strategy in particular it is easy to prove convergence to the entropy solution

a projection method for locally refined grids a numerical method for the solution of the two dimensional euler equatons for incompressible flow on locally refined grids is presented the method is a second order godunov projection method adapted from bell colella and glaz second order accuracy of the numerical method in time and space is established through numerical experiments the main contributions of this work concern the formulation and implementation of a projection for refined grids a discussion of the adjointness relation between gradient and divergence operators for a refined grid mac projection is presented and a refined grid approximate projection is developed an efficient multigrid method which exactly solves the projection is developed and a method for casting certain approximate projections as mac projections on refined grids is presented

predictive turbulence modeling by variational closure we show that a variational implementation of probability density function pdf closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible this possibility exists because of realizability conditions on effective potential functions for general turbulence statistics these potentials measure the cost for fluctuations to occur away from the ensemble mean value in empirical time averages of the given variable and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system navier stokes dynamics approximations of the effective potentials can be calculated within pdf closures by an efficient rayleigh ritz algorithm the failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged the systematic use of these novel realizability conditions within pdf closures is shown in a simple 3 mode system of lorenz to result in a statistically improved predictive ability in certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche