Results 1  10
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22
Local Error Estimates For Discontinuous Solutions Of Nonlinear Hyperbolic Equations
 SIAM J. Numer. Anal
, 1992
"... . Let u(x; t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u " (x; t) is the solution of an approximate viscosity regularization, where " ? 0 is the small viscosity amplitude. We show that by postprocessing the small viscosi ..."
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Cited by 66 (17 self)
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. Let u(x; t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u " (x; t) is the solution of an approximate viscosity regularization, where " ? 0 is the small viscosity amplitude. We show that by postprocessing the small viscosity approximation u " , we can recover pointwise values of u and its derivatives with an error as close to " as desired. 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) Econdition of the forward problem in order to deduce a W 1;1 energy estimate for the discontinuous backward transport equation; this, in turn, leads us to "uniform estimate on moments of the error u " \Gamma u. Our approach does not `follow the characteristics' and, therefore, applies mutatis mutandis to other approximate solutions such as Edi...
Approximate Solutions of Nonlinear Conservation Laws and Related Equations
, 1997
"... During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical ..."
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Cited by 36 (11 self)
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During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^1compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools  the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finitedifference schemes; error estimates derived from the onesided stability of Godunovtype methods for convex conservation laws (and their multidimensional analogue  viscosity solutions of demiconcave HamiltonJacobi equations); we outline, in the onedimensional case, the convergence proof of finiteelement streamlinediffusion and spectral viscosity schemes based on the divcurl lemma; we also address the questions of convergence and error estimates for multidimensional finitevolume schemes on nonrectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finitevolume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.
The Convergence Rate Of Approximate Solutions For Nonlinear Scalar Conservation Laws
, 1992
"... . Let fv " (x; t)g "?0 be a family of approximate solutions for the nonlinear scalar conservation law u t + f(u)x = 0 with C 1 0 initial data. Assume that fv " (x; t)g are Lip + stable in the sense that they satisfy Oleinik's Eentropy condition. It is shown that if these approximate solut ..."
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Cited by 33 (13 self)
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. Let fv " (x; t)g "?0 be a family of approximate solutions for the nonlinear scalar conservation law u t + f(u)x = 0 with C 1 0 initial data. Assume that fv " (x; t)g are Lip + stable in the sense that they satisfy Oleinik's Eentropy condition. It is shown that if these approximate solutions are Lip 0 consistent, i.e., if kv " (\Delta; 0) \Gamma u(\Delta; 0)k Lip 0 (x) + kv " t + f(v " )x k Lip 0 (x;t) = O("), then they converge to the entropy solution, and the convergence rate estimate kv " (\Delta; t) \Gamma u(\Delta; t)k Lip 0 (x) = O(") 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 finitedifference and Glimm's schemes. Key Words. Conservation laws, entropy stability, weak consistency, error estimates,postprocessing, finitedifference approximations, Glimm scheme AMS(MOS) subject classification. 35L65, 65M10,65M15. 1. Intro...
HighResolution Nonoscillatory Central Schemes For HamiltonJacobi Equations
"... In this paper, we construct secondorder central schemes for multidimensional HamiltonJacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory secondorder Godunovtype schemes based ..."
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Cited by 28 (5 self)
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In this paper, we construct secondorder central schemes for multidimensional HamiltonJacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory secondorder Godunovtype schemes based on global projection operators. Numerical experiments are performed; L 1 /L # errors and convergence rates are calculated. For convex Hamiltonians, numerical evidence confirms that our central schemes converge with secondorder rates, when measured in the L 1 norm advocated in our recent paper [Numer Math, to appear]. The standard L # norm, however, fails to detect this secondorder rate.
Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients
 Math. Comp
, 2000
"... Abstract. Conservative linear equations arise in many areas of application, including continuum mechanics or highfrequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In ..."
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Cited by 17 (1 self)
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Abstract. Conservative linear equations arise in many areas of application, including continuum mechanics or highfrequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finitedifference numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied. 1.
A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems
, 1997
"... this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational ..."
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Cited by 17 (4 self)
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this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational implementation of the a posteriori error bounds into adaptive finite element algorithms
Uniqueness Via The Adjoint Problem For Systems Of Conservation Laws
, 1993
"... . We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws, that includes in particular the psystem of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy ..."
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Cited by 13 (5 self)
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. We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws, that includes in particular the psystem of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition or WEC, for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (resp. forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. Content 1. Introduction 2. Statement of the main results 3. Uniqueness via existence for the backward adjoint problem 4. Uniqueness via uniqueness for the forward adjoint problem 5. Linear hyperbolic systems w...
Shock Capturing, Level Sets and PDE Based Methods in Computer Vision and Image Processing: A Review of Osher's Contributions
 J. Comput. Phys
, 2001
"... In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start wit ..."
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Cited by 12 (0 self)
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In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the EngquistOsher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for HamiltonJacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to HamiltonJacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.
On the piecewise smoothness of entropy solutions to scalar conservation laws, Commun
 in PDEs
, 1993
"... The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well kn0a.n that such entropy solutioris consist of at most countable number of C1smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of ..."
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Cited by 12 (2 self)
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The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well kn0a.n that such entropy solutioris consist of at most countable number of C1smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C1smoothness regions. These bounds enable us to measure the hzgh order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cnsemi norm localized to the smooth part of the entropy solution. and we show that the entropy solution is stable with respect to this norm. \Ye also address the question regarding the number of C1smoothness pieces, we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of pzecewise smoothness which is assumed sometime implicitly in many finitedimensional computations for such discontinuous problems.
On wavewise entropy inequalities for highresolution schemes I : the semidiscrete case
 Math. of Comput
, 1996
"... The goal of this supplement is to prove the third WEI criterion, Theorem 3.13. We argue by contradiction: Assume that a certain selfsimilar scheme (1.2)(1.4) satisfying Assumption 3.3 fails to converge. We need to show that there exists a sequence of numerical solutions {uk} of the scheme that har ..."
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Cited by 4 (0 self)
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The goal of this supplement is to prove the third WEI criterion, Theorem 3.13. We argue by contradiction: Assume that a certain selfsimilar scheme (1.2)(1.4) satisfying Assumption 3.3 fails to converge. We need to show that there exists a sequence of numerical solutions {uk} of the scheme that harbors an ATES. This highly nontrivial task is fulfilled in three steps, which extend from §6 to §8. First, in §6 we develop a new technique, the extremum tracking. We introduce the notions of extremum paths and approximate extremum paths of a numerical solution of the aforementioned scheme. It turns out that a sequence of approximate extremum paths associated with a sequence of numerical solutions in � Ψw−,w+,B is necessarily a sequence of {εk}–paths, a building block of the ATES. Next, in §7, we analyze the asymptotic waves of the sequences of numerical solutions in a nonempty � Ψw−,w+,B. Roughly speaking, a wave of a numerical solution is the generic structure of the numerical solution between two ( approximate) extremum paths, and an asymptotic wave is a sequence of waves of a sequence of numerical solutions. Using similarity transforms and selecting subsequences, we show that there exists a sequence in � Ψw−,w+,B such that in a compact domain, the transition regions and the boundaries of all sufficiently strong asymptotic waves converge to x = st, which is the path of the discontinuity of the limit W of � Ψw−,w+,B. Hence, these asymptotic waves must be ATWs. We then split these ATWs into ATDs. Finally, in §8, using Theorem 3.2 and Lemma 3.10, we complete the proof of Theorem 3.13 by showing that if W is a traveling expansion shock, then one of the ATDs must be an ATES. 6. EXTREMUM TRACKING In this section we introduce and analyze the notions of extremum paths and approximate extremum paths of a numerical solution u. Theses paths will serve as the boundaries of the transition regions of the waves of the numerical solution. We begin with the following simple observation: Lemma 6.1. If u is a uniformly bounded numerical solution of a scheme (1.2)(1.4), then there are two positive constants C1 and C2 such that (6.1)  duj(t)