optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

a nonlinear functional for general scalar hyperbolic conservation laws abstract a generalized entropy functional was introduced in t p liu t yang a new entropy functional for scalar conservation laws comm pure appl math 52 1999 1427 1442 for the scalar hyperbolic conservation laws with convex flux function this functional was crucially used in the functional approach to the l 1 stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate however how to construct the generalized entropy functional for scalar conservation laws with general flux and then how to apply the functional approach to the l 1 study on general systems are still open in this paper we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction

on the cellular convexity of complexes in this paper we discuss cellular convexity of complexes a new definition of cellular convexity is given in terms of a geometric property then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image hence the definition of cellular convexity by sklansky 7 is equivalent to the new definition for the case of regular complexes the definition of minsky and papert 4 is shown to be equivalent to our definition therefore au definitions are virtually equivalent it is shown that a regular complex is cellularly convex if and only if its minimum perimeter polygon does not meet the boundary of the complex a 0 n time algorithm is presented to determine the cellular convexity of a complex when it resides in n m cells and is represented by the run length code

the finite element method with nodes moving along the characteristics for convection diffusion equations we examine the convergence properties of the finite element method with nodes moving along the characteristics for one dimensional convection diffusion equations for linear elements we demonstrate optimal rates of convergence in thel 2 h 1 andl norms both linear and nonlinear problems are considered

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

convergence results for the flux identification in a scalar conservation law here we study an inverse problem for a quasilinear hyperbolic equation we start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function then we use a lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation despite the fact that the lagrangian formulation is formal and that the cost function is not necessarily differentiable a viscous perturbation and a numerical approximation of the problem allow us to justify this computation when the adjoint problem for the quasi linear equation admits a smooth solution then the perturbed adjoint states can be proved to converge to that very solution the sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdifferential of the cost function we evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identification of isotherms for conservation laws modeling distillation or chromatography they are illustrated by numerical examples

two a posteriori error estimates for one dimensional scalar conservation laws in this paper we propose a posteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws we first consider the schemes for which a consistent in cell entropy inequality can be derived then we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities as an illustration we show several numerical tests on the burgers equation and we propose an adaptive algorithm for the selection of the mesh

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

an error bound for the polygonal approximation of conservation laws with source terms abstract we give in this study an error bound for the polygonal approximation of hyperbolic conservation laws including a dissipative source term this is a generalization of results in the case of conservation laws without source terms to the inhomogeneous scalar conservation laws

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

l 1 framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations we develop a general l 1 framework for deriving continuous de pendence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the chen perthame kinetic approach 9 we ap ply our l 1 framework to establish an explicit estimate for continuous depen dence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method without imposition of bounded variation of the approximate solutions finally as an example of a direct application of this framework to numerical methods we focus on a linear convection diusi on model equation and derive an l 1 error estimate for an upwind central finite dierence scheme

improved error estimation of dynamic finite element methods for second order parabolic equations dynamic finite element schemes are analyzed for second order parabolic problems these schemes permit different finite element spaces at different time levels in order to efficiently capture time changing localized phenomena such as moving sharp fronts or layers the dynamical change of grids and interpolation polynomials is necessary and essential to many large scale transient problems standard characteristic and mixed finite element methods with dynamic function spaces are considered for linear and nonlinear problems in a unified framework with improved a priori error estimates and convergence results

a best approximation property of the moving finite element method the moving finite element method mfe for the solution of time dependent partial differential equations pdes is a numerical solution scheme which allows the automatic adaption of the finite element approximation space with time an analysis of how this method models the steady solutions of a general class of parabolic linear source equations is presented it is shown that under certain conditions the steady solutions of the mfe problem can correspond to best free knot spline approximations to the true steady solution of the differential equation when using the natural norm associated with the problem hence a quantitative measure of the advantages of the mfe method over the usual fixed grid galerkin method is produced for these equations a number of numerical examples are included to illustrate these results

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

systems of hyperbolic conservation laws with a resonant moving source in this paper we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source compared with the previous results w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 t p liu nonlinear stability and instability of transonic flows through a nozzle comm math phys 83 2 1982 243 260 on this stability problem in this paper the transonic ith shock is assumed to be relatively strong and stable in the sense of majda then the framework of m lewicka l1 stability of patterns of non interacting large shock waves indiana univ math j 49 4 2000 1515 1537 m lewicka stability conditions for patterns of noninteracting large shock waves siam j math anal 32 5 2001 1094 1116 electronic can be applied a new criterion is obtained to test whether such a shock is time asymptotically stable or not and by constructing the liu yang functional one can prove the l1 stability of the shock under the stability condition this is an extension of the result s y ha t yang l1 stability for systems of hyperbolic conservation laws with a resonant moving source siam j math anal 34 5 2003 1226 1251 electronic w c lien hyperbolic conservation laws with a moving source comm pure appl math 52 9 1999 1075 1098 to a more general case

a class of n nonlinear hyperbolic conservation laws the riemann problem for a class of nonlinear systems of first order hyperbolic conservation laws is studied the class consists of systems where the derivative of the flux function is a lower triangular matrix there are no assumptions on genuine nonlinearity and strict hyperbolicity existence and uniqueness are proved except in a set with measure zero in the phase space and a set with measure zero in the flux function space where there is a one parameter family of solutions travelling waves are used as an entropy condition and examples show that the lax or liu entropy conditions are not sufficient an example shows that the solution does not necessarily depend continuously on the data the model may be used to describe three phase and tracer flow and flow in a neighborhood of a heterogeneity in porous media

on the convergence rate of operator splitting for hamilton jacobi equations with source terms we establish a rate of convergence for a semidiscrete operator splitting method applied to hamilton jacobi equations with source terms the method is based on sequentially solving a hamilton jacobi equation and an ordinary differential equation the hamilton jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit euler method we prove that the l infty error associated with the operator splitting method is bounded by mathcal o delta t where delta t is the splitting or time step this error bound is an improvement over the existing mathcal o sqrt delta t bound due to souganidis nonlinear anal 9 1985 pp 217 257 in the one dimensional case we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous hamilton jacobi equations it is proved that this fully discrete splitting method possesses a linear convergence rate moreover numerical results are presented to illustrate the theoretical convergence results

on nonlocal monotone difference schemes for scalar conservation laws analyse d erreur d une classe de methodes aux differences finies monotones pour la solution approchee des lois de conservation hyperboliques

stability of approximate solutions constructed by the wave front tracking method abstract we prove the stability of approximate solutions constructed by the wave front tracking method for the cauchy problem of scalar conservation laws the ideas of the proof are based upon an application of the proof technique of the uniqueness theorem to the entropy solution given by kru kov and a local generalization of the method by liu and yang

viscous splitting approximation of mixed hyperbolic parabolic convection diffusion equations we first analyse a semi discrete operator splitting method for nonlinear possibly strongly degenerate convection diffusion equations due to strong degeneracy solutions can be discontinuous and are in general not uniquely determined by their data hence weak solutions satisfying an entropy condition are sought we then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic parabolic convection diffusion equations

existence and uniqueness of entropy solution to initial boundary value problem for the inviscid burgers equation this paper is concerned with the existence and uniqueness of the entropy solution to the initial boundary value problem for the inviscid burgers equation to apply the method of vanishing viscosity to study the existence of the entropy solution we first introduce the initial boundary value problem for the viscous burgers equation and as in evans 1998 partial differential equations providence ri american mathematical society and hopf 1950 commun pure appl math 3 201 30 give the formula of the corresponding viscosity solutions by hopf cole transformation secondly we prove the convergence of the viscosity solution sequences and verify that the limiting function is an entropy solution finally we give an example to show how our main result can be applied to solve the initial boundary value problem for the burgers equation

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

the moving grid method for nonlinear hyperbolic conservation laws numerically moving grids are often found to improve the resolution of approximate solutions to nonlinear hyperbolic conservation laws in this paper we show how to modify many standard numerical flux functions to incorporate this idea a generalized godunov type scheme is shown in the scalar case to converge to the entropy satisfying solution as the mesh size tends to zero a courant number of one half is found necessary in general and proved to be sufficient to obtain convergence convergence for methods based upon any two point conservative and monotone numerical flux function is also shown in the scalar case

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

stability of solutions of quasilinear parabolic equations abstract we bound the difference between solutions u and v of u t a u div x f h and v t b v div x g k with initial data and respectively by u t v t l p e a e t l r n 2 p b t a b x f x g f u g u h k p e p here all functions a f and h are smooth and bounded and may depend on u x r n and t the functions a and h may in addition depend on u identical assumptions hold for the functions that determine the solutions v furthermore e r n is assumed to be a bounded set and p and p are fractions that depend on n and p the diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth

continuous dependence on the nonlinearities of solutions of degenerate parabolic equations explicit estimates for the continuous dependence inl 0 t l1 rd of solutions of the equation lv v v in 0 rdwith initial conditionv 0 h with respect to the nonlinear continuously differential functions and are established

solving a class of lp problems with a primal dual logarithmic barrier method applying a higher order primal dual logarithmic barrier method for solving large real life linear programming problems is addressed in this paper the efficiency of an interior point algorithm on these problems is compared with the one of the state of the art simplex code minos version 5 3 based on such experience a wide class of lp problems is identified for which logarithmic barrier approach seems advantageous over the simplex one additionally some practical rules for model builders are derived that should allow them to create problems that can easily be solved with logarithmic barrier algorithms

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

on the stability functional for conservation laws this note is devoted to the explicit construction of a functional defined on all pairs of l 1 functions with small total variation which is equivalent to the l 1 distance and non increasing along the trajectories of a given system of conservation laws the present definition of this functional does not need any construction of approximate solutions

an improved front tracking method for the euler equations an improved front tracking method for hyperbolic conservation laws is presented the improved method accurately resolves discontinuities as well as continuous phenomena the method is based on an improved front interaction model for a physically more accurate modeling of the euler equations as compared to standard front tracking methods the resulting algorithm is also more efficient than existing front tracking methods the improved front tracking method is applied to the euler equations for one dimensional unsteady flow and two dimensional steady supersonic flow the results are compared to results of a standard front tracking method and a finite volume method

a new nonlinear functional for general scalar conservation laws abstract the approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well posedness theories of hyperbolic conservation laws especially in one space dimensional case in particular a generalized entropy functional was constructed in t p liu t yang a new entropy functional for scalar conservation laws comm pure appl math 52 1999 1427 1442 for the l 1 stability of weak solutions however this generalized functional is so far only defined for scalar equations with convex flux function in this paper we introduce a new nonlinear functional which is motivated by the new glimm functional introduced in j l hua z h jiang t yang a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws preprint for general scalar conservation laws this functional improves the one given in h x liu t yang a nonlinear functional for general scalar hyperbolic conservation laws j differential equations 235 2 2007 658 667 and it can be viewed as a better attempt for the generalized entropy functional for general equations

a stable adaptive numerical scheme for hyperbolic conservation laws a new adaptive finite difference scheme for scalar hyperbolic conservation laws is introduced a key aspect of the method is a new automatic mesh selection algorithm for problems with shocks we show that the scheme is l 1 stable in the sense of kuznetsov and that it generates convergent approximations for linear problems numerical evidence is presented that indicates that if an error of size varepsilon is required our scheme takes at most o varepsilon 3 operations standard monotone difference schemes can take up to o varepsilon 4 calculations for the same problems

piecewise self similar solutions and a numerical scheme for scalar conservation laws the solution of the riemann problem was a building block for general cauchy problems in conservation laws a cauchy problem is approximated by a series of riemann problems in many numerical schemes but since the structure of the riemann solution holds locally in time only and furthermore a riemann solution is not piecewise constant in general there are several fundamental issues in this approach such as the stability and the complexity of computation in this article we introduce a new approach which is based on piecewise self similar solutions the scheme proposed in this article solves the problem without the time marching process the approximation error enters in the step for the initial discretization only which is given as a similarity summation of base functions the complexity of the scheme is linear convergence to the entropy solution and the error estimate are shown the mechanism of the scheme is introduced in detail together with several interesting properties of the scheme

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

a new entropy functional for a scalar conservation law in this paper we introduce a new entropy functional for a scalar convex conservation law that generalizes the traditional concept of entropy of the second law of thermodynamics the generalization has two aspects the new entropy functional is defined not for one but for two solutions it is defined in terms of the l1 distance between the two solutions as well as the variations of each separate solution in addition it is decreasing in time even when the solutions contain no shocks and is therefore stronger than the traditional entropy even in the case when one of the solutions is zero 1999 john wiley sons inc

some results on fuzzy digital convexity abstract the definition of fuzzy convexity is reviewed and some results on projections of convex and fuzzy convex sets are established digital fuzzy convexity is defined and relationships among alternative definitions are investigated

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound we consider 3 point numerical schemes that resolve scalar conservation laws that are oscillatory either to their dispersive or anti diffusive nature the spatial discretization is performed over non uniform adaptively redefined meshes we provide a model for studying the evolution of the extremes of the oscillations we prove that proper mesh reconstruction is able to control the oscillations we provide bounds for the total variation tv of the numerical solution we moreover prove under more strict assumptions that the increase of the tv due to the oscillatory behavior of the numerical schemes decreases with time hence proving that the overall scheme is tv increase decreasing tvi d we finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique 1 outline mesh adaptation techniques have been employed by several authors in the past it is worth noting the seminal works dd87 for88 hh83 luc85 luc86 tt03 where several properties of mesh adaptation were studied it was noticed in akm01 amt04 ams10 sfa09 that proper use of non uniform adaptively redefined meshes is capable of taming oscillations hence improving the stability properties of the numerical schemes to study the stabilization properties of mesh adaptation techniques we analyze the effect they have on the oscillations that oscillatory unstable numerical schemes produce the setting is the one dimensional scalar riemann problem ut f u x 0 x a b with the flux function f being smooth and convex for initial conditions we consider the single jump u0 x x 0 x0 x with x x0 0 1 we discretize spatially over a non uniform adaptively redefined mesh the mesh adaptation and the time evolution of the numerical solution are combined into the main adaptive scheme definition 1 1 main adaptive scheme mas given at time step n the mesh m x a x1 xn b and the approximations u u1 un the steps of the mas are 1 mesh reconstruction construct new mesh m x a x 1 x n b 2 solution update use the old mesh m x the approximations u n and the new mesh m x received by the editor september 19 2009 and in revised form september 2 2011 2010 mathematics subject classification primary 65 xx c 2012 american mathematical society reverts to public domain 28 years from publication

qhopdm a higher order primal dual method for large scale convex quadratic programming in 1992 we prepared hopdm a modularly structured library of fortran subroutines for large scale linear optimization the code has been developed to be more effective by gondzio further we broadened this software to use it for a quadratic programming problem generated by a cost effective sulphur emission reduction model success with a real life model induced us to prepare a more general quadratic solver we followed other authors and extended our lp procedure to the case of convex quadratic objectives now we are presenting qhopdm a library for convex quadratic optimization with linear constraints

a front tracking method for conservation laws in one dimension abstract we present a front tracking technique for conservation laws in one dimension the method is based on approximations to the solution of riemann problems where the solution is represented by piecewise constant states separated by discontinuities the discontinuities are tracked until they interact at this point a new riemann problem is solved and so on no finite differences are used this method is tested on the system of nonstationary gas dynamics defined by the euler equations and three test cases are presented

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

a priori error estimates for numerical methods for scalar conservation laws part i the general approach we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original kuznetsov approximation theory as a first application of this general technique we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution thus we obtain optimal error estimates for the engquist osher scheme without using the fact i that the solution is uniformly bounded ii that the scheme is total variation diminishing and iii that the discrete semigroup associated with the scheme has the l 1 contraction property which guarantees an upper bound for the modulus of continuity in time of the approximate solution

hopdm version 2 12 a fast lp solver based on a primal dual interior point method hopdm is an implementation of the primal dual interior point method for solving large scale linear programming lp problems hopdm stands for higher order primal dual method its newest version 2 12 of may 1995 is a robust and efficient code that compares favourably with today s commercial lp solvers

a numerical method for first order nonlinear scalar conservation laws in one dimension abstract a numerical method for first order nonlinear scalar hyperbolic conservation laws in one dimension is presented using an idea by dafermos in this paper it is proved that it may be used as a numerical method for a general flux function and a general initial value it is possible to give explicit error estimates for the numerical method the error in the method is far smaller than in any other method the numerical method is illustrated in an example

convergence of a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper a new explicit finite element method for numerically solving the drift diffusion semiconductor device equations is introduced and analyzed the method uses a mixed finite element method for the approximation of the electric field a finite element method using discontinuous finite elements is used to approximate the concentrations which may display strong gradients the use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable in this paper we carry out the analysis of the model method which employs a continuous piecewise linear approximation to the electric field and a piecewise constant approximation to the electron concentration in a model problem namely the so called unipolar case with the diffusion terms neglected the resulting system of equations is equivalent to a conservation law whose flux the electric field depends globally on the solution the concentration of electrons by exploiting the similarities of this system with classical scalar conservation laws the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme the scheme considered as a scheme for the electron concentration is shown to satisfy a maximum principle and to be total variation bounded its convergence to the unique weak solution is proven numerical experiments displaying the performance of the scheme are shown

efficient implementation of the shock fitting algorithm for the lighthill whitham richards traffic flow model this paper firstly presents the existence and uniqueness properties of the intersection time between two neighboring shocks or between a shock and a characteristic for the analytical shock fitting algorithm that was proposed to solve the lighthill whitham richards lwr traffic flow model with a linear speed density relationship in accordance with the monotonicity properties of density variations along a shock which have greatly improved the robustness of the analytical shock fitting algorithm then we discuss the efficient evaluation of the measure of effectiveness moe of the analytical shock fitting algorithm we develop explicit expressions to calculate the moe which is the total travel time that is incurred by travelers within the space time region that is encompassed by the shocks and or characteristic lines a numerical example is used to illustrate the effectiveness and efficiency of the proposed method compared with the numerical solutions that are obtained by a fifth order weighted essentially non oscillatory scheme

error estimates for a finite element method for the drift diffusion semiconductor device equations the zero diffusion case in this paper new error estimates for an explicit finite element method for numerically solving the so called zero diffusion unipolar model a one dimensional simplified version of the drift diffusion semiconductor device equations are obtained the method studied in a previous paper combines a mixed finite element method using a continuous piecewise linear approxima tion of the electric field with an explicit upwinding finite element method using a piecewise constant approximation of the electron concentration by using a suitable extension of kuznetsov approximation theory for scalar conservation laws it is proved that under proper hypotheses on the data the l j error between the approximate and exact electron concentrations of the zero diffusion unipolar model is of order ax1 2 these estimates are sharp

second order front tracking for the euler equations a second order front tracking method is developed for solving the hyperbolic system of euler equations of inviscid fluid dynamics numerically meshless front tracking methods are usually limited to first order accuracy since they are based on a piecewise constant approximation of the solution here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution the linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately in order to construct a physically correct linearization the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model this front interaction model is also extended to include front numbers used in the wave decomposition it is illustrated numerically for sod s riemann problem the two interacting blast waves problem and a two dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions