suppression of oscillations in godunov s method for a resonant non strictly hyperbolic system we prove the stability of the 2 x 2 godunov numerical method in a resonant nonlinear system of conservation laws the system we study provides one of the simplest settings in which wave speeds can coincide in a nonlinear problem namely an inhomogeneous equation in which the inhomogeneity is treated as an unknown variable ut f a u x o at o this is a model for resonance in more complicated systems such as transonic flow in a variable area duct and certain resonant problems in multiphase flow and elasticity we show that the total variation of the conserved quantities can grow at most linearly in weak solutions generated by the 2 2 godunov method under the assumption that a x satisfies the threshold smoothness condition that the total variation of a x is finite we show by counterexample that the condition is sharp in the sense that there is no bound on the growth rate based on the c norm of this variable this is the most complicated setting in which the stability of the 2 2 godunov method has been demonstrated and our results provide the first such result for a numerical method that is essentially based on the solution of the riemann problem for a resonant non strictly hyperbolic system the solution of the riemann problem is more interesting when wave speeds coincide because the coordinate system of riemann invariants is singular and there exists a multiplicity of possible time asymptotic wave patterns for solutions as a consequence numerical methods that are based on the solution of the riemann problem can introduce spurious oscillations in the approximate solutions indeed counterexamples show that the total variation in u of the waves in the riemann problem step of the 2 x 2 godunov method can tend to infinity as ax 0 even when the initial total variation in u is zero thus our results verify that the averaging step in the godunov method wipes out the numerical oscillations that can occur in the riemann problem solution step of the method we interpret this as demonstrating that numerical methods based on the riemann problem are viable in this non strictly hyperbolic setting

optimal results on tv bounds for scalar conservation laws with discontinuous flux abstract this paper is concerned with the total variation of the solution of scalar conservation law with discontinuous flux in one space dimension one of the main unsettled questions concerning conservation law with discontinuous flux was the boundedness of the total variation of the solution near interface in 1 it has been shown by a counter example at t 1 that the total variation of the solution blows up near interface but in that example the solution become of bounded variation after time t 1 so the natural question is what happens to the bv ness of the solution for large time here we give a complete picture of the bounded variation of the solution for all time for a uniform convex flux with only l data we obtain a natural smoothing effect in bv for all time t t 0 also we give a counter example even for a bv data to show that the assumptions which have been made are optimal

interactive natural image segmentation via spline regression this paper presents an interactive algorithm for segmentation of natural images the task is formulated as a problem of spline regression in which the spline is derived in sobolev space and has a form of a combination of linear and green s functions besides its nonlinear representation capability one advantage of this spline in usage is that once it has been constructed no parameters need to be tuned to data we define this spline on the user specified foreground and background pixels and solve its parameters the combination coefficients of functions from a group of linear equations to speed up spline construction k means clustering algorithm is employed to cluster the user specified pixels by taking the cluster centers as representatives this spline can be easily constructed the foreground object is finally cut out from its background via spline interpolation the computational complexity of the proposed algorithm is linear in the number of the pixels to be segmented experiments on diverse natural images with comparison to existing algorithms illustrate the validity of our method

averaging dynamic equations on time scales abstract the aim of this paper is to generalize the classical theorems on averaging of differential equations we focus on dynamic equations on time scales and prove both periodic and nonperiodic version of the averaging theorem as well as a related theorem on the existence of periodic solutions

massera s theorems for various types of equations with discontinuous solutions abstract we present new massera type theorems for various types of equations with periodic right hand sides we deal with generalized ordinary differential equations measure differential equations impulsive equations all of which might have discontinuous solutions as well as dynamic equations on time scales for scalar nonlinear equations we find sufficient conditions guaranteeing that each bounded solution is asymptotic to a periodic solution for linear systems we show that the existence of a bounded solution implies the existence of a periodic solution we include some examples to illustrate our results

stability of a resonant system of conservation laws modeling polymer flow with gravitation abstract we prove l 1 uniqueness and stability for a resonant 2 2 system of conservation laws that arise as a model for two phase polymer flow in porous media the analysis uses the equivalence of the eulerian and lagrangian formulation of this system and the results are first established for an auxiliary scalar equation our methods are based on front tracking approximations for the auxiliary equation and the kru kov entropy condition for scalar conservation laws

convergence of a difference scheme for conservation laws with a discontinuous flux convergence is established for a scalar finite difference scheme based on the godunov or engquist osher eo flux for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient other works in this direction have established convergence for methods employing the solution of 2 2 riemann problems the algorithm discussed here uses only scalar riemann solvers satisfaction of a set of kruzkov type entropy inequalities is established for the limit solution from which geometric entropy conditions follow assuming a piecewise constant coefficient it is shown that these conditions imply l1 contractiveness for piecewise c1 solutions thus extending a well known theorem

a model of continuous sedimentation of flocculated suspensions in clarifier thickener units the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation consolidation processes of flocculated suspensions in clarifier thickener units this model appears in two variants for cylindrical and variable cross sectional area units respectively models 1 and 2 in both cases the governing equation is a scalar strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable the initial value problem for this equation is analyzed for model 1 we introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition a limited analysis of steady states as desired stationary modes of operation is performed numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier thickeners

the immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources we extend the immersed interface method of leveque and li to find numerical solutions of one dimensional parabolic partial differential equations of the form u t beta x t u x x lambda x t u x kappa u u x f x where beta u beta u x and f may have known discontinuities at a known location x alpha at each time step a large weakly nonlinear system is set up using a difference scheme which is standard away from x alpha and which is derived for grid points near alpha by solving small linear systems which are determined from the jumps at x alpha the time stepping is done with a crank nicholson scheme and the nonlinear systems are solved with a levenberg marquardt method as an example we consider the flow of cars on a one lane highway with an entrance or exit where traffic is treated as a continuous fluid numerical examples show that we can compute solutions to these equations with second order accuracy

an engquist osher type scheme for conservation laws with discontinuous flux adapted to flux connections we consider scalar conservation laws with the spatially varying flux h x f u 1 h x g u where h x is the heaviside function and f and g are smooth nonlinear functions adimurthi mishra and veerappa gowda j hyperbolic differ equ 2 2005 pp 783 837 pointed out that such a conservation law admits many l 1 contraction semigroups one for each so called connection a b here we define entropy solutions of type a b involving kruzkov type entropy inequalities that can be adapted to any fixed connection a b it is proved that these entropy inequalities imply the l 1 contraction property for l infty solutions in contrast to the piecewise smooth setting of adimurthi mishra and veerappa gowda for a fixed connection these entropy inequalities include a single adapted entropy of the type used by audusse and perthame proc roy soc edinburgh sect a 135 2005 pp 253 265 we prove convergence of a new difference scheme that approximates entropy solutions of type a b for any connection a b if a few parameters are varied the scheme relies on a modification of the standard engquist osher flux is simple as no 2 times2 riemann solver is involved and is designed such that the steady state solution connecting a to b is preserved in contrast to most analyses of similar problems our convergence proof is not based on the singular mapping or compensated compactness methods but on standard spatial variation estimates away from the flux discontinuity some numerical examples are presented

convergence of godunov type methods for a conservation law with a spatially varying discontinuous flux function we deal with single conservation laws with a spatially varying and possibly discontinuous coefficient this equation includes as a special case sin gle conservation laws with conservative and possibly singular source terms we extend the framework of optimal entropy solutions for these classes of equations based on a two step approach in the first step an interface connection vector is used to define infinite classes of entropy solutions we show that each of these classes of solutions is stable in l1 this allows for the possibility of choosing one of these classes of solutions based on the physics of the problem in the second step we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient this method leads to optimal entropy solutions that are consistent with physically observed solutions in two phase flows in heterogeneous porous media another central aim of this paper is to develop suitable numerical schemes for these equations we develop and analyze a set of godunov type finite volume methods that are based on exact solutions of the corresponding riemann problem numerical experiments are shown comparing the performance of these schemes on a set of test problems

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

solutions of hamilton jacobi equations and scalar conservation laws with discontinuous space time dependence abstract we establish a unique stable solution to the hamilton jacobi equation u t h k x t u x 0 x t 0 with lipschitz initial condition where k x t is allowed to be discontinuous in the x t plane along a finite number of possibly intersecting curves parameterized by t we assume that for fixed k h k p is convex in p and lim p h k p p the solution is determined by showing that if k is made smooth by convolving k in the x direction with the standard mollifier then the control theory representation of the viscosity solution to the resulting hamilton jacobi equation must converge uniformly as the mollification decreases to a lipschitz continuous solution with an explicit control theory representation this also defines the unique stable solution to the corresponding scalar conservation law u t f k x t u x 0 x t 0 with k discontinuous

central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions in this paper two existing one dimensional mathematical models one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions are combined into a model of continuous separation of polydisperse mixtures this model can be written as a first order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable this application motivates the extension of the kurganov tadmor central difference scheme to systems with discontinuous flux the new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws these additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step or solved exactly that is the discontinuity parameters are kept constant with respect to time numerical examples and an l1 error study show that the kurganov tadmor scheme with first order in time discretization produces spurious oscillations whereas its semi discrete version discretized by a second order runge kutta scheme produces good results the scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved numerical examples illustrate the application to separation of polydisperse suspensions

a difference scheme for conservation laws with a discontinuous flux the nonconvex case in a previous work by the author convergence was established for a simple difference scheme approximating a scalar conservation law where the flux was concave and had a discontinuous spatially varying coefficient j d towers siam j numer anal 38 2000 pp 681 698 the main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points additionally spatially varying source terms are allowed the spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing tvd in time

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

periodic averaging theorems for various types of equations abstract we prove a periodic averaging theorem for generalized ordinary differential equations and show that averaging theorems for ordinary differential equations with impulses and for dynamic equations on time scales follow easily from this general theorem we also present a periodic averaging theorem for a large class of retarded equations

measure functional differential equations and functional dynamic equations on time scales abstract we study measure functional differential equations and clarify their relation to generalized ordinary differential equations we show that functional dynamic equations on time scales represent a special case of measure functional differential equations for both types of equations we obtain results on the existence and uniqueness of solutions continuous dependence and periodic averaging

discontinuous conservation laws for production networks with finite buffers in this study we use discontinuous conservation laws to model production networks with finite buffers a network formulation is characterized by the interaction between several production units described by discontinuous conservation laws under the assumption that no parts are lost therefore we discuss appropriate riemann problems and coupling conditions which are derived from a regularized model to solve the discontinuous problem in an accurate and fast way we introduce a novel algorithm based on different settings the new method is compared numerically with standard approaches

semidiscrete central upwind scheme for conservation laws with a discontinuous flux function in space abstract in this paper a modified semidiscrete central upwind scheme is derived for the scalar conservation laws with a discontinuous flux function in space the new scheme is based on dealing with the phase transition at the stationary discontinuity where the unknown variable function is not continuous but the flux function is continuous the main advantages of the new scheme are the same as them of the original semidiscrete central upwind scheme numerical results are displayed to illustrate the efficiency of the methods

interactive image segmentation via adaptive weighted distances an interactive algorithm for soft segmentation of natural images is presented in this paper the user first roughly scribbles different regions of interest and from them the whole image is automatically segmented this soft segmentation is obtained via fast linear complexity computation of weighted distances to the user provided scribbles the adaptive weights are obtained from a series of gabor filters and are automatically computed according to the ability of each single filter to discriminate between the selected regions of interest we present the underlying framework and examples showing the capability of the algorithm to segment diverse images

convergence via oslc of the godunov scheme for a scalar conservation law with time and space flux discontinuities this paper deals with a godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time we extend the definition of vanishing viscosity solution of karlsen and towers j hyperbolic differ equ 14 671 702 2017 which applies to a flux with a spatial discontinuity in order to accommodate the addition of temporal flux discontinuities and prove that this extended definition implies uniqueness we prove convergence of the godunov approximations to the unique vanishing viscosity solution as the mesh size converges to zero thus establishing well posedness for the problem the novel aspect of this paper is the use of a discrete one sided lipschitz condition oslc in the discontinuous flux setting in the classical setting where flux discontinuities are not present the oslc is well known to produce an immediate regularizing effect with a local spatial variation bound resulting at any positive time we show that the oslc also produces a regularizing effect at any finite distance from the spatial flux discontinuity this regularizing effect is not materially affected by temporal flux discontinuities when combined with a cantor diagonal argument these regularizing effects imply convergence of the godunov approximations with this new method it is possible to forgo certain assumptions about the flux that seem to be required when using two commonly used convergence methods

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

averaging for retarded functional differential equations abstract we consider retarded functional differential equations in the setting of kurzweil henstock integrable functions and we state an averaging result for these equations our result generalizes previous ones

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

an engquist osher type finite difference scheme with a discontinuous flux function in space abstract an engquist osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient the new monotone difference scheme is based on introducing a new interface numerical flux function which is called a generalized engquist osher flux by means of this scheme the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods

dynamic equations on time scales and generalized ordinary differential equations abstract the aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by j kurzweil we also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales

convergence of an engquist osher scheme for a multi dimensional triangular system of conservation laws we consider a multi dimensional triangular system of conservation laws this system arises as a model of three phase flow in porous media and includes multi dimensional conservation laws with discontinuous coefficients as a special case the system is neither strictly hyperbolic nor symmetric we propose an engquist osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution numerical examples are also presented

a relaxation scheme for conservation laws with a discontinuous coefficient lwe study a relaxation scheme of the jin and xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k x if k e bv we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution the murat tartar compensated compactness method is used to establish convergence we present numerical experiments with the relaxation scheme and comparisons are made with a front tracking scheme based on an exact 2 x 2 riemann solver

convergence of finite volume schemes for triangular systems of conservation laws we consider non strictly hyperbolic systems of conservation laws in triangular form which arise in applications like three phase flows in porous media we device simple and efficient finite volume schemes of godunov type for these systems that exploit the triangular structure we prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero some numerical examples are presented one of which is related to flows in porous media

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

godunov type numerical methods for a model of granular flow we propose and analyze finite volume godunov type methods based on discontinuous flux for a 2 2 system of non linear partial differential equations proposed by hadeler and kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table the scheme is made well balanced by modifying the flux function locally by including source term as a part of the convection term its extension to multi dimensions is not straightforward for which an approach has been introduced here based on transport rays this approach is compared with another approach for inclusion of source term which uses the idea of inverting the divergence operator relying on the curl free component of the helmholtz decomposition of the source term numerical experiments are presented to illustrate the efficiency of the proposed scheme for both unsteady and steady state calculations and to make comparisons with the previously studied finite difference and semi lagrangian approaches by falcone and finzi vita

explicit hopf lax type formulas for hamilton jacobi equations and conservation laws with discontinuous coefficients we deal with a hamilton jacobi equation with a hamiltonian that is discontinuous in the space variable this is closely related to a conservation law with discontinuous flux recently an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection in this paper we characterize these infinite classes of solutions in terms of explicit hopf lax type formulas which are obtained from the viscosity solutions of the corresponding hamilton jacobi equation with discontinuous hamiltonian this also allows us to extend the framework of infinitely many classes of solutions to the hamilton jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law we have considered the case where both the hamiltonians are convex concave furthermore we also deal with the less explored case of sign changing coefficients in which one of the hamiltonians is convex and the other concave in fact in convex concave case we cannot expect always an existence of a solution satisfying rankine hugoniot condition across the interface therefore the concept of generalised rankine hugoniot condition is introduced and prove existence and uniqueness

the solution of nonstrictly hyperbolic conservation laws may be hard to compute some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to l 1 perturbations of the initial data one of the examples shows that a piecewise constant approximation generates an approximate solution with erroneous qualitative behaviour another example shows that the approximate solution is not well defined for both of these examples a classical stable entropy solution can be given by an explicit formula these properties may cause difficulties for numerical methods based on the solution of riemann problems

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

well posedness in bv t and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units we consider a scalar conservation law modeling the settling of particles in an ideal clarifier thickener unit the conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters we suggest to use a kru kov type notion of entropy solution for this conservation law and prove uniqueness l1 stability of the entropy solution in the bvt class functions w x t with tw being a finite measure the existence of a bvt entropy solution is established by proving convergence of a simple upwind finite difference scheme of the engquist osher type a few numerical examples are also presented

conservation law with the flux function discontinuous in the space variable ii we deal with a single conservation law with discontinuous convex concave type fluxes which arise while considering sign changing flux coefficients the main difficulty is that a weak solution may not exist as the rankine hugoniot condition at the interface may not be satisfied for certain choice of the initial data we develop the concept of generalized entropy solutions for such equations by replacing the rankine hugoniot condition by a generalized rankine hugoniot condition the uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi group in l 1 existence follows by showing that a godunov type finite difference scheme converges to the generalized entropy solution the scheme is based on solutions of the associated riemann problem and is neither consistent nor conservative the analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely numerical results reporting the performance of the scheme are presented

viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape from shading equations with discontinuous intensities the shape from shading sfs equation relating u y r the unknown angular height of a surface to i y r the known synthetic aperture radar sar intensity data from the surface is i frac u r 2 sqrt 1 u r 2 u y 2 where y and r are axial and radial cylindrical coordinates unlike the more common eikonal sfs equation which relates surface height in cartesian coordinates to optical photographic intensity data the above radar equation can be transformed into hamilton jacobi cauchy form ur g i uy 0 we explore the case where i is a discontinuous function which occurs commonly in radar data by considering sequences of continuous intensity functions that converge to i we obtain corresponding sequences of viscosity solutions we prove that these sequences must converge we also establish conditions that guarantee that these sequences converge to a common limit which we define as the solution to the radar equation finally we establish and demonstrate that when this common limit exists monot

the l 1 error estimates for a hamiltonian preserving scheme for the liouville equation with piecewise constant potentials we study the l 1 error of a hamiltonian preserving scheme developed in s jin and x wen commun math sci 3 2005 pp 285 315 for the liouville equation with a piecewise constant potential in one space dimension this problem has important applications in computations of the semiclassical limit of the linear schrodinger equation through barriers and of the high frequency waves through interfaces we use the l 1 error estimates established in x wen and s jin j comput math 26 2008 pp 1 22 x wen convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients ii some related binomial coefficient inequalities preprint 2008 for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients we prove that the scheme with the dirichlet incoming boundary conditions is l 1 convergent for a class of bounded initial data and derive the half order l 1 error bounds with explicit coefficients the initial conditions can be satisfied by applying the decomposition technique proposed in s jin h l liu s osher and r tsai j comput phys 205 2005 pp 222 241 for solving the liouville equation with measure valued initial data which arises in the semiclassical limit of the linear schrodinger equation

well posedness results for abstract generalized differential equations and measure functional differential equations abstract in the first part of the paper we consider nonlinear generalized ordinary differential equations whose solutions take values in infinite dimensional banach spaces and prove new theorems concerning the existence of solutions and continuous dependence on initial values and parameters in the second part we apply these results in the study of nonlinear measure functional differential equations and impulsive functional differential equations with infinite delay

a comparison of convergence rates for godunov s method and glimm s method in resonant nonlinear systems of conservation laws we obtain time independent bounds on derivatives prove convergence and establish a rate of convergence for godunov s numerical method as applied to the initial value problem for a resonant inhomogeneous conservation law which is treated as a 2 times 2 nonstrictly hyperbolic system we compare the results with a corresponding analysis of glimm s method and see that our analysis gives equivalent sharp convergence rates in the strictly hyperbolic setting but an improvement is seen in godunov s method over glimm s method in the nonstrictly hyperbolic resonant regime the 2 times 2 glimm and godunov methods are the only methods for which we can obtain time independent bounds on derivatives these bounds represent a purely nonlinear phenomenon because there are no corresponding time independent bounds for the linearized equations which blow up at a linear rate in time