on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

a finite volume method for a two phase multicomponent polymer flooding multicomponent polymer flooding used in enhanced oil recovery is governed by a system of coupled non strictly hyperbolic conservation laws in the presence of gravity the flux functions need not be monotone and hence designing godunov type upwind schemes is difficult and computationally expensive to overcome this difficulty we use the basic idea of discontinuous flux to reduce the coupled system into an uncoupled system of scalar conservation laws with discontinuous coefficients for these scalar equations we use the dflu flux developed in 5 to construct a second order scheme the scheme is shown to satisfy a maximum principle and the performance of the scheme is shown on both one and two dimensional test problems

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

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

vanishing viscosity solutions for conservation laws with regulated flux abstract in this paper we introduce a concept of regulated function v t x of two variables which reduces to the classical definition when v is independent of t we then consider a scalar conservation law of the form u t f v t x u x 0 where f is smooth and v is a regulated function possibly discontinuous w r t both t and x by adding a small viscosity one obtains a well posed parabolic equation as the viscous term goes to zero the uniqueness of the vanishing viscosity limit is proved relying on comparison estimates for solutions to the corresponding hamilton jacobi equation as an application we obtain the existence and uniqueness of solutions for a class of 2 2 triangular systems of conservation laws with hyperbolic degeneracy

roots of unity and unbounded motions of an asymmetric oscillator r 2 z the general questions that we have in mind are under what conditions ona b and f t can we say that 1 has unbounded solutions are there equationsin the class 1 where unbounded and periodic solutions can coexist equation 1 was first considered by dancer in 2 3 and 4 and byfucik in 7 they studied the periodic and dirichlet boundary valueproblems for 1 and looked at this equation as a model of the so called equations with jumping nonlinearities we sum up some of the results in 2 3 consider the family of curves in the a b parameter space 1 a 1 b 2p p 1 2 2 when a b is not in any of these curves then 1 has at least one 2 periodic solution however if a b lies in one of these curves there existfunctions f t for which 1 has no periodic solution more results on theperiodic problem for 1 have been obtained by lazer and mckenna 8 fabry 5 and fabry and fonda 6 the known results on the existenceof periodic solutions already shed some light on the boundedness problem actually the second theorem of massera implies that if 1 has no periodic

constructing solutions of a single conservation law on construit la solution d une seule loi de conservation u t f u x 0 t 0 x u x o u 0 x x on suppose que f est reguliere et f ne s annulent qu en des points isoles

scalar conservation laws with discontinuous flux function i the viscous profile condition the equation whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data the equation can be written as a triangular 2 2 non strictly hyperbolic system this augmentation is non unique and a natural definition is given by means of viscous profiles by a viscous profile we mean a stationary solution ofu t f x u xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed in terms of the 2 2 system the discontinuity atx 0 is either a regular lax an under or overcompressive a marginal under or overcompressive or a degenerate shock wave in some cases depending onf andg there is a unique viscous profile e g undercompressive and regular lax waves and in some cases there are infinitely many e g overcompressive waves the main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx 0 and the viscous profile condition

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

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

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

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

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

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

weak asymptotic methods for scalar equations and systems abstract in this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions this construction produces weak asymptotic methods which are issued from maslov asymptotic analysis we obtain a sequence of functions which tend to satisfy the equation s in the weak sense in the space variable and in the strong sense in the time variable to this end we reduce the partial differential equations to a family of ordinary differential equations in a classical banach space for scalar equations we prove that the initial value problem is well posed in the l 1 sense for the approximate solutions we construct then we prove that this method gives back the widely accepted solutions when they are known for systems we obtain existence in the general case and uniqueness in the analytic case using an abstract cauchy kovalevska theorem

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

comments on solution of the system of linear algebraic equations by decreasing dimension we show that the decreasing dimension method proposed by h wang and j jiang in solution of the system of linear algebraic equations by decreasing dimension appl math comput 109 2000 51 is a type of schur complement domain decomposition method the decreasing dimension method is more expensive than the standard schur complement domain decomposition method for solving any linear systems

scalar conservation laws with discontinuous flux function ii on the stability of the viscous profiles the equation frac partial u partial t frac partial partial x left h x f u left 1 h x right g u right 0 whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data by a viscous profile of this discontinuity we mean a stationary solution ofu t f x eu xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed we present some results on the stability of the viscous profiles which means that a small disturbance tends to zero uniformly ast this is done by weighted energy methods where the weight depending onf andg plays a crucial role

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

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

dynamic and steady state behavior of continuous sedimentation continuous sedimentation of solid particles in a liquid takes place in a clarifier thickener unit which has one feed inlet and two outlets the process can be modeled by a nonlinear scalar conservation law with point source and discontinuous flux function this paper presents existence and uniqueness results in the case of varying cross sectional area and a complete classification of the steady state solutions when the cross sectional area decreases with depth the classification is utilized to formulate a static control strategy for the large discontinuity called the sludge blanket that appears in steady state operation a numerical algorithm and a few simulations are also presented

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

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

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

bouncing solutions of an equation with attractive singularity for any n m 2 in it is proved the existence of 2m periodic solutions with n bouncings in each period for a second order forced equa tion with attractive singularity by using the approach of successor map and poincare birkho twist theorem

finite difference methods for approximating heaviside functions we present a finite difference method for discretizing a heaviside function h u x where u is a level set function u r email protected r that is positive on a bounded region email protected r n there are two variants of our algorithm both of which are adapted from finite difference methods that we proposed for discretizing delta functions in j d towers two methods for discretizing a delta function supported on a level set j comput phys 220 2007 915 931 j d towers discretizing delta functions via finite differences and gradient normalization preprint at http www miracosta edu home jtowers j d towers a convergence rate theorem for finite difference approximations to delta functions j comput phys 227 2008 6591 6597 we consider our approximate heaviside functions as they are used to approximate integrals over w we prove that our first approximate heaviside function leads to second order accurate quadrature algorithms numerical experiments verify this second order accuracy for our second algorithm numerical experiments indicate at least third order accuracy if the integrand f and email protected are sufficiently smooth numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations we mostly focus on smooth f and u by this we mean that f is smooth in a neighborhood of w u is smooth in a neighborhood of email protected and the level set u x 0 is a manifold of codimension one however our algorithms still give reasonable results if either f or u has jumps in its derivatives numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way assuming that the level set u x 0 is a manifold numerical experiments indicate that dependence on the placement of w with respect to the grid is quite small for our algorithms specifically a grid shift results in an o h p change in the computed solution where p is the observed rate of convergence

iterative decreasing dimension algorithm in this study we have given an iterative decreasing dimension method iddm which decreases by one dimension at every step for solution of a system of linear algebraic equations without any pre processing and an iterative decreasing dimension algorithm idda which is based on this method we have also given numerical examples using this algorithm

infinity of subharmonics for asymmetric duffing equations with the lazer leach dancer condition abstract in this paper based on a generalized version of the poincare birkhoff twist theorem by franks we establish the existence of infinitely many subharmonics for the asymmetric duffing equation with the classical lazer leach dancer condition as a consequence of our result we obtain a sufficient and necessary condition for existence of arbitrarily large amplitude periodic solutions for a class of asymmetric duffing equations at resonance

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

existence and nonexistence of tv bounds for scalar conservation laws with discontinuous flux for the scalar conservation laws with discontinuous flux an infinite family of a b interface entropies are introduced and each one of them is shown to form an l1 contraction semigroup see 2 one of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution away from the interface boundedness of total variation of the solution has been proved in a recent paper 6 in this paper we discuss this particular issue in detail and produce a counterexample to show that the solution in general has unbounded total variation near the interface in fact this example illustrates that smallness of the bv norm of the initial data is immaterial we hereby settle the question of determining for which of the aforementioned a b pairs the solution will have bounded total variation in the case of strictly convex fluxes 2010 wiley periodicals inc

discrete shock profiles for scalar conservation laws with discontinuous fluxes abstract the paper considers a scalar conservation law with a discontinuous flux f of the form f x u h x g u 1 h x h u where h x is the heaviside function herein the fluxes g and h are supposed to have one minimum and no maximum and at most one crossing in the interior of the domain of definition the aim is to verify a weak solution of such a problem in the following way we are looking for discrete shock profiles for its continuously differentiable perturbation with a parameter e and godunov s scheme for conservation laws with spatially varying flux functions the obtained discrete shock profile satisfies a discrete entropy condition of kruzkhov type and after letting e 0 approaches an entropy weak solution of the original equation

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

a semi implicit monotone difference scheme for an initial boundary value problem of a strongly degenerate parabolic equation modeling sedimentation consolidation processes we prove the convergence of a semi implicit monotone finite difference scheme approximating an initial boundary value problem for a spatially one dimensional quasilinear strongly degenerate parabolic equation which is supplied with two different inhomogeneous flux type boundary conditions this problem arises in the modeling of the sedimentation consolidation process we formulate the definition of entropy solution of the model in the sense of kruzkov and prove convergence of the scheme to the unique bv entropy solution of the problem up to satisfaction of one of the boundary conditions

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

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

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

the wave front tracking algorithm for hughes model of pedestrian motion we code the wave front tracking algorithm for hughes model of pedestrian motion with generalized running cost this model displays a nonclassical dynamic at the splitting point between the two directions of motion the wave front tracking scheme provides us with reference solutions for testing numerically the convergence of classical finite volume schemes which do not treat explicitly the dynamics at the turning point

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

fast reliable simulations of secondary settling tanks in wastewater treatment with semi implicit time discretization the bio kinetic and sedimentation processes of wastewater treatment plants can be modelled by a large system of coupled nonlinear ordinary and partial differential equations odes and pdes the subprocess of continuous sedimentation which contains concentration discontinuities is modelled by a degenerate parabolic conservation pde with spatially discontinuous coefficients a spatial discretization of this pde described in brger etal 2013 results in a large system of method of lines odes for the entire plant and simulation can be performed by integration in time in practice standard time integration methods available in commercial simulators are often used shortages of such methods are here shown such as the smearing of shock waves by rungekutta methods and long execution times a semi implicit time discretization which is described in detail provides substantially shorter computational times and is more efficient than standard methods

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 conservation law with point source and discontinuous flux function modelling continuous sedimentation continuous sedimentation of solid particles in a liquid takes place in a clarifier thickener unit which has one feed inlet and two outlets the aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions the conservation of mass yields the scalar conservation law frac partial u x t partial t frac partial partial x f u x t x s t delta x where delta is the dirac measure s is a source and f is a flux function which is discontinuous at three points in the one dimensional space coordinate x under certain regularity assumptions a procedure for constructing a solution locally in time is presented the nonlinear phenomena are complicated and so is the general construction of a solution the problem of nonuniqueness due to the discontinuities of f u cdot is handled by a generalized entropy condition an advantage of this approach is that the a priori boundary conditions at the discontinuities

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

an existence result for conservation laws having bv spatial flux heterogeneities without concavity abstract we prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence for the type of flux considered here where the spatial dependence occurs in a very general form a uniqueness result is known but the existence question was open until the recent work of piccoli and tournus 2018 22 piccoli and tournus proved existence using approximate solutions generated by the wave front tracking algorithm a concavity assumption plays a simplifying role in their analysis the main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption we accomplish this via finite difference approximations

godunov type methods for conservation laws with a flux function discontinuous in space scalar conservation laws with a flux function discontinuous in space are approximated using a godunov type method for which a convergence theorem is proved the case where the flux functions at the interface intersect is emphasized a very simple formula is given for the interface flux a numerical comparison between the godunov numerical flux and the upstream mobility flux is presented for two phase flow in porous media a consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration furthermore for regular solutions uniqueness has been shown

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

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

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

convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function we consider the scalar conservation law with flux function discontinuous in the space variable i e begin eqnarray label eq1 u t h x f u 1 h x g u x 0 quad mbox in r times r nonumber u 0 x u 0 x quad mbox in r label 0 1 end eqnarray where h is the heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex the existence of a weak solution of ref eq1 is proved by showing that upwind finite difference schemes of godunov and enquist osher type converge to a weak solution uniqueness follows from a kruzkhov type entropy condition we also provide explicit solutions to the riemann problem for ref eq1 at the level of numerics we give easy to implement numerical schemes of godunov and enquist osher type the central feature of this paper is the modification of the singular mapping technique the main analytical tool for these types of equations which allows us to show that the numerical schemes converge equations of type ref eq1 with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux begin equation label eq2 begin array email protected l u t k x f u x 0 u 0 x u 0 x end array end equation with f convex and k of indefinite sign

periodic solutions of semilinear duffing equations with impulsive effects abstract in this paper we are concerned with the existence of periodic solutions for semilinear duffing equations with impulsive effects firstly for its autonomous equation any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time under the influence of impulses these two motions are likely to be quite different we introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one basing on poincare birkhoff twist theorem we prove the existence of infinitely many periodic solutions secondly as for the nonautonomous equation where the autonomous case is taken as an auxiliary one we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by poincare birkhoff twist theorem lastly an example with special impulses satisfying the above assumptions is given

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

solution of the system of linear algebraic equations by decreasing dimension a new method is proposed to solve the system of linear algebraic equations in the paper the dimension of the system of linear algebraic equations can be decreased freely by means of linear transformation the concept is clear and the result is precise the method is effective especially to band matrix equations and large sparse matrix equations

convergence of a godunov scheme to an audusse perthame adapted entropy solution for conservation laws with bv spatial flux in this article we consider the initial value problem for a scalar conservation law in one space dimension with a spatially discontinuous flux there may be infinitely many flux discontinuities and the set of discontinuities may have accumulation points thus the existence of traces cannot be assumed in audusse and perthame proc r soc edinb sect a 135 253 265 2005 proved a uniqueness result that does not require the existence of traces using adapted entropies we generalize the godunov type scheme of adimurthi et al siam j numer anal 42 1 179 208 2004 for this problem with the following assumptions on the flux function i the flux is bv in the spatial variable and ii the critical point of the flux is bv as a function of the space variable we prove that the godunov approximations converge to an adapted entropy solution thus providing an existence result and extending the convergence result of adimurthi jaffre and gowda

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

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

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

on the cauchy problems for polymer flooding with gravitation abstract we study two systems of conservation laws for polymer flooding in secondary oil recovery one with gravitation force and one without for each model we prove global existence of weak solutions for the cauchy problems under rather general assumptions on the flux functions approximate solutions are constructed through a front tracking algorithm and compactness is achieved through the bound on suitably defined wave strengths as the main technical novelty we introduce some new nonlinear functionals that yield a uniform bound on the total variation of the flux function

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

coupling techniques for nonlinear hyperbolic equations i self similar diffusion for thin interfaces we investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and in particular we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows the main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost as a consequence non uniqueness of weak solutions is observed for the initial value problem and these solutions need to be supplemented with further admissibility conditions this paper is devoted to investigating these issues in the setting of self similar vanishing viscosity approximations to the riemann problem for general hyperbolic systems following earlier works by joseph lefloch and tzavaras we establish an existence theorem for the riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns

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

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

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

finite volume schemes for locally constrained conservation laws this paper is devoted to the numerical analysis of the road traffic model proposed by colombo and goatin j differ equ 234 2 654 675 2007 the model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x 0 modelling a road light a toll gate etc we first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function as developed by adimurthi et al j hyperbolic differ equ 2 4 783 837 2005 and burger et al siam j numer anal 47 3 1684 1712 2009 we reformulate accordingly the notion of entropy solution introduced by colombo and goatin j differ equ 234 2 654 675 2007 and extend the well posedness results to the l framework then starting from a general monotone finite volume scheme for the non constrained conservation law we produce a simple scheme for the constrained problem and show its convergence the proof uses a new notion of entropy process solution numerical examples modelling a green wave are presented

a relaxation scheme forcontinuous sedimentation in ideal clarifier thickener units we prove convergence for a simple relaxation scheme for a scalar conservation law modeling the settling of particles in an ideal clarifier thickener unit the conservation law has a flux which is spatially dependent on two discontinuous parameters we prove convergence of the relaxation approximations to a weak solution of the clarifier thickener model and present numerical examples using our scheme

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

hyperbolic conservation laws with space dependent flux i characteristics theory and riemann problem in the paper a kind of one dimensional scalar hyperbolic conservation laws with flux functions dependent on space variable is discussed and analyzed a better understanding about the behavior of wave propagation of the kind problems is presented especially some sufficient and necessary conditions that ensure the unique physically relevant solution to the riemann problem are proposed because the numerical flux obtained from the riemann s solver is theoretically correct and exact to the problem it must also be of high resolution in its nature for comparison some convincing numerical examples from traffic flow problems are given at the end of the paper

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

stable numerical methods for conservation laws with discontinuous flux function abstract we develop numerical methods for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients using a relaxation approximation the nonlinear equation is transformed to a semilinear diagonalizable problem with linear characteristic variables eulerian and lagrangian methods are used for the advection stage while an implicit explicit scheme solves the relaxation stage the main advantages of this approach are neither riemann problem solvers nor linear iterations are required during the solution process moreover the characteristic based relaxation method is unconditionally stable such that no cfl conditions are imposed on the selection of time steps numerical results are shown for models on traffic flows and two phase flows

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

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

computational uncertainty quantification for some strongly degenerate parabolic convection diffusion equations abstract strongly degenerate parabolic convection diffusion equations arise as governing equations in a number of applications such as traffic flow with driver reaction and anticipation distance and sedimentation of solid liquid suspensions in mineral processing and wastewater treatment in these applications several parameters that define the convective flux function and the degenerating diffusion coefficient are subject to stochastic variability a method to evaluate the variability of the solution of the governing partial differential equation in response to that of the parameters is presented to this end a generalized polynomial chaos gpc expansion of the solution is approximated by its projection onto a finite dimensional space of piecewise polynomial functions defined on a suitable discretization of the stochastic domain according to the basic principle of the hybrid stochastic galerkin hsg approach this approach is combined with a finite volume fv method resulting in a so called fv hsg method to compute the sought deterministic coefficient functions of the truncated polynomial chaos based expansion of the solution since the stochastic parameter space is now spanned by piecewise polynomial functions one may employ the numerical result to compute the reconstruction of the numerical solution for arbitrary values of the random variables the expectation the variance or other stochastic quantities of the solution as functions of time and position can also be computed from these coefficient functions the method is illustrated by a number of numerical examples

two new moving boundary problems for scalar conservation laws the application of the theory of scalar conservation laws to semiconductor device fabrication is described this application is the source of a stefan problem and another moving boundary problem for a class of such equations the analogue of the riemann problem for these problems is analyzed and solved conditions on the boundary values that characterize physically correct solutions are derived

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

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