existence and approximation of robust solutions of variational inequality problems over polytopes we study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept extending an earlier concept proposed by yang z yang siam j control optim 34 1996 pp 491 506 on the unit simplex we will introduce the concept of the robust stationary point which is a refinement of the concept of the stationary point though a stationary point need not be robust it is shown that every continuous function on a polytope has a robust stationary point we develop a simplicial algorithm to compute a robust stationary point of a continuous function on a polytope the algorithm can be briefly stated as follows starting with any point in the relative interior of a polytope the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps moreover we also discuss several numerical examples and apply the new concept to noncooperative games and economic equilibrium problems

numerical solutions to compressible flows in a nozzle with variable cross section compressible flows in a nozzle can be modeled by the gas dynamics equations in one dimensional space with source terms it turns out that along stationary waves the entropy is conserved investigating properties of the system leads us to the determination of stationary waves relying on this analysis we construct a numerical scheme which takes into account the use of stationary waves our scheme is shown to be capable of maintaining equilibrium states this demonstrates the efficiency of the new scheme over classical ones which usually give unsatisfactory results when reducing the refinement of the mesh size moreover our scheme converges much faster than the classical ones in most cases

the riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross sectional area we present a full investigation of the riemann problem for a nonisentropic polytropic fluid in a nozzle with piecewise constant cross section first we introduce the concept of elementary waves which turn out to make up riemann solutions second we study a procedure to select admissible stationary waves relying on the monotone criterion by projecting all the wave curves in the p u plane we construct riemann solutions existence of riemann solutions can be obtained for large initial data furthermore we establish the uniqueness of riemann solutions in strictly hyperbolic domains our argument can lead to estimate regions where the riemann problem admits a unique solution

nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

volume integrals for boundary element methods abstract we consider the numerical approximation of volume integrals over bounded domains d x r 3 h x 0 where h r 2 r is a suitable decidability function the integrands may be smooth maps or singular maps such as those arising in the volume potentials for boundary element methods an adaptive integration method is described it utilizes an automatic simplicial subdivision of the domain the integration step is based on ideas similar to those recently given by georg 1991 for surface integrals several examples illustrate the performance of the method

the exponential integrator scheme for stochastic partial differential equations pathwise error bounds we present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions we discretise in space by a galerkin method and in time by using a stochastic exponential integrator we show that for spatially regular smooth noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces and the noise becomes rougher

a simplicial approach to the determination of an integer point of a simplex we propose an algorithm to find an integer point of a simplex the algorithm is based on an integer labeling rule and a triangulation of the space starting at an integer point the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension within a finite number of iterations the algorithm either yields an integer point of the simplex or proves that no such point exists

unsteady mixed flows in non uniform closed water pipes a full kinetic approach we recall the pressurized and free surface model constructed for the modeling of unsteady mixed flows in closed water pipes where transition points between the free surface and pressurized flow are treated as a free boundary associated to a discontinuity of the gradient of pressure then we present a numerical kinetic scheme for the computations of unsteady mixed flows in closed water pipes this kinetic method that we call fka for full kinetic approach is an easy and mathematically elegant way to deal with multiple transition points when the changes of state between free surface and pressurized flow occur we use two approaches namely the ghost waves approach and the full kinetic approach to treat these transition points we show that this kinetic numerical scheme has the following properties it is wet area conservative under a cfl condition it preserves the wet area positive it treats naturally the flooding zones and most of all it is very easy to implement it finally numerical experiments versus laboratory experiments are presented and the scheme produces results that are in a very good agreement we also present a numerical comparison with analytic solutions for free surface flows in non uniform pipes the numerical scheme has a very good behavior a code to code comparison for pressurized flows is also conducted and leads to a very good agreement we also perform a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method

a godunov type method for the shallow water equations with discontinuous topography in the resonant regime abstract we investigate the riemann problem for the shallow water equations with variable and possibly discontinuous topography and provide a complete description of the properties of its solutions existence uniqueness in the non resonant regime multiple solutions in the resonant regime this analysis leads us to a numerical algorithm that provides one with a riemann solver next we introduce a godunov type scheme based on this riemann solver which is well balanced and of quasi conservative form finally we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime except in the limiting situation when riemann data precisely belong to the resonance hypersurface

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

on the use of piecewise linear models in nonlinear programming this paper presents an active set algorithm for large scale optimization that occupies the middle ground between sequential quadratic programming and sequential linear quadratic programming methods it consists of two phases the algorithm first minimizes a piecewise linear approximation of the lagrangian subject to a linearization of the constraints to determine a working set then an equality constrained subproblem based on this working set and using second derivative information is solved in order to promote fast convergence a study of the local and global convergence properties of the algorithm highlights the importance of the placement of the interpolation points that determine the piecewise linear model of the lagrangian

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

boldmath topological and epsilon entropy for large volume limits of discretized parabolic equations we consider semidiscrete and fully discrete approximations of nonlinear parabolic equations in the limit of unbounded domains which by a scaling argument is equivalent to the limit of vanishing viscosity we define the spatial density of varepsilon entropy topological entropy and dimension for the attractors and show that these quantities are bounded we also provide practical means of computing lower bounds on them the proof uses the property that solutions lie in gevrey classes of analyticity which we define in a way that does not depend on the size of the spatial domain as a specific example we discuss the complex ginzburg landau equation

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

the riemann problem for the one dimensional free surface shallow water equations with a bed step theoretical analysis and numerical simulations in this paper the solution of the riemann problem for the one dimensional free surface shallow water equations over a bed step is analyzed both from a theoretical and a numerical point of view particular attention has been paid to the wave that is generated at the location of the bed discontinuity starting from the classical shallow water equations considering the bed level as an additional variable and adding to the system an equation imposing its time invariance we show that this wave is a contact wave across which one of the riemann invariants namely the energy is not constant this is due to the fact that the relevant problem is nonconservative we demonstrate that in this type of system riemann invariants do not generally hold in contact waves furthermore we show that in this case the equations that link the flow variables across the contact wave are the generalized rankine hugoniot relations and we obtain these for the specific problem from the numerical point of view we present an accurate and efficient solver for the step riemann problem to be used in a finite volume godunov type framework through a two step predictor corrector procedure the solver is able to provide solutions with any desired accuracy the predictor step uses a well balanced generalized roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution in order to show the effectiveness and the accuracy of the proposed approach we consider several step riemann problems and compare the exact solutions with the numerical results obtained by using a standard roe approach far from the step and the novel two step algorithm for the fluxes over the step achieving good results

exact solution of the riemann problem for the shallow water equations with discontinuous bottom geometry in this paper we present the exact solution of the riemann problem for the non linear shallow water equations with a step like bottom the solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step the resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave we provide examples of possible wave patterns numerical solution of a first order dissipative scheme as well as an implementation of our riemann solver in the second order upwind method are compared with the proposed exact riemann problem solution a practical implementation of the proposed exact riemann solver in the framework of a second order upwind tvd method is also illustrated

finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws we develop here a new class of finite volume schemes on unstructured meshes for scalar conservation laws with stiff source terms the schemes are of equilibrium type hence with uniform bounds on approximate solutions valid in cell entropy inequalities and exact for some equilibrium states convergence is investigated in the framework of kinetic schemes numerical tests show high computational efficiency and a significant advantage over standard cell centered discretization of source terms equilibrium type schemes produce accurate results even on test problems for which the standard approach fails for some numerical tests they exhibit exponential type convergence rate in two of our numerical tests an equilibrium type scheme with 441 nodes on a triangular mesh is more accurate than a standard scheme with 50002 grid points

a trigonometric method for the linear stochastic wave equation a fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented a standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation this explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step size restriction as in the often used stormer verlet leap frog scheme moreover it enjoys a trace formula as does the exact solution of our problem these favorable properties are demonstrated with numerical experiments

on the solution to the riemann problem for the compressible duct flow the quasi one dimensional euler equations in a duct of variable cross section form probably one of the simplest nonconservative systems we consider the riemann problem for it and discuss its properties in particular for some initial conditions the solution to the riemann problem appears to be nonunique in order to rule out the nonphysical solutions we provide two dimensional computations of the euler equations in a duct of corresponding geometry and compare them with the one dimensional 1d results then the physically relevant 1d solutions satisfy a kind of entropy rate admissibility criterion

the riemann problem for the baer nunziato two phase flow model we consider the riemann problem for the two phase flow model proposed by baer and nunziato int j multiphase flows 12 1986 861 it describes the flame spread and the deflagration to detonation transition ddt in gaspermeable reactive granular materials the model is given by a non strictly hyperbolic non conservative system of partial differential equations we investigate the structure of the riemann problem and construct the exact solution for it we establish that the solution across one wave is not unique and propose to use the evolutionarity criterion to single out the admissible solution due to special structure of the riemann problem for the baer nunziato model we are able to introduce a notion of a weak solution for it finally we propose a number of test cases based on the exact solution to the riemann problem for the baer nunziato model

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

singular continuation generating piecewise linear approximations to pareto sets via global analysis we propose a strategy for approximating pareto optimal sets based on the global analysis framework proposed by smale global analysis and economics i pareto optimum and a generalization of morse theory in dynamical systems academic press new york 1973 pp 531 544 the method highlights and exploits the underlying manifold structure of the pareto sets approximating pareto optima by means of simplicial complexes the method distinguishes the hierarchy between singular set pareto critical set and stable pareto critical set and it can handle the problem of superposition of local pareto fronts occurring in the general nonconvex case furthermore a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples

efficient well balanced hydrostatic upwind schemes for shallow water equations the proposed work concerns the numerical approximations of the shallow water equations with varying topography the main objective is to introduce an easy and systematic technique to enforce the well balance property and to make the scheme able to deal with dry areas to access such an issue the derived numerical method is obtained by involving the free surface instead of the water height and this produces the scheme well balanced independently from the numerical flux function associated with the homogeneous problem as a consequence we obtain an easy well balanced scheme which preserves non negative water height when compared with the well known hydrostatic reconstruction the presented topography discretization does not involve any max function known to introduce some numerical errors as soon as the topography admits very strong variations or discontinuities a second order muscl accurate reconstruction is adopted the proposed hydrostatic upwind scheme is next extended for considering 2d simulations performed over unstructured meshes several 1d and 2d numerical experiments are performed to exhibit the relevance of the scheme

on the complexity of an expanded tarski s fixed point problem under the componentwise ordering abstract let be a finite lattice of integer points in a box of r n and f an increasing mapping in terms of the componentwise ordering from to itself the well known tarski s fixed point theorem asserts that f has a fixed point in a simple expansion of f from to a larger lattice c of integer points in a box of r n yields that the smallest point in c is always a fixed point of f an expanded tarski s fixed point problem by introducing an integer labeling rule and applying a cubic triangulation of the euclidean space we prove in this paper that the expanded tarski s fixed point problem is in the class ppa when f is given as an oracle it is shown in this paper that nash equilibria of a bimatrix game can be reformulated as fixed points different from the smallest point in c of an increasing mapping from c to itself this implies that the expanded tarski s fixed point problem has at least the same complexity as that of the nash equilibrium problem as a byproduct we also present a homotopy like simplicial method to compute a tarski fixed point of f the method starts from an arbitrary lattice point and follows a finite simplicial path to a fixed point of f

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

the riemann problem for the shallow water equations with discontinuous topography the wet dry case abstract in this paper we consider riemann problems for the shallow water equations with discontinuous topography whose initial conditions correspond to a wet dry front at time t 0 there is vacuum on the right or on the left of the step besides the theoretical interest of this analysis the results may be useful to design numerical methods and or to produce reference solutions to compare different schemes we show that depending on the state at the wet side 0 1 or 2 self similar solutions can be constructed by composing simple waves in problems with 0 solutions the step acts as an obstacle for the fluid and physically meaningful solutions can be constructed by interpreting the problem as a partial riemann problems for the homogeneous shallow water system some numerical results are shown where different numerical methods are compared in particular it is shown that in the non uniqueness cases the numerical solutions can converge to one or to the other solution what is the reason that explains the huge differences observed when different numerical methods are applied to the shallow water system with abrupt changes in the bottom moreover problems with zero solutions will be reinterpreted as partial riemann problems for the homogeneous system what will allow us to build a physically solution when one side of the step is wet and the other one is dry we will specify the regions where we can find zero one or two solutions giving the form of the solution when it is possible and giving an alternative when it is not possible

a fully well balanced positive and entropy satisfying godunov type method for the shallow water equations this work is devoted to the derivation of a fully well balanced numerical scheme for the well known shallow water model during the last two decades several well balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so called lake at rest by fully well balanced we mean here that the proposed godunov type method is also able to preserve stationary states with non zero velocity the numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality

piecewise linear methods for nonlinear equations and optimization piecewise linear methods had their beginning in the mid 1960s with lemke s algorithm for calculating solutions to linear complementarity problems in the 1970s and 1980s activity moved on to computing fixed points of rather general maps and economic equilibria more recently they have been used to approximate implicitly defined manifolds with applications being made to computer graphics and approximations of integral over implicitly defined manifolds in this paper we present the basic ideas of piecewise linear algorithms and a selection of applications further references to the literature on piecewise linear algorithms are indicated

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

an algorithm for piecewise linear approximation of an implicitly defined manifold a simplicial method is used to approximate the solution manifold to a system of nonlinear equations h x theta where h mathbb r n k to mathbb r n the method begins at a point x 0 in the solution set where the derivative dh x 0 is of full rank given any varepsilon 0 a piecewise linear manifold is constructed along which h x infty varepsilon an algorithm is presented to carry out this construction in an efficient fashion

an algorithm for piecewise linear approximation of implicitly defined two dimensional surfaces let h mathbb r n 2 to mathbb r n be a smooth map and let x 0 in h 1 0 be a regular point of h we present an algorithm for obtaining a global piecewise linear approximation to a component m of the surface h 1 0 in particular for a specific type of triangulation t of mathbb r n 2 we construct a component m t subset h t 1 0 where h t is the piecewise linear map that interpolates h on the vertices of t error bounds on h x infty for x in m t are given in terms of the mesh size of t newton type iterative refinement techniques are given for obtaining points on m starting from points on m t sample numerical examples with computer graphical output are given

the octahedral algorithm a new simplicial fixed point algorithm a new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented it uses the restrart technique of merrill to improve the accuracy of the solution the algorithm is shown to converge quadratically under certain conditions the algorithm should be efficient and relatively easy to implement

reliability of first order numerical schemes for solving shallow water system over abrupt topography we compare some first order well balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography we show that the space step required to obtain a prescribed error depends on the method moreover the solutions given by the numerical scheme can be significantly different if not enough space resolution is used we shall pay special attention to the well known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem

the uniqueness of the exact solution of the riemann problem for the shallow water equations with discontinuous bottom abstract the riemann problem for the shallow water equations with discontinuous topography is considered in a general case the exact solution of this problem is not unique which complicates the application of an exact riemann solver in numerical methods since it is not clear which solution should be chosen in the present work it is shown that involving an additional physical assumption makes it possible to prove the existence and uniqueness of the solution the assumption is that the discharge at the bottom discontinuity should continuously depend on the initial conditions the proven uniqueness opens up a possibility to use an exact riemann solver for a numerical solution of the shallow water equations with complex discontinuous topography

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

a phase decomposition approach and the riemann problem for a model of two phase flows abstract we present a phase decomposition approach to deal with the generalized rankine hugoniot relations and then the riemann problem for a model of two phase flows by investigating separately the jump relations for equations in conservative form in the solid phase we show that the volume fractions can change only across contact discontinuities then we prove that the generalized rankine hugoniot relations are reduced to the usual form it turns out that shock waves and rarefaction waves remain on one phase only and the contact waves serve as a bridge between the two phases by decomposing riemann solutions into each phase we show that riemann solutions can be constructed for large initial data furthermore the riemann problem admits a unique solution for an appropriate choice of initial data

why many theories of shock waves are necessary kinetic functions equivalent equations and fourth order models we consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena these models admit under compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation building on earlier work by lefloch and collaborators we investigate the numerical approximation of these models by high order finite difference schemes and uncover several new features of the kinetic function associated with physically motivated second and third order regularization terms especially viscosity and capillarity terms on one hand the role of the equivalent equation associated with a finite difference scheme is discussed we conjecture here and demonstrate numerically that the numerical kinetic function associated with a scheme approaches the analytic kinetic function associated with the given model especially since its equivalent equation approaches the regularized model at a higher order on the other hand we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized camassa holm model finally we investigate to what extent a kinetic function can be associated with the equations of van der waals fluids whose flux function admits two inflection points

numerical continuation methods a perspective in this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development notably homotopy and degree theory simplicial complexes and mappings submanifolds defined by submersions and singularity and foldpoint theory

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

why many theories of shock waves are necessary convergence error in formally path consistent schemes we are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics and for solutions containing shock waves we investigate the convergence of finite difference schemes applied to such systems according to dal maso lefloch and murat s theory a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space in the present paper we consider schemes that are formally consistent with a given family of paths and we investigate their limiting behavior as the mesh is refined we first generalize to systems a property established earlier by hou and lefloch for scalar conservation laws and we prove that nonconservative schemes generate at the level of the limiting hyperbolic system an convergence error source term which provided the total variation of the approximations remains uniformly bounded is a locally bounded measure this convergence error measure is supported on the shock trajectories and as we demonstrate here is usually small in the special case that the scheme converges in the sense of graphs a rather strong convergence property often violated in practice then this measure source term vanishes we also discuss the role of the equivalent equation associated with a difference scheme here the distinction between scalar equations and systems appears most clearly since for systems the equivalent equation of a scheme that is formally path consistent depends upon the prescribed family of paths the core of this paper is devoted to investigate numerically the approximation of several simplified or full hyperbolic models arising in fluid dynamics this leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields the convergence error vanishes for more general models this measure is evaluated very accurately especially by plotting the shock curves associated with each scheme under consideration as we demonstrate plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme

error analysis of exponential integrators for oscillatory second order differential equations in this paper we analyse a family of exponential integrators for second order differential equations in which high frequency oscillations in the solution are generated by a linear part conditions are given which guarantee that the integrators allow second order error bounds independent of the product of the step size with the frequencies our convergence analysis generalizes

weak second order multirevolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise we introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise we prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations which permits to use large time steps the approach is based on the micro macro framework of multi revolution composition methods recently introduced for deterministic problems and inherits its geometric features in particular to design integrators preserving exactly quadratic first integral numerical experiments including the stochastic nonlinear schrodinger equation with space time multiplicative noise illustrate the performance and versatility of the approach

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

a riemann problem in gas dynamics with bifurcation abstract we construct the solution of the riemann problem for 2 2 isothermal gas dynamics in a duct with discontinuous diameter besides shocks and rarefaction there are standing waves the solution exists globally it is obtained as an asymptotic solution for an appropriate cauchy problem with continuous data in certain cases bifurcation occurs and there are three solutions one of which is unstable this is an example of a riemann problem whose solution depends discontinuously on the initial data

a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section we present a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section the model is of nonconservative form making it hard for standard numerical discretizations of the nonconservative term in particular the error for a standard numerical scheme with a usual discretization of the nonconservative term may become larger as the mesh size gets smaller we first re investigate the riemann problem of the model pointing out several interesting properties of the wave curves and establishing specific existence domain for each type of solutions then we incorporate local riemann solutions to build a godunov type scheme for the model the scheme is constructed in subsonic and supersonic regions where the system is strictly hyperbolic tests show that our scheme can capture standing waves so that it is well balanced furthermore tests also show that our godunov type scheme can give a good accuracy for numerical approximations of exact solutions our godunov type scheme can resolve the difficulty of other existing schemes for similar models of fluid flows with nonconservative terms

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

convergence analysis of trigonometric methods for stiff second order stochastic differential equations we study a class of numerical methods for a system of second order sde driven by a linear fast force generating high frequency oscillatory solutions the proposed schemes permit the use of large step sizes have uniform global error bounds in the position i e independent of the large frequencies present in the sde and offer various additional properties this new family of numerical integrators for sde can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems

the resonant cases and the riemann problem for a model of two phase flows abstract the resonant cases in the riemann problem for a model of two phase flows with resonance are investigated which complete the construction of riemann solutions the model is given by a nonconservative hyperbolic system of balance laws the phase decomposition method is developed to construct solutions in each phase in a relatively separated way but still glued to each other via the contact discontinuities composite wave curves can be built and intersections of wave curves in each phase plane help to determine various configurations of riemann solutions in particular solutions containing multiple waves propagating with the same shock speed are constructed

piecewise linear homotopy algorithms for sparse systems of nonlinear equations when piecewise linear homotopy algorithms are applied to the problem of approximating a zero of a sparse function f r n to r n a large piece of linearity can be traversed in one step by using a suitable linear system the linear system has n rows and n 1 columns but is subject to a number of inequalities depending on the sparsity pattern of f we show how an algorithm can be implemented using these large pieces in particular we demonstrate how to update the linear system corresponding to one large piece to obtain the appropriate system for an adjacent large piece one measure of the complexity of such an implementation is the number of inequalities that may be required for any one piece we prove that there can be no more than o n 3 2 such inequalities and that this bound is essentially tight the argument is purely combinatorial finally we provide guidelines on when such a large piece implementation should be used instead of much simpler small piece implementations for piecew

constructing a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section is presented the model has the form of a nonconservative system of balance laws which poses many challenging questions for study first exact riemann solvers in computing form are described in both subsonic and supersonic regions second the computable exact solutions of local riemann problem are incorporated into a godunov type scheme third the scheme is shown to be well balanced in the sense that it can capture exactly stationary waves finally numerical tests for data belong to both subsonic and supersonic regions are presented these tests show that the scheme has a very fine accuracy especially the scheme can give very good approximations to the exact solutions even in the resonant phenomenon where the solution contains three waves of the same speed

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier