vanishing viscosity with short wave long wave interactions for multi d scalar conservation laws we consider a system coupling a multidimensional semilinear schrodinger equation and a multidimensional nonlinear scalar conservation law with viscosity which is motivated by a model of short wave long wave interaction introduced by benney 1977 we prove the global existence and uniqueness of the solution of the cauchy problem for this system we also prove the convergence of the whole sequence of solutions when the viscosity e and the interaction parameter approach zero so that o e1 2 we also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of schrodinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled

finite element methods with numerical quadrature for elliptic problems with smooth interfaces the purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients due to low global regularity of the solution it seems difficult to achieve optimal order of convergence with classical finite element methods z chen j zou finite element methods and their convergence for elliptic and parabolic interface problems numer math 79 1998 175 202 we derive error estimates in finite element method with quadrature for elliptic interface problems in a two dimensional convex polygonal domain optimal order error estimates in l 2 and h 1 norms are shown to hold even if the regularity of the solution is low on the whole domain finally numerical experiment for two dimensional test problem is presented in support of our theoretical findings

lp independence of spectral bounds of feynman kac semigroups by continuous additive functionals abstract we establish conditions for the l p independence of spectral bounds of feynman kac semigroup by continuous additive functionals whose revuz measures are smooth measures of kato class having non negative order green tightness our continuous additive functionals do not necessarily admit bounded variation in general examples of cauchy principal value and hilbert transform of brownian local time and for relativistic symmetric stable processes are presented

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

a class of nonlinear nonhyperbolic systems of conservation laws with well posed initial value problems abstract with mild restrictions on the initial data we show well posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector the obtained weak solutions assume values as borel measures at each time after a smooth solution ceases to exist the concept of entropy density flux functions is extended to such systems finally we show that systems with well posed initial value problems admitting weak solutions of this type necessarily satisfy these structural assumptions

a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces we present a finite volume method for the solution of the two dimensional elliptic equation s x u x f x with variable discontinuous coefficients and solution discontinuities on irregular domains the method uses bilinear ansatz functions on cartesian grids for the solution u x resulting in a compact nine point stencil the resulting linear problem has been solved with a standard multigrid solver singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two step asymptotic approach the method achieves second order of accuracy in the l and l2 norm

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

vanishing viscosity with short wave long wave interactions for systems of conservation laws motivated by benney s general theory we propose new models for short wave long wave interactions when the long waves are described by nonlinear systems of conservation laws we prove the strong convergence of the solutions of the vanishing viscosity and short wave long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems we analyze several of the representative examples such as scalar conservation laws general symmetric systems nonlinear elasticity and nonlinear electromagnetism

well posedness and existence of bound states for a coupled schrodinger gkdv system abstract we derive some new results concerning the cauchy problem and the existence of bound states for a class of coupled nonlinear schrodinger gkdv systems in particular we obtain the existence of strong global solutions for initial data in the energy space h 1 r h 1 r generalizing previous results obtained in tsutsumi 1993 11 corcho and linares 2007 13 and dias et al submitted for publication 14 for the nonlinear schrodinger kdv system

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

gaugeability and conditional gaugeability new kato classes are introduced for general transient borel right processes for which gauge and conditional gauge theorems hold these new classes are the genuine extensions of the green tight measures in the classical brownian motion case however the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability we show that gaugeability conditional gaugeability and the subcriticality for the associated schrodinger operators are equivalent for transient borel right processes with strong duals analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric markov processes these analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases connections with the positivity of the spectral radii of the associated schrodinger operators are also established

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

limit theorems for branching markov processes abstract we establish almost sure limit theorems for a branching symmetric hunt process in terms of the principal eigenvalue and the ground state of an associated schrodinger operator here the branching rate and the branching mechanism can be state dependent in particular the branching rate can be a measure belonging to a certain kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying hunt process x the almost sure limit theorems are established under the assumption that the associated schrodinger operator of x has a spectral gap such an assumption is satisfied if the underlying process x is a brownian motion a symmetric stable like process on r n or a relativistic symmetric stable process on r n

l p independence of spectral radius for generalized feynman kac semigroups under mild conditions on measures used in the perturbation we establish the l p independence of spectral radius for generalized feynman kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric markov processes these results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized feynman kac functionals developed by the second and third authors

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

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

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

local discontinuous galerkin methods for parabolic interface problems with homogeneous and non homogeneous jump conditions abstract in this paper we present the local discontinuous galerkin ldg methods for solving second order parabolic interface problems in two dimensional convex polygonal domains with homogeneous and non homogeneous jump conditions respectively we start analyzing from the homogeneous problems after giving the semi discrete ldg method we derive its stability and a prior error estimate then after a series of derivation we obtain the ldg method for the non homogeneous problems which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface so its analysis can be easily obtained within the framework developed for the homogeneous case it is an advantage of the ldg method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the ldg methods have the optimal and suboptimal convergence rate in the l 2 norms for the potential and the flux respectively in addition they also indicate that the ldg methods possess the same convergence properties in the l norms what is more to avoid the strict time step restriction of explicit schemes the implicit integration factor iif method is employed which not only relaxes the time step to t o h m i n but also remains the important property of ldg method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do we provide numerical examples to illustrate that the numerical scheme combined ldg method with iif method indeed reduces the computational cost greatly

partially penalized immersed finite element methods for elliptic interface problems this article presents new immersed finite element ife methods for solving the popular second order elliptic interface problems on structured cartesian meshes even if the involved interfaces have nontrivial geometries these ife methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in ife functions with the enhanced stability due to the added penalty not only can these ife methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity but also numerical results indicate that their convergence rates in both the h 1 norm and the l 2 norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic ife methods in some situations trace inequalities are established for both linear and bilinear ife functions that are not only critical for the error analysis of these new ife methods but are also of a great potential to be useful in error analysis for other

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws we consider the riemann problem for a system of two decoupled nonstrictly hyperbolic burgers like conservation equations with added artificial viscosity we analytically establish two different vanishing viscosity limits for the solution of this system which correspond to the two cases where one of the viscosities vanishes much faster than the other this is done without altering the initial condition as is necessary with travelling wave methods numerical evidence is then provided to show that when the two viscosities vanish at the same rate the solution converges to a limit that lies strictly between the two previously established limits finally we use control theory to explain the mechanism behind this nonuniqueness behavior which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur

optimal error estimates for linear parabolic problems with discontinuous coefficients a finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients due to low global regularity of the solution it seems difficult to achieve optimal order of convergence with classical finite element methods numer math 79 1998 pp 175 202 in this paper we have used a finite element discretization where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods optimal order error estimates in l2 and h1 norms are shown to hold even if the regularity of the solution is low on the whole domain while the continuous time galerkin method is discussed for the spatially discrete scheme the discontinuous galerkin method is analyzed for the fully discrete scheme the interfaces and boundaries of the domains are assumed to be smooth for our purpose

existence of the entropy solution for a viscoelastic model in this paper we consider the cauchy problem for a viscoelastic model with relaxationut x 0 f u t 1 f u 0with discontinuous large initial data where 0 0 are constants when the system is nonstrictly hyperbolic under the additional assumptionv0x l the system is reduced to an inhomogeneous scalar balance law by employing the special form of the system itself after introducing a definition of entropy solutions to the system we prove the existence uniqueness and continuous dependence of the global entropy solution for the system when the system is strictly hyperbolic some special entropy pairs of the lax type are constructed in which the progression terms are functions of a single variable and the necessary estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove the existence of the global entropy solutions for the corresponding cauchy problem by applying the method of compensated compactness

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

on positivity criticality and the spectral radius of the shuttle operator for elliptic operators in this paper we study the shuttle operator for a second order linear elliptic operator p on a noncompact manifold x zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the laplacian in ir n 3 zhao was motivated by works of chung and varadhan which consider one dimensional schrodinger operators the main purpose of the paper is to extend the above studies we prove that in the general case the spectral radius of the shuttle operator is strictly less equal or strictly greater than 1 if and only if the operator p is respectively subcritical critical or supercritical in x we demonstrate the usefulness of the characterization for proving theorems our approach is purely analytic this research was partially supported by the israel science foundation administered by the israel academy of sciences and humanities and by the fund for the promotion of research at the technion

a sharp interface finite volume method for elliptic equations on cartesian grids we present a second order sharp interface finite volume method for the solution of the three dimensional elliptic equation x u x f x with variable coefficients on cartesian grids in particular we focus on interface problems with discontinuities in the coefficient the source term the solution and the fluxes across the interface the method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27 point stencil singularities associated with vanishing partial volumes of intersected grid cells are removed by a two term asymptotic approach in contrast to the 2d method presented by two of the authors in m oevermann r klein a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal of computational physics 219 2006 749 769 we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz this simplifies the treatment of the different cut cell types and avoids additional special operations for degenerated interface topologies the resulting set of linear equations has been solved with a bicgstab solver preconditioned with an algebraic multigrid in various testcases including large ratios and non smooth interfaces the method achieves second order of accuracy in the l and l 2 norm

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

the cauchy problem for the systems tz x zy 0 in this paper we solve the cauchy problem for the systems tz xz 0 where z u iv c and 1 2 these systems are nonstrictly hyperbolic possessing an isolated umbilic point at z 0 we use the vanishing viscosity method with the help of the theory of compensated compactness uniform bounds in l for the solutions of the viscous systems are not available but such bounds can be found in l2 this makes necessary the use of the generalized young measures and improvements in known techniques of the compensated compactness theory applied to conservation laws

radially symmetric weak solutions for a quasilinear wave equation in two space dimensions abstract we prove the convergence of the radially symmetric solutions to the cauchy problem for the viscoelasticity equations tt div 1 3 2 t as 0 with radially symmetric initial data x 0 0 r t x 0 1 r r x 1 2 x 2 2 1 2 where 0 r 0 r 1 1 to a weak solution of the cauchy problem for the corresponding limit equation with 0 and initial data x 0 0 r t x 0 1 r our analysis is based on energy estimates and the method of compensated compactness closely following serre and shearer convergence with physical viscosity for nonlinear elasticity 1993 unpublished

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

general analytic characterization of gaugeability for feynman kac functionals we give a general analytic characterization of the gaugeability for generalized feynman kac functionals involving continous additive functionals locally of zero energy generalizing the earlier work on the analytic characterization of the gaugeability of feynman kac functional for absorbing brownian motion on a bounded domain given by aizenman simon commun pure appl math 35 2 209 273 1982 the result also extends the previous known results on the analytic characterization of the gaugeability for generalized feynman kac functionals in the framework of symmetric markov processes satisfying irreducibility condition and absolute continuity condition

efficient semi implicit schemes for stiff systems when explicit time discretization schemes are applied to stiff reaction diffusion equations the stability constraint on the time step depends on two terms the diffusion and the reaction the part of the stability constraint due to diffusion can be totally removed if the linear diffusions are treated exactly using integration factor if or exponential time differencing etd methods for systems with severely stiff reactions those methods are not efficient because the reaction terms in if or etd are still approximated with explicit schemes in this paper we introduce a new class of semi implicit schemes which treats the linear diffusions exactly and explicitly and the nonlinear reactions implicitly a distinctive feature of the scheme is the decoupling between the exact evaluation of the diffusion terms and implicit treatment of the nonlinear reaction terms as a result the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points it only depends on the number of original equations unlike the case in which standard implicit temporal schemes are directly applied to the reaction diffusion system the stability region for this class of methods is much larger than existing methods using an explicit treatment of reaction terms in particular the one with second order accuracy is unconditionally linearly stable with respect to both diffusion and reaction direct numerical simulations on test equations as well as morphogen systems from developmental biology show the new semi implicit schemes are efficient robust and accurate

an almost sure scaling limit theorem for dawson watanabe superprocesses abstract we establish a scaling limit theorem for a large class of dawson watanabe superprocesses whose underlying spatial motions are symmetric hunt processes where the convergence is in the sense of convergence in probability when the underling process is a symmetric diffusion with c b 1 coefficients or a symmetric levy process on r d whose levy exponent is bounded from below by c for some c 0 and 0 2 when is large a stronger almost sure limit theorem is established for the superprocess our approach uses the principal eigenvalue and the ground state for some associated schrodinger operator the limit theorems are established under the assumption that an associated schrodinger operator has a spectral gap

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

differentiability of spectral functions for symmetric alpha stable processes let be a signed radon measure in the kato class and define a schrodinger type operator h 1 2 2 on r d we show that its spectral bound c inf h is differentiable if d 2 and is green tight

conditional gaugeability and subcriticality of generalized schrodinger operators abstract we obtain a necessary and sufficient condition for conditional gaugeability and show the equivalence between conditional gaugeability and subcriticality of generalized schrodinger type operators we apply the condition to concrete examples

analytic characterization of conditional gaugeability for non local feynman kac transforms abstract an analytic characterization of gaugeability and conditional gaugeability is given for non local or discontinuous feynman kac transforms of general symmetric markov processes this analytic characterization is very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases

central limit theorem for branching brownian motions in random environment we introduce a model of branching brownian motions in time space random environment associated with the poisson random measure we prove that if the randomness of the environment is moderated by that of the brownian motion the population density satisfies a central limit theorem and the growth rate of the population size is the same as its expectation with strictly positive probability we also characterize the diffusive behavior of our model in terms of the decay rate of the replica overlap on the other hand we show that if the randomness of the environment is strong enough the growth rate of the population size is strictly less than its expectation almost surely to do this we use a connection between our model and the model of brownian directed polymers in random environment introduced by comets and yoshida

a high order hybridizable discontinuous galerkin method for elliptic interface problems we present a high order hybridizable discontinuous galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface the hybridizable discontinuous galerkin method is endowed with several distinct characteristics first they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries thereby leading to a significant reduction in the global degrees of freedom second they provide for elliptic problems with polygonal interfaces approximations of all the variables that converge with the optimal order of kc1 in the l norm where k denotes the polynomial order of the approximation spaces third they possess some superconvergence properties that allow the use of an inexpensive element by element postprocessing to compute a new approximate solution that converges with order kc2 however for elliptic problems with finite jumps in the solution across the curvilinear interface the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric the discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence to recover the optimal convergence for the approximate solution and gradient we propose to use superparametric elements at the interface copyright 2012 john wiley sons ltd

plane waves in nonlinear electromagnetism on enonce un theoreme d existence globale pour le probleme de cauchy des ondes planes les donnees initiales sont supposees mesurables et bornees mais pas necessairement petites

an exact riemann solver for compressible two phase flow models containing non conservative products in this article we present a new numerical procedure for solving exactly the riemann problem of compressible two phase flow models containing non conservative products these products appear in the expressions for the interactions between the two phases thus in the compressible limit the governing equations are hyperbolic but can not be written as conservation laws i e in divergence form in general the solution to the riemann problem of these models contains six distinct centered waves according to the relative position of these waves in the x t plane the possible solutions can be classified into four principal configurations the riemann solver we propose herein investigates sequentially each of these configurations until an admissible solution is calculated special configurations corresponding to coalescence of waves are also analyzed and included in the solver further we examine the accuracy and robustness of three known methods for the integration of the non conservative products via a series of numerical tests finally the issue of existence and uniqueness of solutions to the riemann problem is discussed