hllc type and path conservative schemes for a single velocity six equation two phase flow model a comparative study abstract the present article deals with the numerical integration of a six equation single velocity two phase flow model with stiff mechanical relaxation this model can be employed to approximate efficiently the well known single velocity single pressure five equation model of kapila et al 2001 work in the literature has shown the efficiency of the six equation model in simulating complex two phase flows involving cavitation and evaporation processes the aim of this work is to present and discuss various numerical schemes for this two phase model focusing on the integration of the nonconservative terms appearing in the phasic energy equations in fact previous work has suggested that the choice of the discretization method for the nonconservative terms often does not play a significant role two new methods are proposed a path conservative hllc type scheme that is based on the dal maso lefloch murat theory and a generalized hllc type scheme that is based on a suliciu s riemann solver the latter scheme has the important property of preserving the positivity of the intermediate states of the conserved quantities moreover we also approximate solutions of the six equation model by applying two path conservative schemes recently proposed in the literature which have been derived from the osher and hllem riemann solvers we show comparisons of the different numerical schemes for several test cases including cavitation problems and shock tubes an efficiency study for first and second order schemes is also presented numerical results show that different methods corresponding to different numerical treatments of the nonconservative terms give analogous results and they are all able to produce accurate approximations of solutions of the kapila s five equation model except as expected for shocks in two phase mixtures with very high pressure ratios

high order exactly well balanced numerical methods for shallow water systems abstract this paper is concerned with the development of high order methods for the shallow water system that preserve every stationary solution of the system and not only the water at rest ones the pursued strategy is based on the generalized hydrostatic reconstruction introduced in 10 and on the technique proposed in 8 to obtain high order well balanced reconstruction operators the more difficult step in the reconstruction procedure is the first one given a cell value a stationary solution of the system has to be found whose average at the cell equals the given value the implementation of this step is discussed in detail as well as the similarities and differences with some previous approaches to obtain well balanced methods for the shallow water system finally the well balanced properties of the developed numerical methods are checked in some test cases

well balanced bicharacteristic based scheme for multilayer shallow water flows including wet dry fronts the aim of this paper is to present a new well balanced finite volume scheme for two dimensional multilayer shallow water flows including wet dry fronts the ideas presented here for the two layer model can be generalized to a multilayer case in a straightforward way the method developed here is constructed in the framework of the finite volume evolution galerkin fveg schemes the fveg methods couple a finite volume formulation with evolution operators the latter are constructed using the bicharacteristics of multidimensional hyperbolic systems however in the case of multilayer shallow water flows the required eigenstructure of the underlying equations is not readily available thus we approximate the evolution operators numerically this approximation procedure can be used for arbitrary hyperbolic systems we derive a well balanced approximation of the evolution operators and prove that the fveg scheme is well balanced for the multilayer lake at rest states even in the presence of wet dry fronts several numerical experiments confirm the reliability and efficiency of the new well balanced fveg scheme

numerical investigation of a modified family of centered schemes applied to multiphase equations with nonconservative sources systems of hyperbolic partial differential equations are frequently used to model the flow of multiphase mixtures these equations often contain sources referred to as nozzling terms that cannot be posed in divergence form and have proven to be particularly challenging in the development of finite volume methods upwind schemes have recently shown promise in properly resolving the steady wave solution of the associated multiphase riemann problem however these methods require a full characteristic decomposition of the system eigenstructure which may be either unavailable or computationally expensive central schemes such as the kurganov tadmor kt family of methods require minimal characteristic information which makes them easily applicable to systems with an arbitrary number of phases however the proper implementation of nozzling terms in these schemes has been mathematically ambiguous the primary objectives of this work are twofold first an extension of the kt family of schemes is proposed that formally accounts for the nonconservative nozzling sources this modification results in a semidiscrete form that retains the simplicity of its predecessor and introduces little additional computational expense second this modified method is applied to multiple but equivalent forms of the multiphase equations to perform a numerical study by solving several one dimensional test problems both ideal and mie gruneisen equations of state are used with the results compared to an analytical solution this study demonstrates that the magnitudes of the resulting numerical errors are sensitive to the form of the equations considered and suggests an optimal form to minimize these errors finally a separate modification of the wave propagation speeds used in the kt family is also suggested that can reduce the extent of numerical diffusion in multiphase flows

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

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

a musta scheme for a nonconservative two fluid model we present a multistage centered scheme of the kind proposed by toro appl numer math 56 2006 pp 1464 1479 for numerically resolving the simultaneous flow of two fluids through a transport pipeline this model contains nonconservative terms in both the temporal and spatial derivatives and an extension of the standard numerical framework for conservation laws is needed in this paper we rewrite the model in an equivalent mathematical form eliminating the nonconservative time derivatives this allows us to use the framework described by pares siam j numer anal 44 2006 pp 300 321 we develop force and musta type schemes which are consistent with pares formalism numerical simulations demonstrate a high degree of stability of our proposed schemes comparisons with the roe and rusanov schemes indicate that convergence to near identical solutions is obtained when the nonconservative terms are discretized with respect to the same evaluation of the path dependent integrals however if the schemes are not mutually formally path consistent in the sense of pares different converged solutions are obtained

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 many theories of shock waves are necessary kinetic relations for non conservative systems for a class of nonconservative hyperbolic systems of partial dif ferential equations endowed with a strictly convex mathematical entropy we formulate the initial value problem by supplementing the equations with a ki netic relation prescribing the rate of entropy dissipation across shock waves our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by abeyaratne knowles and truskinovsky for subsonic phase transitions and by lefloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems the proposed kinetic relation for non conservative systems turns out to be equivalent for the class of systems under consideration at least to dal maso lefloch and murat s definition based on a prescribed family of lipschitz continuous paths in agreement with previous theories the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconserva tive system we illustrate with several examples that nonconservative systems arising in the applications fit in our framework and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function

testing for autoregressive against moving average errors in the linear regression model abstract this paper presents a new approach to hypotheses testing problems which are non nested in the classical sense and which concern the covariance matrix of the disturbance vector of the linear regression model in particular the application of the approach to testing for ar 1 disturbances against ma 1 disturbances is explored in some detail practical difficulties are discussed and selected upper bounds for the test s five percent significance points are tabulated the small sample power of four versions of the new test are compared empirically and a clear conclusion is made in regard to the best overall test

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

constraint consistent runge kutta methods for one dimensional incompressible multiphase flow new time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one dimensional two fluid model the methodology is based on halfexplicit runge kutta methods being explicit for the mass and momentum equations and implicit for the volume constraint these half explicit methods are constraint consistent i e they satisfy the hidden constraints of the two fluid model namely the volumetric flow incompressibility constraint and the poisson equation for the pressure a novel analysis shows that these hidden constraints are present in the continuous semi discrete and fully discrete equations next to constraint consistency the new methods are conservative the original mass and momentum equations are solved and the proper shock conditions are satisfied efficient the implicit constraint is rewritten into a pressure poisson equation and the time step for the explicit part is restricted by a cfl condition based on the convective wave speeds and accurate achieving high order temporal accuracy for all solution components masses velocities and pressure highorder accuracy is obtained by constructing a new third order runge kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time dependent boundary conditions two test cases kelvin helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank show that for time independent boundary conditions the half explicit formulation with a classic fourth order runge kutta method accurately integrates the two fluid model equations in time while preserving all constraints a third test case ramp up of gas production in a multiphase pipeline shows that our new third order method is preferred for cases featuring time dependent boundary conditions

high order wave propagation algorithms for hyperbolic systems we present a finite volume method that is applicable to hyperbolic pdes including spatially varying and semilinear nonconservative systems the spatial discretization like that of the well known clawpack software is based on solving riemann problems and calculating fluctuations not fluxes the implementation employs weighted essentially nonoscillatory reconstruction in space and strong stability preserving runge kutta integration in time the method can be extended to arbitrarily high order of accuracy and allows a well balanced implementation for capturing solutions of balance laws near steady state this well balancing is achieved through the f wave riemann solver and a novel wave slope weno reconstruction procedure the wide applicability and advantageous properties of the method are demonstrated through numerical examples including problems in nonconservative form problems with spatially varying fluxes and problems involving near equilibrium solutions of balance laws

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

aggregation kinetics of popularity abstract the tools of aggregation kinetics are applied to the popularity phenomena of single lane traffic clustering and to the growth of a network that mimics citations of scientific publications in the latter the network is built by introducing papers new nodes one at a time with preferential linking to more popular previously existing nodes from the rate equations the distribution of node degree as well as various global properties can be determined easily a simple extension of the model appears to describe the degree distributions of the world wide web

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

spr based markov chain method for degree distributions of evolving networks abstract in this paper we develop a stochastic process rules spr based markov chain method to calculate the degree distributions of evolving networks this new approach overcomes two shortcomings of shi chen and liu s use of the markov chain method shi et al 2005 21 in addition we show how an spr based markov chain method can be effectively used to calculate degree distributions of random birth and death networks which we believe to be novel first spr are introduced to replace traditional evolving rules tr making it possible to compute degree distributions in one sample space then the spr based markov chain method is introduced and tested by using it to calculate two kinds of evolving network finally and most importantly the spr based method is applied to the problem of calculating the degree distributions of random birth and death networks

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

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

short note a comment on the computation of non conservative products we are interested in the solution of non conservative hyperbolic systems and consider in particular the so called path conservative schemes see e g 2 3 which rely on the theoretical work in 1 the example of the standard euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach

a new hydrostatic reconstruction scheme based on subcell reconstructions a key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom solu

a new approximate point optimal test of a composite null hypothesis in this paper we use the generalized neyman pearson lemma to introduce a new approximate point optimal test that can be used for testing a composite null hypothesis against a composite alternative the new test involves finding multiple critical values two methods for obtaining these critical values are outlined we report simulations of the application of this test to two composite non nested testing problems namely testing for first order moving average ma 1 errors against first order autoregressive ar 1 errors in the linear regression model and testing for ar 1 errors against integrated ma 1 ima 1 1 errors in the linear model we compare the performance of the new test with silvapulle and king s 1991 approximate point optimal test and some asymptotic tests and find that the new test has a clear advantage over the other tests particularly for the second testing problem

a new comment on the computation of non conservative products using roe type path conservative schemes abstract we are interested in the numerical approximation of the discontinuous solutions of non conservative hyperbolic systems we more precisely consider a non conservative formulation of the usual gas dynamics equations and show how to slightly modify the so called roe type path conservative schemes to properly capture the underlying shock discontinuities numerical evidences are proposed the present note follows a first comment on the computation of non conservative products in 1

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

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