invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

second order boltzmann schemes for compressible euler equations in one and two space dimensions a class of second order numerical schemes for the compressible euler equations is described and their l 1 stability i e rho geqq 0 t geqq 0 is proved following van leer s approach the solution rho u sqrt t here is represented as piecewise linear functions the necessity of a slope limitation appears naturally in the derivation of the schemes but it can be less strict than the slope reconstructions usually used these schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space the upwinding is obtained through a very generalized notion of characteristics the kinetic one

why the muscl hancock scheme is l 1 stable the finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws they are developed in the framework of first order numerical schemes several approaches are proposed to increase the order of accuracy the van leer methods are interesting ways one of them namely the muscl hancock scheme is full time and space second order accuracy in the present work we exhibit relevant conditions to ensure the l1 stability of the method a cfl like condition is established and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion in addition we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables numerical experiments are performed to show the interest and the robustness of the method

a minimum entropy principle of high order schemes for gas dynamics equations the entropy solutions of the compressible euler equations satisfy a minimum principle for the specific entropy tadmor in appl numer math 2 211 219 1986 first order schemes such as godunov type and lax friedrichs schemes and the second order kinetic schemes khobalatte and perthame in math comput 62 119 131 1994 also satisfy a discrete minimum entropy principle in this paper we show an extension of the positivity preserving high order schemes for the compressible euler equations in zhang and shu j comput phys 229 8918 8934 2010 and zhang et al j scientific comput in press to enforce the minimum entropy principle for high order finite volume and discontinuous galerkin dg schemes

stochastic averaging of strongly non linear oscillators under bounded noise excitation abstract a stochastic averaging method for strongly non linear oscillators under external and or parametric excitation of bounded noise is proposed by using the so called generalized harmonics functions the method is then applied to study the primary resonance of duffing oscillator with hardening spring under external excitation of bounded noise the stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase subsequently the method is applied to study the dynamical instability and parametric resonance of duffing oscillator with hardening spring under parametric excitation of bounded noise the primary unstable region is delineated by evaluating the lyapunov exponent of linearized system and the response and jump of non linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

robustness of muscl schemes for 2d unstructured meshes we consider second order accuracy muscl schemes to approximate the solutions of hyperbolic system of conservation laws in the context of the 2d unstructured grids we propose a limitation procedure on the gradient reconstruction to enforce several stability properties we establish that the muscl scheme preserves the invariant domains and satisfy a set of entropy inequalities a conservation assumption is not useful in the present work to define the piecewise linear approximations and the proposed limitation can be understood as a systematic correction of the standard gradient reconstruction procedure the numerical method is applied to the compressible euler equations the gradient reconstruction is performed using the characteristic variables several numerical tests exhibit stability and robustness of the scheme

construction of bgk models with a family of kinetic entropies for a given system of conservation laws we introduce a general framework for kinetic bgk models we assume to be given a system of hyperbolic conservation laws with a family of lax entropies and we characterize the bgk models that lead to this system in the hydrodynamic limit and that are compatible with the whole family of entropies this is obtained by a new characterization of maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies we deduce a general procedure to construct such bgk models and we show how classical examples enter the framework we apply our theory to isentropic gas dynamics and full gas dynamics and in both cases we obtain new bgk models satisfying all entropy inequalities

maximum principle on the entropy and second order kinetic schemes we consider kinetic schemes for the multidimensional inviscid gas dynamics equations compressible euler equations we prove that the discrete maximum principle holds for the specific entropy this fixes the choice of the equilibrium functions necessary for kinetic schemes we use this property to perform a second order oscillation free scheme where only one slope limitation for three conserved quantities in 1d is necessary numerical results exhibit stability and strong convergence of the scheme introduction we consider the gas dynamics equations in one or two space dimensions atp div pu 0 1 atpu div puju a p 0 j 1 2 9 e div e p u 0 where x xl x2 u ul u2 and the total energy e piu12 2 pt y 1 is related to the pressure by the relation p pt 1 oh and in 11d godunov and lax friedrichs schemes preserve this property at the discretized level because they solve exactly the system 1 a reason why 4 should hold is that s satisfies the meaningless equation ats u vxs 0 received by the editor march 10 1992 and in revised form january 7 1993 1991 mathematics subject classification primary 35l65 76n10 65m06 76po5

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

entropy satisfying flux vector splittings and kinetic bgk models we establish forward and backward relations between entropy satisfying bgk relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws classically to a kinetic bgk model that is compatible with some family of entropies we can associate an entropy flux vector splitting we prove that the converse is true any entropy flux vector splitting can be interpreted by a kinetic model and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes we deduce a new proof of discrete entropy inequalities under a sharp cfl condition that generalizes the monotonicity criterion in the scalar case in particular this gives a stability condition for numerical kinetic methods with noncompact velocity support a new interpretation of general kinetic schemes is also provided via approximate riemann solvers we deduce the construction of finite velocity relaxation systems for gas dynamics and obtain a hllc scheme for which we are able to prove positiveness of density and internal energy and discrete entropy inequalities

structure preserving exponential runge kutta methods exponential runge kutta erk and partitioned exponential runge kutta perk methods are developed for solving initial value problems with vector fields that can be split into conservative and linear nonconservative parts the focus is on linearly damped ordinary differential equations that possess certain invariants when the damping coefficient is zero but in the presence of constant or time dependent linear damping the invariants satisfy linear differential equations similar to the way that runge kutta and partitioned runge kutta methods preserve quadratic invariants and symplecticity for hamiltonian systems erk and perk methods exactly preserve conformal symplecticity as well as decay or growth rates in linear and quadratic invariants under certain constraints on their coefficient functions numerical experiments illustrate the higher order accuracy and structure preserving properties of various erk methods demonstrating clear advantages over classical conservative runge kutta methods a

exponential integrators preserving local conservation laws of pdes with time dependent damping driving forces abstract structure preserving algorithms for solving conservative pdes with added linear dissipation are generalized to systems with time dependent damping driving terms this study is motivated by several pde models of physical phenomena such as korteweg de vries klein gordon schrodinger and camassa holm equations all with damping driving terms and time dependent coefficients since key features of the pdes under consideration are described by local conservation laws which are independent of the boundary conditions the proposed second order in time discretizations are developed with the intent of preserving those local conservation laws the methods are respectively applied to a damped driven nonlinear schrodinger equation and a damped camassa holm equation numerical experiments illustrate the structure preserving properties of the methods as well as favorable results over other competitive schemes

entropy bounded discontinuous galerkin scheme for euler equations an entropy bounded discontinuous galerkin ebdg scheme is proposed in which the solution is regularized by constraining the entropy the resulting scheme is able to stabilize the solution in the vicinity of discontinuities and retains the optimal accuracy for smooth solutions the properties of the limiting operator according to the entropy minimum principle are proofed and an optimal cfl criterion is derived we provide a rigorous description for locally imposing entropy constraints to capture multiple discontinuities significant advantages of the ebdg scheme are the general applicability to arbitrary high order elements and its simple implementation for multi dimensional configurations numerical tests confirm the properties of the scheme and particular focus is attributed to the robustness in treating discontinuities on arbitrary meshes

exponential integrators with quadratic energy preservation for linear poisson systems abstract motivated by the recent paper wang and wu 2018 12 where the authors proposed a class of functionally fitted energy preserving integrators for poisson systems in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear poisson systems by rewriting the linear poisson system in a new formulation we apply the symplectic exponential runge kutta methods and thus obtain a novel class of exponential integrators for the linear poisson system it is proved that the obtained exponential integrator can exactly preserve the quadratic energy in comparison with the functionally fitted energy preserving methods whose numerical performance depends on the fitted frequency the new exponential integrators are independent of the fitted frequency and convenient to use furthermore we investigate the convergence of the fixed point iteration used in the implementation of the proposed exponential integrator and show that the convergence region of the stepsize is independent of the norm of the skew symmetric matrix finally numerical results with the euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation

navier stokes equations with several independent pressure laws and explicit predictor corrector schemes this work is concerned with the numerical capture of stiff viscous shock solutions of navier stokes equations for complex compressible materials in the regime of large reynolds numbers after 2 and 6 a relevant numerical capture is known to require the satisfaction of an extended set of non classical rankine hugoniot conditions due to the non conservation form of the governing pde model here we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate riemann solutions a relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

the behavior of flux difference splitting schemes near slowly moving shock waves abstract an investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of riemann s problem is performed the schemes of godunov roe and osher are examined for a one dimensional model problem consisting of a nearly stationary shock both scalar and systems of equations are examined it is found that for slow shocks there is a significant error generated when solving systems of equations while the scalar results are well behaved this error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme the source of this error is shown and the implications for the performance of these schemes are considered this error may contribute to the slow convergence to steady state reported by many researchers

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

stable lattice boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves we follow the mathematical framework proposed by bouchut 2003 22 and present in this contribution a dual entropy approach for determining equilibrium states of a lattice boltzmann scheme this method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law it appears as a good mathematical framework for establishing a h theorem for the system of equations with discrete velocities the dual entropy approach is used with d1q3 lattice boltzmann schemes for the burgers equation it conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition satisfactory numerical results for strong nonlinear shocks and rarefactions are presented we prove also that the dual entropy approach can be applied with a d1q3 lattice boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics we establish also a negative result with the present framework the dual entropy approach cannot be used for the shallow water equations

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

second order invariant domain preserving ale approximation of hyperbolic systems abstract in this paper we introduce an invariant domain preserving arbitrary lagrangian eulerian method for solving hyperbolic systems the time stepping is explicit and the approximation in space is done with continuous finite elements the method is second order in space and made invariant domain preserving by using a newly introduced convex limiting technique

an entropy satisfying muscl scheme for systems of conservation laws for the high order numerical approximation of hyperbolic systems of conservation laws we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by harten for scalar equations based on this new criterion we derive entropy diminishing projections that ensure both the second order of accuracy and all of the classical discrete entropy inequalities the resulting scheme is a nonlinear version of the classical van leer s muscl scheme strong convergence of this second order entropy satisfying scheme is proved for systems of two equations numerical tests demonstrate the interest of our theory