well balanced unstaggered central schemes for the euler equations with gravitation we consider the euler equations with gravitational source term and propose a new well balanced unstaggered central finite volume scheme which can preserve the hydrostatic balance state exactly the proposed scheme evolves a nonoscillatory numerical solution on a single grid avoids the time consuming process of solving riemann problems arising at the cell interfaces and is second order accurate both in time and space furthermore the numerical scheme follows a well balanced discretization that first discretizes the gravitational source term according to the discretization of the flux terms and then mimics the surface gradient method and discretizes the density and energy according to the discretization of steady state density and energy functions respectively finally several numerical experiments demonstrating the performance of the well balanced schemes in both one and two spatial dimensions are presented the results indicate that the new scheme is accurate simple and robust

a well balanced scheme for the shallow water equations with topography a non negativity preserving and well balanced scheme that exactly preserves all the smooth steady states of the shallow water system including the moving ones is proposed in addition the scheme must deal with vanishing water heights and transitions between wet and dry areas a godunov type method is derived by using a relevant average of the source terms within the scheme in order to enforce the required well balance property a second order well balanced muscl extension is also designed numerical experiments are carried out to check the properties of the scheme and assess the ability to exactly preserve all the steady states

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

a conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws in the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method we propose a new formulation of the first order linear optimal positive n scheme relying on a contour integration of the convective fluxes over the boundaries of an element full conservation is achieved for arbitrary flux functions while the robustness and the monotone shock capturing of the original n scheme is retained the new variant of the n scheme is combined with the conservative second order linear lda scheme to obtain a nonlinear second order monotone b scheme the performance of the new residual distribution schemes is evaluated on problems governed by the euler equations as an application to a more complex system of conservation laws lacking an exact conservative linearization we solve the ideal magnetohydrodynamics equations in two spatial dimensions

a modified central discontinuous galerkin method with positivity preserving and well balanced properties for the one dimensional nonlinear shallow water equations abstract in this paper we present a modified central discontinuous galerkin method for solving the nonlinear shallow water equations over variable bottom topography in one dimensional space the proposed method balances exactly the flux gradient and source term in the still water stationary case by adding a correction term to the base scheme and ensures non negativity of the water depth by using special approximations to the bottom together with a positivity preserving limiter meanwhile reduces the computational cost considerably compared with the standard central discontinuous galerkin method by using l 2 projection to obtain an approximation solution numerical tests are presented to illustrate the accuracy and validity of the proposed schemes

a second order accurate monotone implicit fluctuation splitting scheme for unsteady problems this paper provides an implicit fluctuation splitting scheme which achieves second order accuracy in both space and time while guaranteeing monotone solutions the method is based on a dual time stepping approach which allows to separate the spatial and temporal discretizations a novel distribution scheme of the unsteady fluctuation term coupled with a new limiting procedure for hampering spurious extrema are the key ingredients of the remarkable accuracy of the new method which is applied with success to both scalar advection problems and unsteady inviscid flows in two space dimensions

a numerical method for finding positive solution of diffusive logistic equation abstract using a numerical methods based on sub super solution we will find positive solution for the diffusive logistic equation u au bu 2 0 for x with dirichlet boundary condition

third order accurate fluctuation splitting schemes for unsteady hyperbolic problems this paper provides a two dimensional fluctuation splitting scheme for unsteady hyperbolic problems which achieves third order accuracy in both space and time for a scalar conservation law the sufficient conditions for a stable fluctuation splitting scheme to achieve a prescribed order of accuracy in both space and time are derived then using a quadratic space approximation of the solution over each triangular element based on the reconstruction of the gradient at the three vertices and a four level backward discretization of the time derivative an implicit third order accurate scheme is designed such a scheme is extended to the euler system and is validated versus well known scalar advection problems and inviscid discontinuous flows

singular limits for the compressible euler equation in an exterior domain justification des phenomenes lies au passage des equations des fluides parfaits compressibles a celles des fluides parfaits incompressibles limites singulieres de l equation d euler des fluides compressibles dans un domaine exterieur

on the well balanced numerical discretization of shallow water equations on unstructured meshes we consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography on unstructured meshes relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable instead of the water height we introduce a simple discretization of the bed slope source term together with some suitable conservative variables reconstructions the resulting scheme is automatically consistent and well balanced for any given consistent numerical flux for the homogeneous system we obtain a very simple formulation which do not need to be modified when second order accuracy muscl reconstructions are adopted additionally the positivity of the water height is preserved under a relevant stability condition as soon as the numerical flux for the associated homogeneous system does numerical assessments involving dry areas and complex geometry are performed

nonlinear stability criteria for the hmf model we study the nonlinear stability of a large class of inhomogeneous steady state solutions to the hamiltonian mean field hmf model under a specific criterion we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy to achieve this task we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational vlasov poisson equation explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to hmf which are physically relevant in this context but induces additional difficulties compared to the vlasov poisson system

on the riemann problem for non conservative hyperbolic systems we consider the construction and the properties of the riemann solver for the hyperbolic system

negative specific heat in the canonical statistical ensemble according to thermodynamics the specific heat of boltzmannian short range interacting systems is a positive quantity less intuitive properties are instead displayed by systems characterized by long range interactions in that case the sign of specific heat depends on the considered statistical ensemble negative specific heat can be found in isolated systems which are studied in the framework of the microcanonical ensemble on the other hand it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath for which the canonical ensemble is the appropriate one we demonstrate that the latter assumption is not generally true one can in principle measure negative specific heat also in the canonical ensemble if the system under scrutiny is non boltzmannian and or out of equilibrium

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

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

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

an accurate low mach scheme for a compressible two fluid model applied to free surface flows classical approximate riemann solvers are known to be too much dissipative in the low mach number regime for this reason since the mach number in liquids is generally very small usual upwind schemes may provide inaccurate solutions when applied to the simulation of two phase flows in this paper to circumvent this difficulty while keeping a compressible model for the description of both gas and liquid an original accurate low mach scheme is introduced and theoretically studied extending some ideas already used for the gas dynamics system the proposed scheme is based on a centred formulation for the pressure gradient term in the momentum equation and on the introduction of a stabilising term proportional to the pressure difference between two neighbouring cells the scheme stability is ensured and theoretically proved under a convective cfl like condition by using a semi implicit time discretisation algorithm finally the correct asymptotic behaviour of the scheme in the limit of small mach numbers is assessed on several academic test cases

application of the grp scheme to open channel flow equations the grp generalized riemann problem scheme originally conceived for gasdynamics is reformulated for the numerical integration of the shallow water equations in channels of rectangular cross section variable width and bed profile including a friction model for the fluid channel shear stress this scheme is a second order analytic extension of the first order godunov scheme based on time derivatives of flow variables at cell interfaces resulting from piecewise linear data reconstruction in cells the second order time integration is based on solutions to generalized riemann problems at cell interfaces thus accounting for the full governing equations including source terms the source term due to variable bed elevation is treated in a well balanced way so that quiescent flow is exactly replicated this is done by adopting the surface gradient method sgm several problems of steady or unsteady open channel flow are considered including the terms corresponding to variable channel width and bed elevation as well as to shear stress at the fluid channel interface using the manning friction model in all these examples remarkable agreement is obtained between the numerical integration and the exact or accurate solutions

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

well balanced high order extensions of godunov s method for semilinear balance laws this paper is concerned with the development of well balanced high order numerical schemes for systems of balance laws with a linear flux function whose coefficients may be variable first well balanced first order numerical schemes are obtained based on the use of exact solvers of riemann problems that include both the flux and the source terms godunov s methods so obtained are extended to higher order schemes by using a technique of reconstruction of states the main contribution of this paper is to introduce a reconstruction technique that preserves the well balanced property of godunov s methods some numerical experiments are presented to verify in practice the properties of the developed numerical schemes

entropy satisfying relaxation method with large time steps for euler ibvps this paper could have been given the title how to positively and implicitly solve euler equations using only linear scalar advections the new relaxation method we propose is able to solve euler like systems as well as initial and boundary value problems with real state laws at very low cost using a hybrid explicit implicit time integration associated with the arbitrary lagrangian eulerian formalism furthermore it possesses many attractive properties such as i the preservation of positivity for densities ii the guarantee of min max principle for mass fractions iii the satisfaction of entropy inequality under an expressible bound on the cfl ratio the main feature that will be emphasized is the design of this optimal time step which takes into account data not only from the inner domain but also from the boundary conditions

a family of stable numerical solvers for the shallow water equations with source terms abstract in this work we introduce a multiparametric family of stable and accurate numerical schemes for 1d shallow water equations these schemes are based upon the splitting of the discretization of the source term into centered and decentered parts these schemes are specifically designed to fulfill the enhanced consistency condition of bermudez and vazquez necessary to obtain accurate solutions when source terms arise our general family of schemes contains as particular cases the extensions already known of roe and van leer schemes and as new contributions extensions of steger warming vijayasundaram lax friedrichs and lax wendroff schemes with and without flux limiters we include some meaningful numerical tests which show the good stability and consistency properties of several of the new methods proposed we also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

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

higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the csp method in systems of stiff ordinary differential equations odes both fast and slow time scales are encountered the fast time scales are responsible for the development of low dimensional manifolds on which the solution moves according to the slow time scales in this paper methodologies for constructing highly accurate i expressions describing the manifold and ii simplified non stiff equations governing the slow evolution of the solution on the manifold are developed according to an iterative procedure proposed in the computational singular perturbation csp method it is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the csp vectors spanning the manifold along the solution trajectory here an algorithm is presented which implements these calculations and is validated on the basis of two simple examples

analysis of numerically induced oscillations in 2d finite element shallow water models part i inertia gravity waves the numerical approximation of shallow water models is a delicate problem for most of the discretization schemes the coupling between the momentum and the continuity equations usually leads to anomalous dispersion in the representation of fast waves a dispersion relation analysis is employed here to ascertain the presence and determine the form of spurious modes as well as the dispersive nature of the finite element galerkin mixed formulation of the two dimensional linearized shallow water equations nine popular finite element pairs are considered using a variety of mixed interpolation schemes for each pair the frequency or dispersion relation is obtained and analyzed and the dispersion properties are compared analytically and graphically with the continuous case it is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of fast waves the p nc 1 p 1 and rt0 pairs are identified as a promising compromise provided the grid resolution is high relative to the rossby radius of deformation for the rt0 element the numerical solutions of two test problems to simulate fast waves are in good agreement with the analytical results

a positivity preserving and well balanced dg scheme using finite volume subcells in almost dry regions this paper deals with a subcell finite volume strategy for a discontinuous galerkin dg scheme discretizing the shallow water equations with non constant bottom topography using finite volume subcells within the dg scheme is a recent shock capturing strategy decomposing elements marked by a shock indicator into a specific number of subcells on which a robust fv scheme is applied this work newly considers the subcell approach in the context of wetting and drying shallow water flows here finite volume subcells are introduced for almost dry cells in order to enable an improved resolution of the wet dry front furthermore in this work the theory for positivity preserving and well balanced dg schemes is extended to the dg scheme using finite volume subcells the performance of the resulting novel approach both in the context of preserving lake at rest stationary states and for wetting and drying computations is then numerically demonstrated for challenging standard test cases in two space dimensions on unstructured triangular grids

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

determination of bed elevation in the enhanced lattice boltzmann method for the shallow water equations the bed slope in the shallow water equations reflects the bed topography it is not a flow variable and cannot be determined in the solution to the flow equations an immovable nonflat bed affects a flow as a force term but the flow has no effect on it only when the bed term is correctly represented in a numerical method can it generate an accurate solution in the enhanced lattice boltzmann method for the shallow water equations elabswe using an individual chapman enskog analysis it is found that such a correct representation can be achieved by retaining c 2 in which c is the coefficient for bed elevation in the lattice boltzmann equation and is that for the water depth in the local equilibrium distribution function the finding has been validated through simulations of a water at rest in a dish shaped lake a wind induced shallow flow in the same lake and a steady flow over a two dimensional bed hump

a subsonic well balanced reconstruction scheme for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom in past years efficient well balanced methods have been proposed in order to well resolve solutions close to steady states at rest here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic well balanced scheme preserving exactly all the subsonic steady states it generalizes the now well known hydrostatic solver and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality an application to the euler poisson system is proposed

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

an energy and potential enstrophy conserving numerical scheme for the multi layer shallow water equations with complete coriolis force we present an energy and potential enstrophy conserving scheme for the non traditional shallow water equations that include the complete coriolis force and topography these integral conservation properties follow from material conservation of potential vorticity in the continuous shallow water equations the latter property cannot be preserved by a discretisation on a fixed eulerian grid but exact conservation of a discrete energy and a discrete potential enstrophy seems to be an effective substitute that prevents any distortion of the forward and inverse cascades in quasi two dimensional turbulence through spurious sources and sinks of energy and potential enstrophy and also increases the robustness of the scheme against nonlinear instabilities we exploit the existing arakawa lamb scheme for the traditional shallow water equations reformulated by salmon as a discretisation of the hamiltonian and poisson bracket for this system the non rotating traditional and our non traditional shallow water equations all share the same continuous hamiltonian structure and poisson bracket provided one distinguishes between the particle velocity and the canonical momentum per unit mass we have determined a suitable discretisation of the non traditional canonical momentum which includes additional coupling between the layer thickness and velocity fields and modified the discrete kinetic energy to suppress an internal symmetric computational instability that otherwise arises for multiple layers the resulting scheme exhibits the expected second order convergence under spatial grid refinement we also show that the drifts in the discrete total energy and potential enstrophy due to temporal truncation error may be reduced to machine precision under suitable refinement of the timestep using the third order adams bashforth or fourth order runge kutta integration schemes hamiltonian derived scheme exactly conserves energy and potential enstrophy energy enstrophy conserved at machine accuracy for sufficiently small time step can be used to incorporate full coriolis force into existing isopycnal models

quasistationary states in the self gravitating sheet model we study quasistationary states qsss resulting from violent relaxation in the one dimensional self gravitating sheet model revisiting in particular the question of the adequacy of the theory of lynden bell lb to describe them for waterbag initial conditions characterized by a single phase space density the prediction of this theory is in this model a function of only one parameter which can conveniently be chosen to be the ratio of the energy to that in the degenerate limit studying a class of such initial conditions in which the shape of the initial waterbag is varied we find that the lb predictions are reasonably good always in the low energy region while at higher energies i e in the nondegenerate limit they are generally not even qualitatively correct although certain initial conditions can still be found where they are as good as at low energy we find notably that in line with what has been observed by levin et al in some other models when lb theory does not work the qsss are always characterized by the presence of a degenerate core which these authors explain as the result of dynamical resonances in short lb theory appears to be a good approximation only when violent relaxation is sufficiently gentle and otherwise a degenerate core halo structure results

high order well balanced discontinuous galerkin methods based on hydrostatic reconstruction for shallow water equations abstract in this paper we introduce high order well balanced discontinuous galerkin methods for shallow water equations over non flat bottom topography which preserve the lake at rest steady state to achieve the well balanced property we propose to construct the numerical fluxes based on the hydrostatic reconstruction idea and to be in combination with a novel source term approximation as well as a decomposition algorithm rigorous theoretical analysis and extensive numerical results all verify that the current methods maintain the well balanced property in addition numerical results also indicate that the resulting methods enjoy the ability to accurately capture small perturbations of the lake at rest steady state and keep the genuine high order accuracy for smooth solutions

an explicit asymptotic preserving low froude scheme for the multilayer shallow water model with density stratification we present an explicit scheme for a two dimensional multilayer shallow water model with density stratification for general meshes and collocated variables the proposed strategy is based on a regularized model where the transport velocity in the advective fluxes is shifted proportionally to the pressure potential gradient using a similar strategy for the potential forces we show the stability of the method in the sense of a discrete dissipation of the mechanical energy in general multilayer and non linear frames these results are obtained at first order in space and time and extended using a simple second order muscl extension with the objective of minimizing the diffusive losses in realistic contexts sufficient conditions are exhibited on the regularizing terms to ensure the scheme s linear stability at first and second order in time and space the other main result stands in the consistency with respect to the asymptotics reached at small and large time scales in low froude regimes which governs large scale oceanic circulation additionally robustness and well balanced results for motionless steady states are also ensured these stability properties tend to provide a very robust and efficient approach easy to implement and particularly well suited for large scale simulations some numerical experiments are proposed to highlight the scheme efficiency an experiment of fast gravitational modes a smooth surface wave propagation an initial propagating surface water elevation jump considering a non trivial topography and a last experiment of slow rossby modes simulating the displacement of a baroclinic vortex subject to the coriolis force

balanced finite volume weno and central weno schemes for the shallow water and the open channel flow equations the goal of this work is to extend finite volume weno and central weno schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function in particular we apply proposed schemes to the shallow water and the open channel flow equations where the source term depends on the channel geometry for obtaining stable numerical schemes that are free of spurious oscillations it becomes crucial to use the decomposed source term evaluation which maintains the balancing between the flux gradient and the source term in addition the open channel flow equations contain spatially variable flux function the appropriate definitions of the terms that arise in the source term decomposition in combination with the roe approximate riemann solver that includes the spatial derivative of the flux function lead to the finite volume weno scheme that satisfies the exact conservation property the property of preserving the quiescent flow exactly when the central weno schemes are applied additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the weno reconstruction procedure the proposed central weno schemes also preserve the quiescent flow but only in prismatic channels in various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes as well as with some other first and second order numerical schemes

unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations the finite volume discretisation of the shallow water equations has been the subject of many previous studies most of which deal with a well balanced conservative discretisation of the convective flux and bathymetry however the bed friction discretisation has not been so profusely analysed in previous works while it may play a leading role in certain applications of shallow water models in this paper we analyse the numerical discretisation of the bed friction term in the two dimensional shallow water equations and we propose a new unstructured upwind finite volume discretisation for this term the new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions the relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations is also analysed in the paper it is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations the most important implication is that a well balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields but has to consider also the way in which the hydrodynamic equations have been solved

asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes the following work is devoted to the construction and validation of a numerical scheme for the 2d shallow water system on unstructured meshes supplemented by topography and friction source terms approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws involving the water free surface and some fractions of water accounting for the topography variations the discretization of the friction source terms relies on the use of a modified riemann solver for the flux computation the resulting scheme is then corrected in order to achieve an asymptotic regime preservation a muscl reconstruction is also performed to increase the space order of accuracy the overall numerical approach is shown to be consistent well balanced and to satisfy a domain invariant principle these results are assessed through several benchmark tests involving complex geometry and varying bathymetry in the presence of dry areas special interest is given to the wave front speed computation highlighting the stability of the method even when implementing the asymptotic preserving correction

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

transport of pollutant in shallow flows a space time ce se scheme abstract this paper focuses on implementation of space time ce se scheme for computing the transport of a passive pollutant by a flow the flow model comprises of the saint venant system of shallow water equations and the pollutant propagation is described by a transport equation the one dimensional and two dimensional flow equations are numerically investigated using the ce se scheme a number of test problems are presented to check the accuracy and efficiency of the proposed scheme the results of ce se scheme are compared with the central scheme both the schemes are found to be in close agreement however our proposed ce se scheme accurately captures shocks and discontinuous profiles

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

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 staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms we propose large time step and asymptotic preserving schemes for the gas dynamics equations with external forces and friction terms by asymptotic preserving we mean that the numerical scheme is able to reproduce at the discrete level the parabolic type asymptotic behavior satisfied by the continuous equations by large time step we mean that the scheme is stable under a cfl stability condition driven by the slow material waves and not by the fast acoustic waves customary in godunov type schemes numerical evidence is proposed and shows a gain of several orders of magnitude in both accuracy and efficiency

exactly well balanced discontinuous galerkin methods for the shallow water equations with moving water equilibrium hyperbolic conservation laws with source terms often admit steady state solutions where the fluxes and source terms balance each other to capture this balance and near equilibrium solutions well balanced methods have been introduced and performed well in many numerical tests shallow water equations have been extensively investigated as a prototype example in this paper we develop well balanced discontinuous galerkin methods for the shallow water system which preserve not only the still water at rest steady state but also the more general moving water equilibrium the key idea is the recovery of well balanced states a special source term approximation and the approximation of the numerical fluxes based on a generalized hydrostatic reconstruction we also study the extension of the positivity preserving limiter presented in 40 in this framework numerical examples are provided at the end to verify the well balanced property and good resolution for smooth and discontinuous solutions

precise determination of the nonequilibrium tricritical point based on lynden bell theory in the hamiltonian mean field model existence of a nonequilibrium tricritical point has been revealed in the hamiltonian mean field model by a nonequilibrium statistical mechanics this statistical mechanics gives a distribution function containing unknown parameters and the parameters are determined by solving simultaneous equations depending on a given initial state due to difficulty in solving these equations pointwise numerical detection of the tricritical point has been unavoidable on a plane characterizing a family of initial states in order to look into the tricritical point we expand the simultaneous equations with respect to the order parameter and reduce them to one algebraic equation the tricritical point is precisely identified by analyzing coefficients of the reduced equation reentrance to an ordered phase in a high energy region is revisited around the obtained tricritical point

a discontinuous galerkin algorithm for the two dimensional shallow water equations a new high resolution finite element scheme is introduced for solving the two dimensional 2d depth integrated shallow water equations swe via local plane approximations to the unknowns bed topography data are locally approximated in the same way as the flow variables to render an instinctive well balanced scheme a finite volume fv wetting and drying technique that reconstructs the riemann states by ensuring non negative water depth and maintaining well balanced solution is adjusted and implemented in the current finite element framework meanwhile a local slope limiting process is applied and those troubled slope components are restricted by the minmod fv slope limiter the inter cell fluxes are upwinded using the hllc approximate riemann solver friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients boundary conditions are derived and reported in details the present model is validated against several test cases including dam break flows on regular and irregular domains with flooding and drying

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

enhancement of the labswe for shallow water flows in the lattice boltzmann method for the shallow water equations labswe the force term involves the first order derivative related to a bed slope which has to be determined using the centred scheme for an accurate solution however such calculation contradicts the spirit of only simple arithmetic calculations required in the lattice boltzmann method this paper shows how to remove this drawback from the labswe by incorporating the bed level into the lattice boltzmann equation in a consistent manner with the lattice boltzmann approach three numerical tests have been presented to verify the proposed method

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

a well balanced scheme for the shallow water equations with topography or manning friction we consider the shallow water equations with manning friction or topography as well as a combination of both these source terms the main purpose of this work concerns the derivation of a non negativity preserving and well balanced scheme that approximates solutions of the system and preserves the associated steady states including the moving ones in addition the scheme has to deal with vanishing water heights and transitions between wet and dry areas to address such issues a particular attention is paid to the study of the steady states related to the friction source term then a godunov type scheme is obtained by using a relevant average of the source terms in order to enforce the required well balance property an implicit treatment of both topography and friction source terms is also exhibited to improve the scheme while dealing with vanishing water heights a second order well balanced muscl extension is designed as well as an extension for the two dimensional case numerical experiments are performed in order to highlight the properties of the scheme

energy stable staggered schemes for the shallow water equations abstract we focus on the development and analysis of staggered schemes for the two dimensional non linear shallow water equations with varying bathymetry semi implicit and fully explicit time discretizations are proposed particular attention is paid on non linear stability results principally considered here through discrete energy dissipation arguments to address such an issue specific convective fluxes are employed implying diffusive terms relying on the pressure gradient in addition of providing an explicit control of the discrete energy budget the method is shown to preserve motionless steady states as well as the positivity of the water height these properties are still satisfied in a fully explicit context provided an appropriate discretization of the pressure gradient these stability results make the approach particularly robust and efficient for both coastal flows and low froude number regimes as a result in addition of a great ease of implementation the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows

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

local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws this paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws based on numerical fluxes with a total variation diminishing tvd runge kutta evolution in time using the idea recently introduced by c w shu and s j osher for essentially nonoscillatory eno methods the constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell to design the method a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy the method becomes third order accurate in smooth regions of the solution except at local extrema where it may degenerate to o h 3 2 thus giving better accuracy than tvd methods at local extrema the main advantage of this method lies on the property that is more local than that of eno and tvd upwind

the incompressible limit of the non isentropic euler equations we study the euler equations for slightly compressible fluids that is after rescaling the limits of the euler equations of fluid dynamics as the mach number tends to zero in this paper we consider the general non isentropic equations and general data we first prove the existence of classical solutions for a time independent of the small parameter then on the whole space d we prove that the solution converges to the solution of the incompressible euler equations

the generalized riemann problem method for the shallow water equations with bottom topography this paper extends the generalized riemann problem method grp to the system of shallow water equations with bottom topography the main contribution is that the generalized riemann problem method j comput phys 1984 55 1 1 32 is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes and at the same step the source term is discretized with an interface method in which only mid point values are plugged in this scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method sgm j comput phys 2001 168 1 1 25 into data reconstruction step and it is also suitable for both steady and unsteady flow simulations we illustrate the accuracy of this scheme by several 1 d and 2 d numerical experiments copyright 2005 john wiley sons ltd

spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the vlasov equation for the hamiltonian mean field model stability of spatially inhomogeneous solutions to the vlasov equation is investigated for the hamiltonian mean field model to provide the spectral and formal stability criteria in the form of necessary and sufficient conditions these criteria determine stability of spatially inhomogeneous solutions whose stability has not been decided correctly by using a less refined formal stability criterion it is shown that some of such solutions can be found in a family of stationary solutions to the vlasov equation which is parametrized with macroscopic quantities and has a two phase coexistence region in the parameter space

weak solutions for partial differential equations with source terms application to the shallow water equations weak solutions of problems with m equations with source terms are proposed using an augmented riemann solver defined by m 1 states instead of increasing the number of involved equations these weak solutions use propagating jump discontinuities connecting the m 1 states to approximate the riemann solution the average of the propagated waves in the computational cell leads to a reinterpretation of the roe s approach and in the upwind treatment of the source term of vazquez cendon it is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities as usually done in the homogeneous case positivity requirements over the values of the intermediate states are the only way to control the global stability of the method also it is shown that the definition of well balanced equilibrium in trivial cases is not sufficient to provide correct results it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded the one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study the stability region is shown to differ from the one defined for the case without source terms and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet dry front is a particular case of the wet wet fronts the consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case starting from this result 1d and 2d numerical schemes are developed for both quadrilateral and triangular grids enforcing conservation and positivity over the solution allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size

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

numerical procedures for a boundary value problem with a non linear boundary condition numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial boundary value problem for the two dimensional parabolic partial differential equation pde with a non linear boundary condition this class of parabolic pdes plays a very important role in many branches of science and engineering the non linear condition is in the form of a double integral giving the specification of mass in the solution domain not only the problem has both neumann and dirichlet boundary conditions but the dirichlet boundary condition is in a non standard form while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case due to the structure of the boundary conditions a reduced one dimensional parabolic equation for the new unknown v y t email protected 1 0u x y t dx can be formulated the resulting problem has a non local boundary condition the new two dimensional parabolic pde with neumann s boundary conditions will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the article concludes with the results of some numerical experiments

well balanced unstaggered central schemes for one and two dimensional shallow water equation systems abstract we propose a new well balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms in particular we construct a new one and two dimensional finite volume method for the numerical solution of shallow water equations on flat variable bottom topographies the proposed scheme evolves a non oscillatory numerical solution on a single grid avoids the time consuming process of solving riemann problems arising at the cell interfaces and is second order accurate both in space and time furthermore the numerical scheme follows a well balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level the resulting scheme exactly satisfies the c property at the discrete level the proposed scheme is then applied and classical one and two dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved the obtained numerical results are in good agreement with corresponding ones appearing in the recent literature thus confirming the potential and efficiency of the proposed method

a fast adaptive quadtree scheme for a two layer shallow water model this paper presents a dynamically adaptive quadtree grid generation system for the solution of a two dimensional two layer shallow water model roe type two layer shallow water solvers require numerical approximation of the system eigenvalues as well as numerical balancing which increase computational cost considerably when a regular grid is used in order to improve computational efficiency we consider a dynamically adaptive quadtree grid generation system capable of increasing local resolution where high gradients occur in the physical flow variables test results show that satisfactory convergence can be obtained using the present scheme with the adaptive grid generator at a fraction of the cost incurred by a regular grid

centered potential regularization for the advection upstream splitting method this paper is devoted to a centered imex scheme in a multidimensional framework for a wide class of multicomponent and isentropic flows the proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations the stability of the scheme is ensured by the dissipation of mechanic energy which stands for a mathematical entropy under an advective cfl condition the main physical properties such as positivity conservation of the total momentum and conservation of the steady state at rest are satisfied in addition asymptotic preserving properties in the regimes incompressible and acoustic are analyzed finally several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension

the surface gradient method for the treatment of source terms in the shallow water equations a novel scheme has been developed for data reconstruction within a godunov type method for solving the shallow water equations with source terms in contrast to conventional data reconstruction methods based on conservative variables the water surface level is chosen as the basis for data reconstruction this provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a riemann solver the main advantages are 1 a simple centered discretization is used for the source terms 2 the scheme is no more complicated than the conventional method for the homogeneous terms 3 small perturbations in the water surface elevation can be accurately predicted and 4 the method is generally suitable for both steady and unsteady shallow water problems the accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems excellent agreement has been obtained between the numerical predictions and analytical solutions the results indicate that the new scheme is accurate simple efficient and robust

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

moving water equilibria preserving partial relaxation scheme for the saint venant system we develop a new moving water equilibria preserving numerical scheme for the saint venant system the new scheme is designed in two major steps first the geometric source term is incorporated int

reduced order finite difference extrapolation model based on proper orthogonal decomposition for two dimensional shallow water equations including sediment concentration abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order finite difference fd extrapolating model with very few degrees of freedom for two dimensional shallow water equations that include the sediment concentration we provide estimates of the error between the accurate solution and classical fd solutions as well as those between the accurate solution and the pod based reduced order fd solutions moreover we present two numerical simulation experiments to demonstrate that the pod based reduced order fd extrapolating model can greatly reduce the computational load thus we validate both the feasibility and efficiency of the pod based reduced order fd extrapolating model

multi dimensional conservative semi lagrangian method of characteristics cip for the shallow water equations a new characteristic approach that guarantees conservative property is proposed and is applied to the shallow water equations cip csl constrained interpolation profile conservative semi lagrangian interpolation is applied to the cip method of characteristics in order to enhance the mass conservation of the numerical result although the characteristic formulation is originally derived from non conservative form present scheme achieves complete mass conservation by solving mass conservation simultaneously and reflecting conserving mass in interpolation profile present method has less height error compared to the cip method of characteristics by several orders of magnitude by the enhanced conservation property present scheme is applicable to nonlinear problem such as shock furthermore application to two dimensions including the coriolis term is straightforward with directional splitting technique

ader schemes for the shallow water equations in channel with irregular bottom elevation this paper deals with the construction of high order ader numerical schemes for solving the one dimensional shallow water equations with variable bed elevation the non linear version of the schemes is based on eno reconstructions the governing equations are expressed in terms of total water height instead of total water depth and discharge the eno polynomial interpolation procedure is also applied to represent the variable bottom elevation ader schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed with particular attention to their convergence rates non oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases the resulting schemes can be applied to solve realistic problems characterized by non uniform bottom geometries

a conservative high order discontinuous galerkin method for the shallow water equations with arbitrary topography a conservative high order godunov type scheme is presented for solving the balance laws of the 1d shallow water equations swe the scheme adopts a finite element runge kutta rk discontinuous galerkin dg framework based on an overall third order accurate formulation the model is referred to as rkdg3 treatment of topographic source term is built in the dg approximation simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations owing to the adverse effects caused by using an uncontrolled global limiting process in an rkdg3 scheme rkdg3 gl a new conservative rkdg3 scheme with user parameter free local limiting method rkdg3 ll is designed to gain better accuracy and conservativeness the advantages of the new rkdg3 ll model are demonstrated by applying to several steady and transient benchmark flow tests with irregular either differentiable or non differentiable topography copyright 2010 john wiley sons ltd

wave operators and the incompressible limit of the compressible euler equation in an exterior domain in rn n 2 the solution of the compressible euler equation is shown to converge to that of the incompressible euler equation when the mach number tends to 0 the initial layer appears