hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

high order asymptotic preserving methods for fully nonlinear relaxation problems we study solutions to nonlinear hyperbolic systems with fully nonlinear relaxation terms in the limit of both infinitely stiff relaxation and arbitrary late time in this limit the dynamics is governed by effective systems of parabolic type which may contain degenerate and or fully nonlinear diffusion terms for this class of problems we develop an implicit explicit method based on runge kutta discretization in time and we apply this method to the investigation of several examples of interest in fluid dynamics importantly we impose here a realistic stability condition on the time step and we demonstrate that solutions in the hyperbolic to parabolic regime can be computed numerically with high robustness and accuracy even in the presence of fully nonlinear relaxation

a new analytical and numerical treatment for singular two point boundary value problems via the adomian decomposition method based on the adomian decomposition method a new analytical and numerical treatment is introduced in this research to investigate linear and non linear singular two point bvps the effectiveness of the proposed approach is verified by several linear and non linear examples

a finite volume scheme for nonlinear degenerate parabolic equations we propose a second order finite volume scheme for nonlinear degenerate parabolic equations which admit an entropy functional for some of these models porous media equation drift diffusion system for semiconductors it has been proved that the transient solution converges to a steady state when time goes to infinity the present scheme preserves steady states and provides a satisfying long time behavior moreover it remains valid and second order accurate in space even in the degenerate case after describing the numerical scheme we present several numerical results which confirm the high order accuracy in various regime degenerate and nondegenerate cases and underline the efficiency to preserve the large time asymptotic

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

positivity preserving sixth order implicit finite difference weighted essentially non oscillatory scheme for the nonlinear heat equation abstract this paper presents a class of semi implicit finite difference weighted essentially non oscillatory weno schemes for solving the nonlinear heat equation for the discretization of second order spatial derivatives a sixth order modified weno scheme is directly implemented this scheme preserves the positivity principle and rejects spurious oscillations close to non smooth points in order to admit large time steps a class of implicit runge kutta methods is used for the temporal discretization the implicit parts of these methods are linearized in time by using the local taylor expansion of the flux the stability analysis of the semi implicit weno scheme with 3 stages form is provided finally some comparative results for one two and three dimensional pdes are included to illustrate the effectiveness of the proposed approach

global well posedness for fractional hartree equation on modulation spaces and fourier algebra abstract we study the cauchy problem for fractional schrodinger equation with cubic convolution nonlinearity i t u 2 u k u 2 u 0 with cauchy data in the modulation spaces m p q r d for k x x 0 min d 2 we establish global well posedness results in m p q r d 1 p 2 1 q 2 d d when 2 d 1 and with radial cauchy data when d 2 2 d 2 d 1 2 similar results are proven in fourier algebra f l 1 r d l 2 r d

on the convergence of finite difference methods for a class of two point boundary value problems with periodic boundary conditions abstract in this paper we present a detailed convergence analysis to establish second order convergence for a three point two parameter family of finite difference methods based on one evaluation of f for the class of two point boundary value problems tpbvps y f x y o o x 1 y o y 1 y o y 1 under suitable conditions on f y and for y c 4 0 1 some numerical examples are provided to illustrate the results

the inclusion relation between sobolev and modulation spaces the inclusion relations between the lp sobolev spaces and the modulation spaces is determined explicitly as an application mapping properties of unimodular fourier multiplier ei d between lp sobolev spaces and modulation spaces are discussed

regularization of the burnett equations for rapid granular flows via relaxation abstract in j fluid mech 361 1998 41 sela and goldhirsch have used the chapman enskog expansion to derive constitutive relations for the pressure deviator p heat flux q and rate of energy loss for rapid flows of smooth inelastic spheres unfortunately as in the classical chapman enskog expansion for elastic spheres any truncation of the expansion beyond navier stokes order n 1 will possess unphysical instabilities in this paper we propose a visco elastic relaxation approximation that eliminates the instability paradox for all wave numbers and provide a system of local equations allowing robust numerical approximations of gas dynamics valid to the burnett order this system is weakly parabolic has a linearly hyperbolic convection part and is endowed with a generalized entropy inequality in the case of purely elastic collisions thus it is linearly stable for all wave numbers it agrees with the solution of the boltzmann equation up to the burnett order via the chapman enskog expansion

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

inclusion relations between l p sobolev and wiener amalgam spaces abstract we determined optimal inclusion relations between l p sobolev and wiener amalgam spaces for applications we discuss mapping properties of unimodular fourier multipliers e i d between l p sobolev and wiener amalgam spaces and derive some littlewood paley type inequalities

frequency uniform decomposition method for the generalized bo kdv and nls equations abstract sufficient and necessary conditions for the embeddings between besov spaces b p q s 1 and modulation spaces m p q s 2 are obtained moreover using the frequency uniform decomposition method we study the cauchy problem for the generalized bo kdv and nls equations for which the global well posedness of solutions with the small rough data in certain modulation spaces m 2 1 s is shown

viscous splitting approximation of mixed hyperbolic parabolic convection diffusion equations we first analyse a semi discrete operator splitting method for nonlinear possibly strongly degenerate convection diffusion equations due to strong degeneracy solutions can be discontinuous and are in general not uniquely determined by their data hence weak solutions satisfying an entropy condition are sought we then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic parabolic convection diffusion equations

on lagrangian schemes for porous medium type generalized diffusion equations a discrete energetic variational approach abstract in this paper we present a systematic framework to derive a variational lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach such discrete energetic variational approaches are analogous to energetic variational approaches 39 25 in a semidiscrete level which provide a basis of deriving the semi discrete equations and can be applied to a large class of partial differential equations with energy dissipation laws and kinematic relations the numerical schemes derived by this framework can inherit the variational structure from the continuous energy dissipation law as an illustration we develop two variational lagrangian schemes for the multidimensional porous medium equations pme based on two different energy dissipation laws we focus on the numerical scheme based on the energy dissipation law with 1 2 u 2 d x as the dissipation functional several numerical experiments demonstrate the accuracy of this scheme as well as its ability in capturing the free boundary and estimating the waiting time for the pme in both 1d and 2d

numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type in such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved we develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type

variable mesh methods for the numerical solution of two point singular perturbation problems abstract in this paper variable mesh difference methods of third order are derived for the solution of the two point second order singular perturbation problems y f x y y e these methods are applied to a number of examples and their superiority over the existing methods is shown

an all speed relaxation scheme for gases and compressible materials abstract we present a novel relaxation scheme for the simulation of compressible material flows at all speeds an eulerian model describing gases fluids and elastic solids with the same system of conservation laws is adopted the proposed numerical scheme is based on a fully implicit time discretization easily implemented thanks to the linearity of the transport operator in the relaxation system the spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in all different mach regimes the scheme is validated by simulating gas and liquid flows with different mach numbers the simulation of the deformation of compressible solids is also approached assessing the ability of the scheme in approximating material waves in hyperelastic media

solving second order non linear hyperbolic pdes using generalized finite difference method gfdm abstract the generalized finite difference method gfdm has been proved to be a good meshless method to solve several linear partial differential equations pdes wave propagation advection diffusion plates beams etc the gfdm allows us to use irregular clouds of nodes that can be of interest for modelling non linear parabolic pdes this paper illustrates that the gfd explicit formulae developed to obtain the different derivatives of the pde s are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and taylor series development criteria for convergence of fully explicit method using gfdm for different non linear parabolic pdes are given this paper shows the application of the gfdm to solving different non linear problems including applications to heat transfer acoustics and problems of mass transfer

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

high order relaxation schemes for nonlinear degenerate diffusion problems several relaxation approximations to partial differential equations have been recently proposed examples include conservation laws hamilton jacobi equations convection diffusion problems and gas dynamics problems the present paper focuses on diffusive relaxation schemes for the numerical approximation of nonlinear parabolic equations these schemes are based on a suitable semilinear hyperbolic system with relaxation terms high order methods are obtained by coupling eno and weighted essentially nonoscillatory weno schemes for space discretization with implicit explicit imex schemes for time integration error estimates and a convergence analysis are developed for semidiscrete schemes with a numerical analysis for fully discrete relaxed schemes various numerical results in one and two dimensions illustrate the high accuracy and good properties of the proposed numerical schemes also in the degenerate case these schemes can be easily implemented on parallel computers and applied to more general systems of nonlinear parabolic equations in two and three dimensional cases

nonhomogeneous dirichlet problems for degenerate parabolic hyperbolic equations the aim of the paper is to give a formulation for the initial boundary value problem of parabolic hyperbolic type in the case of nonhomogeneous boundary data a 0 here u u x t with x t q 0 t where is a bounded domain in n with smooth boundary and t 0 the function b is assumed to be nondecreasing allowing degeneration zones where b is constant is locally lipschitz continuous and g l 0 b after introducing the definition of an entropy solution to the above problem in the spirit of otto 14 we prove uniqueness of the solution in the proposed setting moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type specifically the diffusion function b is approximated by functions b that are strictly increasing

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

characterization of inclusion relations between wiener amalgam and some classical spaces abstract in this paper we establish the sharp conditions for the inclusion relations between besov spaces b p q and wiener amalgam spaces w p q s we also obtain the optimal inclusion relations between local hardy spaces h p and wiener amalgam spaces w p q s which completely improve and extend the main results obtained by cunanana kobayashib and sugimotoa in 3 in addition we establish some mild characterizations of inclusion relations between triebel lizorkin and wiener amalgam spaces which relates some modern inequalities to classical inequalities

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

the dilation property of modulation spaces and their inclusion relation with besov spaces abstract we consider the dilation property of the modulation spaces m p q let d f t f t be the dilation operator and we consider the behavior of the operator norm d m p q m p q with respect to our result determines the best order for it and as an application we establish the optimality of the inclusion relation between the modulation spaces and besov spaces which was proved by toft j toft continuity properties for modulation spaces with applications to pseudo differential calculus i j funct anal 207 2004 399 429

some light on littlewood paley theory the purpose of this note is to correct an error in a paper of m cowling g fendler and j j f fournier and to give a counterexample to a conjecture of j l rubio de francia

unimodular fourier multipliers on wiener amalgam spaces abstract we study the boundedness of unimodular fourier multipliers on wiener amalgam spaces for a real valued homogeneous function on r n of degree 2 we show the boundedness of the operator e i d between the weighted wiener amalgam space w s p q and w p q for all 1 p q and s n 2 1 p 1 2 n 1 p 1 q this threshold is shown to be optimal for regions max 1 q 1 2 1 p and min 1 q 1 2 1 p moreover we give sufficient conditions for the boundedness of e i d on w p q for 0 2

explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems we design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection these schemes are based on discrete bgk models where both characteristic velocities and the source term depend singularly on the relaxation parameter general stability conditions are derived and convergence is proved to the entropy solutions for scalar equations

finite difference methods for certain singular two point boundary value problems the aim of this paper is to study the outcome of finite difference methods of mainly second order for the numerical solution of certain singular two point boundary value problems the method is mainly an extension of the method of gustafsson 1973 from the linear to nonlinear differential equation and is particularly suitable when the differential equation in question cannot be reduced easily to an elegant system of first order equations

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

solution of nonlinear diffusion problems by linear approximation schemes an approximation of degenerate parabolic problems of porous medium and stefan type is studied the enthalpy formulation and the variational technique are used nonstandard semidiscretization in time is used and newton like iterations are applied to solve the corresponding elliptic problems some numerical experiments are discussed and compared to other methods and to analytic solutions

convergence of diffusive bgk approximations for nonlinear strongly parabolic systems we study a class of bgk approximations of parabolic systems in one space dimension we prove stability and existence of global solutions for this model moreover under certain conditions we prove a rigorous result of convergence toward the formal limit by using compensated compactness techniques

a numerical approach to degenerate parabolic equations in this work we propose a numerical approach to solve some kind of degenerate parabolic equations the underlying idea is based on the maximum principle more precisely we locally perturb the initial and boundary data instead of the nonlinear diffusion coefficients so that the resulting problem is not degenerate the efficiency of this method is shown analytically as well as numerically the numerical experiments show that this new approach is comparable with the existing ones

a new finite difference method for a class of singular two point boundary value problems we present a new three point finite difference method based on uniform mesh for a class of singular two point boundary value problem x ay f x y y 0 a y 1 b 0 a 1 we show that the method based on uniform mesh provides o h 4 convergent approximations this method is illustrated by two numerical examples one is linear and another is non linear

a class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

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

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method