a stochastic spectral projection method for solving pdes in domains composed by overlapping discs spheres and half spaces a new mesh free stochastic spectral method based on a spectral inversion of the poisson integral formula is suggested we consider the cases of laplace equation and the system of elasticity equations for domains consisting of overlapping discs half planes spheres and half spaces a randomized projection method we developed in sabelfeld and loshchina 2010 21 is used to solve the system of linear equations approximating the integral equations whose phase space is the intersection surfaces of the domains

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

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

the well posedness for fractional nonlinear schrodinger equations abstract in this paper we apply the tools of harmonic analysis to study the cauchy problem for time fractional schrodinger equations the existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed some fundamental properties of operators appearing in the solution of the problem are also discussed

numerical methods for hyperbolic conservation laws with stiff relaxation ii higher order godunov methods a higher order godunov method is presented for hyperbolic systems of conservation laws with stiff relaxing source terms the goal is to develop a godunov method that produces higher order accurate

tractability of algebraic function fields in one variable of genus zero over global fields if the condition described in definition 1 1 on the simultaneous representation of quaternion algebras holds over a field f then f is called tractable in this paper we show that a field f is tractable if and only if any purely transcendental extension of f is tractable next if f is a global field of characteristic unequal to two we consider tractability of an algebraic function field k in one variable of genus zero over f and determine precisely when k is tractable if f has at most one ordering

fractional time quantum dynamics application of the fractional calculus to quantum processes is presented in particular the quantum dynamics is considered in the framework of the fractional time schr odinger equation se which differs from the standard se by the fractional time derivative ensuremath partial ensuremath partial t ensuremath rightarrow ensuremath partial ensuremath alpha ensuremath partial t ensuremath alpha it is shown that for ensuremath alpha 1 2 the fractional se is isospectral to a comb model an analytical expression for the green s functions of the systems are obtained the semiclassical limit is discussed

time dependent schrodinger like equation with nonlocal term we investigate a time dependent schrodinger like equation in presence of a nonlocal term by using the method of variable separation and the green function approach we analyze the green function for different forms of the memory kernel and the nonlocal term results for delta potential energy function are presented distributed order memory kernels are also considered and the asymptotic behaviors of the green function are derived by using tauberian theorem the obtained results for the green function for the considered schrodinger like equation may be transformed to those for the probability distribution function of a diffusion like equation with memory kernel and can be used to explain various anomalous diffusive behaviors

time dependent solutions for a fractional schrodinger equation with delta potentials we investigate for an arbitrary initial condition the time dependent solutions for a fractional schrodinger equation in the presence of delta potentials by using the green function approach the solutions obtained show an anomalous spreading asymptotically characterized by a power law behavior which is governed by the order of the fractional spatial operator present in the schrodinger equation

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

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

a stable numerical method for multidimensional time fractional schrodinger equations in this paper the stability analysis is presented for a first order difference scheme applied to a nonhomogeneous time fractional schrodinger differential equation based on the z transform method stability theorems are proved for the abstract case the stability results are applied on initial boundary value problems for multidimensional time fractional schrodinger differential equations theoretical findings are validated by numerical experiments on one and two dimensional time fractional schrodinger differential equations

the random projection method for stiff detonation capturing in this paper we present a simple and robust random projection method for underresolved numerical simulation of stiff detonation waves in chemically reacting flows this method is based on the random projection method proposed by the authors for general hyperbolic systems with stiff reaction terms w bao and s jin j comput phys 163 2000 pp 216 248 where the ignition temperature is randomized in a suitable domain it is simplified using the equations of instantaneous reaction and then extended to handle the interactions of detonations extensive numerical experiments including interaction of detonation waves and in two dimensions demonstrate the reliability and robustness of this novel method

the validity of the modified equation for nonlinear shock waves abstract the modified model equivalent equation is an important tool in designing and analyzing nonlinear difference schemes in this note the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case

hybrid flux splitting schemes for a two phase flow model in this paper we deal with the construction of hybrid flux vector splitting fvs schemes and flux difference splitting fds schemes for a two phase model for one dimensional flow the model consists of two mass conservation equations one for each phase and a common momentum equation the complexity of this model as far as numerical computation is concerned is related to the fact that the flux cannot be expressed in terms of its conservative variables this is the motivation for studying numerical schemes which are not based on approximate riemann solvers and or calculations of jacobian matrix this work concerns the extension of an fvs type scheme a van leer type scheme and an advection upstream splitting method ausm type scheme to the current two phase model our schemes are obtained through natural extensions of corresponding schemes studied by y wada and m s liou 1997 siam j sci comput 18 633 657 for euler equations we explore the various schemes for flow cases which involve both fast and slow transients in particular we demonstrate that the fvs scheme is able to capture fast propagating acoustic waves in a monotone way while it introduces an excessive numerical dissipation at volume fraction contact steady and moving discontinuities on the other hand the ausm scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves this motivates us to propose other hybrid fvs fds schemes obtained by removing numerical dissipation at contact discontinuities in the fvs and van leer schemes

solving the time fractional schrodinger equation by krylov projection methods the time fractional schrodinger equation is a fundamental topic in physics and its numerical solution is still an open problem here we start from the possibility to express its solution by means of the mittag leffler function then we analyze some approaches based on the krylov projection methods to approximate this function their convergence properties are discussed together with related issues numerical tests are presented to confirm the strength of the approach under investigation

analysis of time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise abstract in this paper we establish the existence and uniqueness of the mild solution and the holder continuity for the time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise a full discrete scheme with spectral galerkin method in space and exponential method in time is proposed it is rigorously proved that the numerical scheme admits an unique solution finally we present the optimal error estimates of the proposed scheme based on the regularity analysis

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

approximating fractional time quantum evolution an algorithm is presented for approximating the arbitrary powers of a black box unitary operation where t is a real number and is a black box implementing an unknown unitary the complexity of this algorithm is calculated in terms of the number of calls to the black box the errors in the approximation and a certain gap parameter for general and large t one should apply a total of t times followed by our procedure for approximating the fractional power an example is also given where for large integers t this method is more efficient than direct application of t copies of further applications and related algorithms are also discussed

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

time fractional schrodinger equation fox s h functions and the effective potential after introducing the formalism of the general space and time fractional schrodinger equation we concentrate on the time fractional schrodinger equation and present new results via the elegant language of fox s h functions we show that the general time dependent part of the wave function for the separable solutions of the time fractional schrodinger equation is the mittag leffler function with an imaginary argument by two different methods after separating the mittag leffler function into its real and imaginary parts in contrast to existing works we show that the total probability is 1 and decays with time introducing the effective potential approach we also write the mittag leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts in the light of these we reconsider the simple box problem

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

exact solutions of fractional schrodinger like equation with a nonlocal term we study the time space fractional schrodinger equation with a nonlocal potential by the method of fourier transform and laplace transform the green function and hence the wave function is expressed in terms of h functions graphical analysis demonstrates that the influence of both the space fractal parameter and the nonlocal parameter on the fractional quantum system is strong indeed the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field

local well posedness of semilinear space time fractional schrodinger equation abstract the semilinear space time fractional schrodinger equation is considered first we give the explicit form for the fundamental solutions by using the fox h functions in order to establish some l s decay estimates after that we give some space time estimates for the mild solutions from which the local well posedness is derived on some proper banach space

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

continuous time random walk as a guide to fractional schrodinger equation we argue that the continuous time random walk approach may be a useful guide to extend the schrodinger equation in order to incorporate nonlocal effects avoiding the inconsistencies raised by jeng et al j math phys 51 062102 2010 as an application we work out a free particle in a half space obtaining the time dependent solution by considering an arbitrary initial condition

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

the random projection method for stiff multispecies detonation capturing in this paper we extend the random projection method proposed for general hyperbolic systems with stiff reaction terms for underresolved numerical simulation of stiff inviscid multispecies detonation waves the key idea in this method is to randomize the ignition temperatures in suitable domains several numerical experiments in both one and two dimensions demonstrate the reliability and robustness of this novel method

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

the random projection method for a model problem of combustion with stiff chemical reactions in this paper we extend the random projection method recently proposed by the author and s jin j comput phys 163 2000 216 for under resolved numerical simulations of a qualitative model problem for combustion with stiff chemical reactions ut f u q0z x 0 zx 1 e u z in this problem the reaction time e is small making the problem numerically stiff a classic spurious numerical phenomenon the incorrect shock speed occurs when the reaction time scale is not properly resolved numerically the random projection method is introduced recently to handle this kind of numerical difficulty the key idea in this method is to randomize the ignition temperature in a suitable domain several numerical experiments demonstrate the reliability and robustness of this method

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

time fractional development of quantum systems in this study the effect of time fractionalization on the development of quantum systems is taken under consideration by making use of fractional calculus in this context a mittag leffler function is introduced as an important mathematical tool in the generalization of the evolution operator in order to investigate the time fractional evolution of the quantum nano systems time fractional forms of motion are obtained for a schroedinger equation and a heisenberg equation as an application of the concomitant formalism the wave functions energy eigenvalues and probability densities of the potential well and harmonic oscillator are time fractionally obtained via the fractional derivative order alpha which is a measure of the fractality of time in the case alpha 1 where time becomes homogenous and continuous traditional physical conclusions are recovered since energy and time are conjugate to each other the fractional derivative order alpha is relevant to time it is understood that the fractionalization of time gives rise to energy fluctuations of the quantum nano systems

zero relaxation and dissipation limits for hyperbolic conservation laws we are interested in hyperbolic systems of conservation laws with relaxation and dissipation particularly the zero relaxation limit such a limit is of interest in several physical situations including gas flow near thermo equilibrium kinetic theory with small mean free path and viscoelasticity with vanishing memory in this article we study hyperbolic systems of two conservation laws with relaxation for the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit for the marginally stable case where the equilibrium speed may equal one of the frozen speeds we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation our analysis includes the construction of suitable entropy pairs to derive energy estimates we need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems the theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit 1993 john wiley sons inc

central schemes for balance laws of relaxation type several models in mathematical physics are described by quasi linear hyperbolic systems with source term and in several cases the production term can become stiff here suitable central numerical schemes for such problems are developed and applications to the broadwell model and extended thermodynamics are presented the numerical methods are a generalization of the nessyahu tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations a second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed numerical tests confirm the accuracy and robustness of the scheme

numerical methods for hyperbolic conservation laws with stiff relaxation i spurious solutions the author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed uniform spatial grid it is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes one criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system a second criterion is that a certain subcharacteristic condition be satisfied by the hyperbolic system this conjecture is supported with analytical and numerical results for a specific example the solution of generalized riemann problems of a model system of equations with a fractional step scheme in which godunov s method is coupled with the backward euler method similar ideas are applied to the numerical solution of stiff detonation problems

second order godunov type scheme for reactive flow calculations on moving meshes the method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered the flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ode part the numerical algorithm is based on the godunov s scheme on deforming meshes with some modification to increase the scheme order in time and space the variational approach is applied to generate the moving adaptive mesh at every time step the functional of smoothness written on the graph of the control function is minimized the grid lines are condensed in the vicinity of the main solution singularities e g precursor shock fire zones intensive transverse shocks and slip lines which allows resolving a fine structure of the reaction domain the numerical examples relating to the chapman jouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions

operators of fractional integration and their applications the main purpose of this paper is to present a systematic and historical account of the investigations carried out by various authors in the field of fractional calculus and its applications several interesting results relevant to the present investigation are also considered

the asymptotic behavior of the time fractional schrodinger equation on hilbert space the purpose of this article is to show the asymptotic behavior of the solution of the abstract schrodinger equation with time fractional derivative of order 0 1 we give a detailed description of the long time behavior of the corresponding operator solution as it is expected in this case the dynamics are not governed neither by a group nor by a semigroup

on the consistency of the solutions of the space fractional schrodinger equation recently it was pointed out that the solutions found in the literature for the space fractional schrodinger equation in a piecewise manner are wrong except the case with the delta potential we re analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise we also discuss effective potential approach and present a free particle solution for the space and time fractional schrodinger equation in general coordinates in terms of fox s h functions

finite volume schemes for nonhomogeneous scalar conservation laws error estimate in this paper we study finite volume schemes for the nonhomogeneous scalar conservation law u t rm div f x t u q x t u with initial condition u x 0 u0 x the source term may be either stiff or nonstiff in both cases we prove error estimates between the approximate solution given by a finite volume scheme the scheme is totally explicit in the nonstiff case semi implicit in the stiff case and the entropy solution the order of these estimates is h frac 1 4 in space time l 1 norm h denotes the size of the mesh furthermore the error estimate does not depend on the stiffness of the source term in the stiff case

generalized fractional schrodinger equation with space time fractional derivatives in this paper the generalized fractional schrodinger equation with space and time fractional derivatives is constructed the equation is solved for free particle and for a square potential well by the method of integral transforms fourier transform and laplace transform and the solution can be expressed in terms of mittag leffler function the green function for free particle is also presented in this paper finally we discuss the relationship between the cases of the generalized fractional schrodinger equation and the ones in standard quantum

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

theoretical and numerical structure for unstable one dimensional detonations the spatio temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods a new high resolution numerical method for computing unstable detonations is developed this method combines the piecewise parabolic method ppm with conservative shock tracking and adaptive mesh refinement a new nonlinear asymptotic theory for the spatio temporal growth of instabilities is also developed this asymptotic theory involves a nonclassical hopf bifurcation because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free boundary problem for a nonlinear hyperbolic equation an interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio temporal mechanisms of nonlinear stability near the transition boundary in particular a quantitative qualitative explanation is developed for the experimentally observed instabilities

the random projection method for hyperbolic conservation laws with stiff reaction terms we propose a random projection method for numerical simulations of hyperbolic conservation laws with stiff source terms arising from chemically reactive flows ut f u x g u y 1 u in this problem the chemical time scales may be orders of magnitude faster than the fluid dynamical time scales making the problem numerically stiff a classic spurious numerical phenomenon the incorrect propagation speeds of discontinuities occurs in underresolved numerical solutions we introduce a random projection method for the reaction term by replacing the ignition temperature with a uniformly distributed random variable the statistical average of this method corrects the spurious shock speed as will be proved with a scalar model problem and demonstrated by a wide range of numerical examples in inviscid detonation waves in both one and two space dimensions

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved