steady states and dynamics of an autocatalytic chemical reaction model with decay abstract the dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates upper lower solution method and bifurcation theory the effects of decay order decay rate and diffusion rates to the dynamical behavior are discussed

effects of random motility on microbial growth and competition in a flow reactor the authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations for pure culture a single population a sharp condition is derived which distinguishes between the two outcomes 1 washout of the population from the reactor or 2 persistence of the population and the existence of a unique single population steady state the simulations suggest that this steady state is globally attracting for the case of two populations competing for scarce nutrient they obtain sufficient conditions for the uniform persistence of the two populations for the existence of a coexistence steady state and for the ability of one population to competitively exclude a rival extensive simulations are reported which suggest that 1 all solutions approach some steady state solution 2 all possible outcomes exhibited by the classical competitive lotka volterra ode model can occur in the model and 3 the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients

a simple model for sustained oscillations in isothermal branched chain or autocatalytic reactions in a well stirred open system i stationary states and local stabilities the behaviour of a simple autocatalytic or branched chain reaction a b rightarrow 2b text rate k 1 ab coupled with a surface controlled decay of the catalyst b rightarrow c text rate k 2 b 1 rb is considered under isothermal well stirred open conditions the c s t r this system has a unique physically acceptable stationary state at any given residence time if the saturation term r is zero this solution is always stable although it may exhibit damped oscillatory responses characteristic of a stable focus small non zero values of r do not introduce qualitative changes however for sufficiently large surface effects such that r k 1 k 2 the stable focus may become unstable at a point of hopf bifurcation this leads to the appearance of sustained oscillations in the concentrations a and b for relatively stable catalysts such that k 2 is non zero but very small compared with k 1 a 0 where a 0 is the concentration of reactant a in the feed to the reactor the governing kinetic equations can be reduced to especially simple asymptotic forms these in turn allow the analytical determination of the point of hopf bifurcation in terms of the residence time t res at which the change in stability occurs numerical solution of the full equations reveals that the asymptotic expressions provide a very good prediction for the dependence of t res on k 1 a 0 k 2 and r even when k 2 approx k 1 a 0 systems which have some autocatalyst b in the feed show an important qualitative change at long residence times there is a second point of hopf bifurcation at which the focal stationary state becomes stable again oscillations die out at this point and in the limit of infinitely long residence times the behaviour approaches that of the corresponding closed system

the use of domain splitting with an explicit hyperbolic solver abstract a method of splitting the solution domain in order to allow faster computation of practical first order hyperbolic problems is described results are presented which show the performance of the method in both one and two dimensions an indication is also given of the likely saving in computer time which can be achieved by using the method

variable fidelity aerodynamic optimization for turbulent flows using a discrete adjoint formulation an aerodynamic shape optimization methodology based on a discrete adjoint solver for navier stokes flows is described the flow solver at the heart of this optimization process is a reynolds averaged navier stokes code for multiblock structured grids it uses osher s approximate riemann solver and the algebraic turbulence model of baldwin lomax a corresponding discrete navier stokes adjoint solver is derived analytically it has to calculate accurately the jacobian including the effect of the turbulence modeling the shape deformations

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

comparison of coordinate invariant and coordinate aligned upwinding for the euler equations a floating shock fitting method for the euler equations has been developed that uses one sided spatial differences along and across streamlines the coordinate invariant formulation of the spatial differences permits automatic capture of shears results are presented for unsteady shocked flow in a duct with a ramp for supercritical flow over a circular cylinder and for subsonic transonic and supersonic 0 3 m 1 5 flow over airfoils in flows with strong shears the coordinate invariant differencing concept appears to yield some gains in accuracy over euler methods that rely on coordinate aligned differencing concepts in applications to transonic airfoils fitted shocks have a tendency to be predicted upstream of captured shocks regardless of whether coordinate invariant or coordinate aligned differencing is used

computing of separated flow using the space marching conservative supra characteristics method steady hypersonic viscous flows over compression corners with streamwise separation have been computed using the space marching conservative supra characteristics method of lombard this method permits stable space marching of the parabolized navier stokes equations through large separated flow regions the present method has been used to compute surface pressure heat transfer and skin friction coefficients for two compression corner cases studied experimentally by holden and moselle the computed results compare favorably with previous navier stokes results and experimental data the present method has also been compared with the conventional beam warming scheme for solving the parabolized navier stokes equations comparisons are made for accuracy computer time and computer storage

numerical experiments with the osher upwind scheme for the euler equations the osher algorithm for solving the euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations a first order accurate version of the numerical method was derived by osher circa 1980 for the one dimensional non isentropic euler equations in cartesian coordinates in this paper the extension of the scheme to arbitrary two dimensional geometries is explained results are then presented for several example problems in one and two dimensions future work will include extension of the method to second order accuracy and the development of implicit time differencing for the osher algorithm

improvements to a dual time stepping method for computing unsteady flows the nonlinear problem to be solved within an implicit time marching method has several differences from a typical steady state problem the initial guess is usually quite good and the degree of convergence required is quite small perhaps a few orders of magnitude reduction in residual therefore algorithms developed for steady problems may not be optimal for use in a dual time context we present some modifications to the diagonal approximately factored algorithm that lead to savings in a dual time stepping context

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

navier stokes analysis of wind tunnel interference on transonic airfoil flutter a time domain numerical method is used for the simulation and investigation of the transonic two degree offreedom bending torsion e utter characteristics of the national aerospace laboratory 7301 section measured in a wind tunnelexperiment animplicit time accurate two dimensional compressible thin layernavier stokeseow solver is coupled with a two degree of freedom structural model the wind tunnel walls are included to model the experimental e ow conditions more closely and to obtain accurate aerodynamic predictions in the computations a three block grid is used for the discretization of the domain including the airfoil and wind tunnel walls solid and porous wall boundary conditions are implemented and tested for the domain boundaries representing the tunnel walls computations of the steady transonic aerodynamic characteristics show good agreement with the experimental results when a 50 porosity wall boundary condition is assumed for the wind tunnel walls including the wall effectsimproved theagreementoftheaeroelasticcomputations which predicted limit cyclee utter amplitudes in closer agreement with the experiment compared to the unbounded e ow calculations

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

inviscid and viscous stationary waves of gas flow through contracting expanding nozzles abstract in this work we study steady states of one dimensional viscous isentropic compressible flows through a contracting expanding nozzle treating the viscosity coefficient as a singular parameter the steady state problem can be viewed as a singularly perturbed system for a contracting expanding nozzle a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory particularly interesting is the existence of a maximal sub to super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub to super portion

viscous standing asymptotic states of isentropic compressible flows through a nozzle in this work we consider a viscous regularization of a well known one dimensional model for isentropic viscous compressible flows through a nozzle for the existence and multiplicity of standing asymptotic states for a certain type of ducts a complete analysis in a framework of dynamical systems is provided as an application of the geometric singular perturbation theory we show that all standing asymptotic states admit viscous profiles

optimal explicit strong stability preserving general linear methods this paper constructs strong stability preserving general linear time stepping methods that are well suited for hyperbolic pdes discretized by the method of lines these methods generalize both runge kutta rk and linear multistep schemes they have high stage orders and hence are less susceptible than rk methods to order reduction from source terms or nonhomogeneous boundary conditions a global optimization strategy is used to find the most efficient schemes that have low storage requirements numerical results illustrate the theoretical findings

a contribution to the numerical prediction of unsteady flows methods based on flux differe nce splitting are investigated with reference to the prediction of unsteady compressible flows they combine the feature of taking into account the wake like nature of these flows and the capability of correctly capturing shock waves two recently proposed approaches and a third one suggested here are compared for one dimensional flow with particular reference to the means of operating the splitting they are extended to the cases of variable area ducts and a general transformation of the independent variables firstand second order accurate schemes are presented and procedures for the computation at the boundaries are shown finally numerical experiments are reported and discussed

modification of the osher upwind scheme for use in three dimensions the pertinent formulas are given by eq 24 for the variation of c kleb with the modified clauser pressure gradient parameter by eq 14 for the variation of ccp with ckleb and by eq 26 for the variation of the clauser factor k with low reynolds number as presented here the relations for the baldwin lomax factors have an indirect experimental validity by being derived from the experimentally confirmed velocity similarity laws finally it should be apparent that for boundary layers in pressure gradients constant values for the baldwin lomax factors are unsatisfactory in satisfying the velocity similarity laws to approach the same accuracy as the parent cebeci smith method the baldwinlomax factors should be varied as indicated

an implicit form for the osher upwind scheme conservative upwind schemes for the euler equations such as the osher scheme accurately resolve flow discontinuities and correctly model the physics of the problem however these schemes require many more arithmetic operations per integration step than simple central difference schemes and hence result in large computing times an implicit version of the first order and second order accurate osher schemes in two spatial dimensions and generalized coordinates is developed in this study because implicit schemes permit the use of large integration steps in many cases they require fewer integration steps to reach steady state especially in calculations on grids with widely varying mesh cell sizes the implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered test cases include quasi one dimensional nozzle flow and supersonic flow past a cylinder

linear stability of the sub to super inviscid transonic stationary wave for gas flow in a nozzle of varying area abstract in this work we consider the linear stability of the sub to super inviscid transonic stationary wave of a one dimensional model of isentropic compressible flows through a nozzle of varying area this sub to super inviscid transonic stationary wave is newly founded by the authors using the geometric singular perturbation theory the main result of this work is to show that the sub to super inviscid transonic stationary wave is physically relevant in the sense that it is l linearly stable on any bounded space interval as long as its velocity is greater than 1 2 of the sound speed

the fundamental solutions of the space space time riesz fractional partial differential equations with periodic conditions in this paper the space time riesz fractional partial differential equations with periodic conditions are considered the equations are obtained from the integral partial differential equation by replacing the time derivative with a caputo fractional derivative and the space derivative with riesz potential the fundamental solutions of the space riesz fractional partial differential equation srfpde and the space time riesz fractional partial differential equation strfpde are discussed respectively using methods of fourier series expansion and laplace transform we derive the explicit expressions of the fundamental solutions for the srfpde and the strfpde respectively

upwind schemes and boundary conditions with applications to euler equations in general geometries abstract a unified treatment of several upwind shock capturing algorithms is presented each algorithm has a riemann initial value problem as its basis the treatment of boundaries involves solving the associated riemann initial boundary value problem the first author s algorithm applied to multidimensional euler equations in general geometries is then presented its worth is verified by various calculations which include mach 8 supersonic flow past a circular cylinder

grid adaptation for functional outputs application to two dimensional inviscid flows this paper presents an error estimation and grid adaptive strategy for estimating and reducing simulation errors in integral outputs functionals of partial differential equations adaptive criteria are derived using an adjoint based error correction technique that relates the local residual errors of both the primal and adjoint solutions to the global error in the prescribed functional this relationship allows local error contributions to be used as indicators in a grid adaptive strategy designed to produce specially tuned grids for accurately estimating the chosen functional in this paper attention is limited to two dimensional inviscid flows using a standard finite volume discretization although the procedure may be readily applied to other types of multidimensional problems and discretizations numerical examples demonstrate that the proposed adaptive procedure compares favorably in terms of accuracy efficiency and robustness relative to a commonly used gradient based adaptive method

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

the evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay abstract in this paper we study a reaction diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step a b 2 b and a decay step b c where a b and c are the reactant the autocatalyst and the inner product respectively previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant a initially distributes uniformly in the whole space then a pair of waves will be generated and will propagate outwards from the initial reaction zone one crucial feature of this phenomenon is that for the strong decay case the formation of waves is independent of the amount of the autocatalyst b introduced into the system it is this phenomenon of kpp type which we would like to address in this paper to study the propagation of reactant and autocatalyst analytically we first use the tail behavior of waves to construct a pair of generalized super sub solutions for the approximate system of the autocatalytic reaction system note that the autocatalytic reaction system does not enjoy comparison principle together with a family of truncated problems we can establish the existence of a family of traveling waves with the minimal speed second we use this pair of generalized super sub solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of aronson weinberger formulation which in turn confirms the aforementioned numerical and experimental results

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

the exact solution of certain differential equations of fractional order by using operational calculus abstract in the present paper the authors first develop an operational calculus for the familiar riemann liouville fractional differential operator this operational calculus is then used here to solve a cauchy boundary value problem for a certain linear equation involving the riemann liouville fractional derivatives relevant connections are also indicated with the special cases of the equation which were solved earlier by using other methods

review lecture instabilities and oscillations in chemical reactions in closed and open systems this lecture is chiefly concerned with isothermal oscillations in homogeneous systems and with finding and validating self consistent models emphasis is laid on satisfying basic physical principles and on preserving chemical reasonableness in as simple a manner as possible the goal is to devise a scheme that can be applied to a wide variety of circumstances to predict new behaviour as well as to interpret what is known and to form a foundation for later developments and elaborations this core of the scheme called the autocatalator combines an autocatalytic step a 2b rightarrow 3b rate k 1 ab 2 with catalyst decay b rightarrow c rate k 2 b these core reactions may display reversibility and they may be accompanied by uncatalysed conversion a rightarrow b text rate k u a particular stress is given to the problem of closed systems in closed systems oscillations cannot be indefinitely sustained in former days it was customary to ignore this inconvenience and to repeal the law of conservation of matter today a less unsatisfactory account is possible

grid independent upwind scheme for multidimensional flow recent advances in the development of a grid independent finite volume scheme for the euler equations of gas dynamics are described in the proposed method flowfield gradient data are reconstructed locally on a triangle using five elementary planar waves and an upwind numerical flux function for grid oblique waves is developed to model the effect of the passage of these waves on the data in a cell numerical examples for several two dimensional test problems are included

a new approach for building bounded skew upwind schemes abstract a new methodology for the construction of streamline bounded high resolution schemes is presented the route followed in the development is to combined existing streamline schemes with the normalized variable formulation nvf bounding approach the procedure is applied to the skew upwind differenting scheme suds to yield a new composite high resolution bounded convective scheme nvf suds the increase in computational cost with nvf suds is very small as compared to the unbounded suds the new scheme is tested and compared with the composite high resolution stoic scheme the upwind scheme and the unbounded suds by solving for three pure convection test problems results generated demonstrate the capability of the new upwind scheme in accurately resolving steep gradients as accurate as the ones obtained with the third order stoic scheme while retaining the solution bounded

fast propagation for kpp equations with slowly decaying initial conditions abstract this paper is devoted to the analysis of the large time behavior of solutions of one dimensional fisher kpp reaction diffusion equations the initial conditions are assumed to be globally front like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function we prove that all level sets of the solutions move infinitely fast as time goes to infinity the locations of the level sets are expressed in terms of the decay of the initial condition furthermore the spatial profiles of the solutions become asymptotically uniformly flat at large time this paper contains the first systematic study of the large time behavior of solutions of kpp equations with slowly decaying initial conditions our results are in sharp contrast with the well studied case of exponentially bounded initial conditions