fourier spectral exponential time differencing methods for multi dimensional space fractional reaction diffusion equations abstract in this paper we propose two numerical methods for solving the space fractional reaction diffusion equations which are based on a fourier spectral approach in space and exponential time differencing schemes in time the advantages of the approaches are that they attain spectral convergence produce a full diagonal representation of the fractional operator and the extension to multiple spatial dimensions is the same as the one dimensional space that can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non locality of the fractional operator which results in full dense matrices moreover the proposed schemes are second order convergent unconditionally stable and highly efficient due to the predictor corrector feature when comparing them with the existing method it is observed that the scheme based on using pade 1 1 approximations to the matrix exponential function introduces oscillations with non smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases however the scheme based on pade 0 2 approximations to the matrix exponential function is oscillation free for any time step numerical experiments for well known models from the literature are performed to show the reliability and effectiveness of the proposed methods

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

hllc type and path conservative schemes for a single velocity six equation two phase flow model a comparative study abstract the present article deals with the numerical integration of a six equation single velocity two phase flow model with stiff mechanical relaxation this model can be employed to approximate efficiently the well known single velocity single pressure five equation model of kapila et al 2001 work in the literature has shown the efficiency of the six equation model in simulating complex two phase flows involving cavitation and evaporation processes the aim of this work is to present and discuss various numerical schemes for this two phase model focusing on the integration of the nonconservative terms appearing in the phasic energy equations in fact previous work has suggested that the choice of the discretization method for the nonconservative terms often does not play a significant role two new methods are proposed a path conservative hllc type scheme that is based on the dal maso lefloch murat theory and a generalized hllc type scheme that is based on a suliciu s riemann solver the latter scheme has the important property of preserving the positivity of the intermediate states of the conserved quantities moreover we also approximate solutions of the six equation model by applying two path conservative schemes recently proposed in the literature which have been derived from the osher and hllem riemann solvers we show comparisons of the different numerical schemes for several test cases including cavitation problems and shock tubes an efficiency study for first and second order schemes is also presented numerical results show that different methods corresponding to different numerical treatments of the nonconservative terms give analogous results and they are all able to produce accurate approximations of solutions of the kapila s five equation model except as expected for shocks in two phase mixtures with very high pressure ratios

state estimation and detectability of probabilistic discrete event systems a probabilistic discrete event system pdes is a nondeterministic discrete event system where the probabilities of nondeterministic transitions are specified state estimation problems of pdes are more difficult than those of non probabilistic discrete event systems in our previous papers we investigated state estimation problems for non probabilistic discrete event systems we defined four types of detectabilities and derived necessary and sufficient conditions for checking these detectabilities in this paper we extend our study to state estimation problems for pdes by considering the probabilities the first step in our approach is to convert a given pdes into a nondeterministic discrete event system and find sufficient conditions for checking probabilistic detectabilities next to find necessary and sufficient conditions for checking probabilistic detectabilities we investigate the convergence of event sequences in pdes an event sequence is convergent if along this sequence it is more and more certain that the system is in a particular state we derive conditions for convergence and hence for detectabilities we focus on systems with complete event observation and no state observation for better presentation the theoretical development is illustrated by a simplified example of nephritis diagnosis

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

the diffusion brusselator equation abstract a solution is found for the reaction diffusion equation called the diffusion brusselator equation 1 using the decomposition method

stability and asymptotic behavior of a numerical solution corresponding to a diffusion reaction equation solved by a finite difference scheme crank nicolson abstract in this paper some partial differential equations arising in biochemistry are studied from the numerical point of view these nonlinear equations based on diffusion reaction processes are solved by using a crank nicolson technique and convergence is proved by using lower and upper solutions stability and asymptotic behavior are also examined a numerical algorithm has been deduced and is presently used by chemists

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

a relaxation projection method for compressible flows part i the numerical equation of state for the euler equations a new projection method is developed for the euler equations to determine the thermodynamic state in computational cells it consists in the resolution of a mechanical relaxation problem between the various sub volumes present in a computational cell these sub volumes correspond to the ones traveled by the various waves that produce states with different pressures velocities densities and temperatures contrarily to godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium the pressure computation with this relaxation process replaces the use of the conventional equation of state eos a simplified relaxation method is also derived and provides a specific eos named the numerical eos the use of the numerical eos gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas eos it is then extended to the computation of interface problems separating fluids with different eos liquid gas interface for example with the euler equations the resulting method is very robust accurate oscillation free and conservative for the sake of simplicity and efficiency the method is developed in a lagrange projection context and is validated over exact solutions in a companion paper f petitpas e franquet r saurel a relaxation projection method for compressible flows part ii computation of interfaces and multiphase mixtures with stiff mechanical relaxation j comput phys submitted for publication the method is extended to the numerical approximation of a non conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a relaxation projection method for compressible flows part ii artificial heat exchanges for multiphase shocks the relaxation projection method developed in saurel et al r saurel e franquet e daniel o le metayer a relaxation projection method for compressible flows part i the numerical equation of state for the euler equations j comput phys 2007 822 845 is extended to the non conservative hyperbolic multiphase flow model of kapila et al a k kapila menikoff j b bdzil s f son d s stewart two phase modeling of deflagration to detonation transition in granular materials reduced equations physics of fluids 13 10 2001 3002 3024 this model has the ability to treat multi temperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures the non conservative character of this model poses however computational challenges in the presence of shocks the first issue is related to the riemann problem resolution that necessitates shock jump conditions thanks to the rankine hugoniot relations proposed and validated in saurel et al r saurel o le metayer j massoni s gavrilyuk shock jump conditions for multiphase mixtures with stiff mechanical relaxation shock waves 16 3 2007 209 232 exact and approximate 2 shocks riemann solvers are derived however the riemann solver is only a part of a numerical scheme and non conservative variables pose extra difficulties for the projection or cell average of the solution it is shown that conventional godunov schemes are unable to converge to the exact solution for strong multiphase shocks this is due to the incorrect partition of the energies or entropies in the cell averaged mixture to circumvent this difficulty a specific lagrangian scheme is developed the correct partition of the energies is achieved by using an artificial heat exchange in the shock layer with the help of an asymptotic analysis this heat exchange takes a similar form as the pseudoviscosity introduced by von neumann and richtmyer j von neumann r d richtmyer a method for the numerical calculation of hydrodynamic shocks j appl phys 21 1950 232 237 the present lagrangian numerical scheme thus combines riemann solvers and artificial heat exchanges an eulerian variant is then obtained by using the relaxation projection method developed earlier by the authors for the euler equations the method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the euler equations in the context of interface problems the method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks

verification of detectability in probabilistic finite automata abstract in this paper we analyze state estimation in stochastic discrete event systems sdes that can be modeled as probabilistic finite automata pfas for a given pfa we obtain the necessary and sufficient conditions that guarantee exact state estimation at least asymptotically with increasing certainty as more information is acquired from observing the behavior of the given pfa by defining the notion of aa detectability and providing necessary and sufficient conditions that can be used to verify it the characterization and analysis of aa detectability is transformed to a problem of classification between two or more pfas which capture the recurrent behavior of an underlying markov process that is obtained by ignoring output behavior and focusing on state transitions in the given pfa our approach combines techniques used in classification between two or more pfas with state estimation methods used in logical discrete event systems des we prove that the proposed verification of aa detectability is of polynomial complexity with respect to the size of the state space of the given pfa

one dimensional front tracking based on high resolution wave propagation methods a simple approach to shock tracking is presented in conjunction with conservative high resolution shock capturing methods in one space dimension an underlying uniform grid is used with additional grid interfaces introduced at appropriate points for tracked shocks conservative high resolution methods based on the large time step wave propagation approach are used on the resulting nonuniform grid this method is stable even if some of the small cells created by the tracked interface are orders of magnitude smaller than the regular cells used to determine the time step a fractional step method is used to handle source terms several calculations are presented to demonstrate the effectiveness of the method including an unstable detonation wave calculation where mesh refinement in the reaction zone is required in addition to shock tracking stability and accuracy results of the method are also shown for some sample problems the basic ideas described here can be extended to two space dimensions as will be

diagnosability of stochastic discrete event systems we investigate diagnosability of stochastic discrete event systems we define the notions of a and aa diagnosability for stochastic automata these notions are weaker than the corresponding notion of diagnosability for logical automata introduced by sampath et al through the construction of a stochastic diagnoser we determine offline conditions necessary and sufficient to guarantee a diagnosability and sufficient to guarantee aa diagnosability we also show how the stochastic diagnoser can be used for on line diagnosis of failure events we illustrate the results through two examples from hvac systems

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

numerical solution of the time fractional black scholes model governing european options when considering the price change of the underlying fractal transmission system a fractional black scholes b s model with an order time fractional derivative is derived in this paper we discuss the numerical simulation of this time fractional black scholes model tfbsm governing european options a discrete implicit numerical scheme with a spatially second order accuracy and a temporally 2 order accuracy is constructed then the stability and convergence of the proposed numerical scheme are analyzed using fourier analysis some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method finally as an application we use the tfbsm and the above numerical technique to price several different european options

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

design and analysis of a dissipative scheme to solve a generalized multi dimensional higgs boson equation in the de sitter space time abstract in this work we design a numerically efficient finite difference technique for the solution of a fractional extension of the higgs boson equation in the de sitter space time the model under investigation is a multidimensional equation with riesz fractional derivatives of orders in 0 1 1 2 which considers a generalized potential and a time dependent diffusion factor an energy integral for the mathematical model is readily available and we propose an explicit and consistent numerical technique based on fractional order centered differences with similar hamiltonian properties as the continuous model a fractional energy approach is used then to prove the properties of stability and convergence of the technique for simulation purposes we consider both the classical and the fractional higgs real valued scalar fields in the 3 1 dimensional de sitter space time and find results qualitatively similar to those available in the literature the present work is the first paper to report on a hamiltonian discretization of the higgs boson equation both fractional and non fractional in the de sitter space time and its numerical analysis more precisely the present manuscript is the first paper of the literature in which a dissipation preserving scheme to solve the multi dimensional fractional higgs boson equation in the de sitter space time is proposed and thoroughly analyzed indeed it is worth pointing out that previous efforts used techniques based on the runge kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

observability of discrete event dynamic systems a finite state automaton is adopted as a model for discrete event dynamic systems deds observations are assumed to be a subset of the event alphabet observability is defined as having perfect knowledge of the current state at points in time separated by bounded numbers of transitions a polynomial test for observability is given it is shown that an observer may be constructed and implemented in polynomial time and space a bound on the cardinality of the observer state space is also presented a notion of resiliency is defined for observers and a test for resilient observability and a procedure for the construction of a resilient observer are presented

boundary particle method for laplace transformed time fractional diffusion equations highlights error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations our method avoids costly convolution integral calculation and dilemmatic selection of time step size a truly boundary only meshless method is applied to laplace transformed inhomogeneous problem our method effectively simulates 3d long time history fractional diffusion systems this paper develops a novel boundary meshless approach laplace transformed boundary particle method ltbpm for numerical modeling of time fractional diffusion equations it implements laplace transform technique to obtain the corresponding time independent inhomogeneous equation in laplace space and then employs a truly boundary only meshless boundary particle method bpm to solve this laplace transformed problem unlike the other boundary discretization methods the bpm does not require any inner nodes since the recursive composite multiple reciprocity technique rc mrm is used to convert the inhomogeneous problem into the higher order homogeneous problem finally the stehfest numerical inverse laplace transform nilt is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding bpm solutions in comparison with finite difference discretization the ltbpm introduces laplace transform and stehfest nilt algorithm to deal with time fractional derivative term which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability consequently it can effectively simulate long time history fractional diffusion systems error analysis and numerical experiments demonstrate that the present ltbpm is highly accurate and computationally efficient for 2d and 3d time fractional diffusion equations

analytical and numerical solutions of the unsteady 2d flow of mhd fractional maxwell fluid induced by variable pressure gradient abstract the nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations this paper studies the analytical and numerical methods of unsteady 2d flow of magnetohydrodynamic mhd fractional maxwell fluid in a rectangular pipe driven by variable pressure gradient the governing equation is formulated with caputo time dependent fractional derivatives whose orders are distributed in interval 0 2 a challenge is to firstly obtain the exact solution by combining modified separation of variables method with mikusinski type operational calculus meanwhile the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed different to the most classical works both the stability and convergence analysis of two dimensional multi term time fractional momentum equation are derived based on numerical analysis the results show that the velocity increases with the rise of the fractional parameter and relaxation time while an increase in the values of hartmann number leads to a slower velocity in the rectangular pipe

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

an acoustic convective splitting based approach for the kapila two phase flow model in this paper we propose a new acoustic convective splitting based numerical scheme for the kapila five equation two phase flow model the splitting operator decouples the acoustic waves and convective waves the resulting two submodels are alternately numerically solved to approximate the solution of the entire model the lagrangian form of the acoustic submodel is numerically solved using an hllc type riemann solver whereas the convective part is approximated with an upwind scheme the result is a simple method which allows for a general equation of state numerical computations are performed for standard two phase shock tube problems a comparison is made with a non splitting approach the results are in good agreement with reference results and exact solutions

detectability in stochastic discrete event systems abstract a discrete event system possesses the property of detectability if it allows an observer to perfectly estimate the current state of the system after a finite number of observed symbols i e detectability captures the ability of an observer to eventually perfectly estimate the system state in this paper we analyze detectability in stochastic discrete event systems sdes that can be modeled as probabilistic finite automata more specifically we define the notion of a detectability which characterizes our ability to estimate the current state of a given sdes with increasing certainty as we observe more output symbols the notion of a detectability is differentiated from previous notions for detectability in sdes because it takes into account the probability of problematic observation sequences that do not allow us to perfectly deduce the system state whereas previous notions for detectability in sdes considered each observation sequence that can be generated by the underlying system we discuss observer based techniques that can be used to verify a detectability and provide associated necessary and sufficient conditions we also prove that a detectability is a pspace hard problem

a novel finite element method for the distributed order time fractional cable equation in two dimensions abstract in this paper the unstructured mesh galerkin finite element method with a weighted and shifted grunwald difference approximation and composite trapezoid formula is presented to solve the nonhomogeneous two dimensional distributed order time fractional cable equation on irregular convex domains the crank nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution the stability and convergence of the numerical scheme are discussed and derived finally some numerical examples on irregular convex domains are given to confirm our theoretical results

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems an alternating direction implicit adi orthogonal spline collocation osc method is described for the approximate solution of a class of nonlinear reaction diffusion systems its efficacy is demonstrated on the solution of well known examples of such systems specifically the brusselator gray scott gierer meinhardt and schnakenberg models and comparisons are made with other numerical techniques considered in the literature the new adi method is based on an extrapolated crank nicolson osc method and is algebraically linear it is efficient requiring at each time level only o n operations where n is the number of unknowns moreover it is shown to produce approximations which are of optimal global accuracy in various norms and to possess superconvergence properties

an eulerian algorithm for coupled simulations of elastoplastic solids and condensed phase explosives the performance of non ideal condensed phase explosives depends on the nature of the confiner material as well as the charge itself so the accurate modelling of this confiner and its interaction with the explosive allows for improved performance predictions traditionally numerical methods for solving such multi material problems have used lagrangian or mixed eulerian lagrangian approaches but recent advances in numerical methods for coupling cfd computational fluid dynamics and cmd computational material dynamics algorithms has made such coupled simulations possible in the eulerian frame of reference however to date the explosive material representation within these simulations has been restricted to the single phase euler equations in the present study we couple a multi phase chemically active model for condensed phase explosives to an elastic plastic model for inert confiner materials in the presented algorithm the ghost fluid method is employed to represent the evolving material interfaces as discontinuities on discrete space the coupling between the materials at these interfaces is achieved by means of a new approximate mixed riemann solver developed as part of this research in addition we present a mixed riemann solver for a simpler transport model which ignores compaction effects at the interface the robustness and accuracy of the developed solvers is demonstrated by comparisons against results from the original ghost fluid method and exact solutions of model riemann problems to allow for more realistic material behaviour the mixed riemann solvers are subsequently extended to handle the shock mie gruneisen equation of state and an iterative procedure is suggested to increase accuracy as required these mixed riemann solvers demonstrate their suitability for explosive solid interactions in two test cases of multi phase detonations confined by an elastic plastic solid

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

an implicit numerical method for the two dimensional fractional percolation equation in this paper an implicit numerical method for the two dimensional fractional percolation equation without the assumption of continued and rigid body motion is proposed consistency stability and convergence of the implicit finite difference method are established finally some numerical examples are given the numerical results demonstrate the effectiveness of the theoretical analysis

an unstructured mesh control volume method for two dimensional space fractional diffusion equations with variable coefficients on convex domains in this paper we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains which to the best of our knowledge is a new contribution to the literature firstly we present the finite volume scheme for the two dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements secondly we explore the property of the stiffness matrix generated by the integral of the space fractional derivative we find that the stiffness matrix is sparse and not regular therefore we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system which is more efficient than using the gaussian elimination method finally we present several examples to verify our method in which we make a comparison of our method with the finite element method for solving a riesz space fractional diffusion equation on a circular domain the numerical results demonstrate that our method can reduce cpu time significantly while retaining the same accuracy and approximation property as the finite element method the numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains

modelling wave dynamics of compressible elastic materials an eulerian conservative hyperbolic model of isotropic elastic materials subjected to finite deformation is addressed it was developed by godunov s k godunov elements of continuum mechanics nauka moscow 1978 in russian and g h miller p colella a high order eulerian godunov method for elastic plastic flow in solids j comput phys 167 2001 131 176 some modifications are made concerning a more suitable form of governing equations they form a set of evolution equations for a local cobasis which is naturally related to the almansi deformation tensor another novelty is that the equation of state is given in terms of invariants of the almansi tensor in a form which separates hydrodynamic and shear effects this model is compared with another hyperbolic non conservative model which is widely used in engineering sciences for this model we develop a riemann solver and determine some reference solutions which are compared with the conservative model the numerical results for different tests show good agreement of both models for waves of very small and very large amplitude however for waves of intermediate amplitude important discrepancies between results are clearly visible

a five equation reduced model for compressible two phase flow problems this paper studies an eulerian diffuse interface model for the simulation of compressible multifluid and two phase flow problems we first show how to derive this model from a seven equation two pressure two velocity model of baer nunziato type using an asymptotic analysis in the limit of zero relaxation time we then study the mathematical properties of the system the structure of the waves the expression of the riemann s invariants and the existence of a mathematical entropy we also describe two different numerical approximation schemes for this system the first one relies on a linearized riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations finally we present some numerical experiments and comparisons with the results obtained by the two pressure two velocity model as well as some test cases and comparisons with another five equation model recently proposed for interface computations between compressible fluids

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

rational approximation to the fractional laplacian operator in reaction diffusion problems this paper provides a new numerical strategy for solving fractional in space reaction diffusion equations on bounded domains under homogeneous dirichlet boundary conditions using the matrix transfer technique the fractional laplacian operator is replaced by a matrix which in general is dense the approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices this leads to a semilinear initial value problem in which the matrices involved are sparse numerical results are presented to verify the effectiveness of the proposed solution strategy

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

modelling evaporation fronts with reactive riemann solvers this work deals with the modelling of permeable fronts and the building of a numerical method allowing the multi dimensional propagation of such fronts a particular attention is given to evaporation waves that appear in cavitating systems these ones are considered as discontinuities through which a non equilibrium liquid turns to a liquid vapor mixture at thermodynamic equilibrium such transformation occurs at finite rate in order to determine this kinetics the evaporation front is assumed to propagate at the maximum admissible speed corresponding to the chapman jouguet deflagration point j r simoes moreira j e shepherd evaporation waves in superheated dodecane j fluid mech 382 1999 63 86 using this particular kinetic relation rankine hugoniot relations are closed at such fronts then it is possible to solve the associated reactive riemann problem however another difficulty is present to solve the multi dimensional propagation of permeable fronts this kind of front is subsonic and a conventional averaging scheme such as godunov scheme is inappropriate to overcome this difficulty the reactive riemann problem solution is embedded into the discrete equations method dem r abgrall r saurel discrete equations for physical and numerical compressible multiphase mixtures j comp phys 186 2003 361 396 r saurel s gavrilyuk f renaud a multiphase model with internal degrees of freedom application to shock bubble interaction j fluid mech 495 2003 283 321 this numerical method necessitates deep extensions that are detailed herein numerical results are shown and validated over experimental data some examples show that the same method may be applied to the propagation of detonation fronts

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

liquid and liquid gas flows at all speeds all speed flows and in particular low mach number flow algorithms are addressed for the numerical approximation of the kapila et al 1 multiphase flow model this model is valid for fluid mixtures evolving in mechanical equilibrium but out of temperature equilibrium and is efficient for material interfaces computation separating miscible and non miscible fluids in this context the interface is considered as a numerically diffused zone captured as well as all present waves shocks expansion waves the same flow model can be used to solve cavitating and boiling flows 2 many applications occurring with liquid gas interfaces and cavitating flows involve a very wide range of mach number from 10 3 to supersonic and even hypersonic conditions with respect to the mixture sound speed it is thus important to address numerical methods free of restrictions regarding the mach number to do this a preconditioned riemann solver is built and embedded into the godunov explicit scheme it is shown that this method converges to exact solutions but needs too small time steps to be efficient an implicit version is then derived first in one dimension and second in the frame of 2d unstructured meshes two phase flow preconditioning is then addressed in the frame of the saurel et al 3 algorithm modifications of the preconditioned riemann solver are needed and detailed convergence of both single phase and two phase numerical solutions are demonstrated with the help of single phase and two phase steady nozzle flow solutions last the method is illustrated by the computation of real cavitating flows in venturi nozzles vapour pocket size and instability frequencies are reproduced by the model and method without using any adjustable parameter

a semi alternating direction method for a 2 d fractional fitzhugh nagumo monodomain model on an approximate irregular domain a fitzhugh nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue in this paper we consider a two dimensional fractional fitzhugh nagumo monodomain model on an irregular domain the model consists of a coupled riesz space fractional nonlinear reaction diffusion model and an ordinary differential equation describing the ionic fluxes as a function of the membrane potential second we use a decoupling technique and focus on solving the riesz space fractional nonlinear reaction diffusion model a novel spatially second order accurate semi implicit alternating direction method siadm for this model on an approximate irregular domain is proposed third stability and convergence of the siadm are proved finally some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two dimensional fractional fitzhugh nagumo model on both an approximate circular and an approximate irregular domain

a second order operator splitting fourier spectral method for fractional in space reaction diffusion equations abstract fractional differential equations have been proved to be valuable tools for modeling diffusive processes associated with anomalous diffusion or spatial heterogeneity however many numerical methods for these equations have limitations in terms of computational efficiency due to the nonlocal nature of the fractional operator which leads to large dense matrices the aim of this paper is to propose a second order operator splitting fourier spectral method as an accurate and efficient approach for solving fractional in space reaction diffusion equations this approach gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power in the problem in order to achieve second order time accuracy we decompose the original equation into linear and nonlinear subequations and combine a half time linear solver and a full time second order nonlinear solver followed by a final half time linear solver we numerically demonstrate the accuracy and efficiency of the proposed method and apply the proposed method to investigate the effect of the fractional power in fractional in space reaction diffusion equations including the allen cahn fitzhugh nagumo and gray scott models

diffuse interface model for compressible fluid compressible elastic plastic solid interaction an eulerian hyperbolic diffuse interface model for elastic plastic solid fluid interaction is constructed the system of governing equations couples euler equations of compressible fluids and a visco plastic model of maxwell type materials the deviatoric part of the stress tensor decreases during plastic deformations in the same manner as models of multicomponent fluids in particular the model is able to create interfaces which were not present initially the model is thermodynamically compatible it verifies the entropy inequality however a numerical treatment of the model is particularly challenging indeed the model is non conservative so a special numerical splitting is proposed to overcome this difficulty the numerical algorithm contains two relaxation procedures one of them is physical and is related to the plastic relaxation mechanism relaxation toward the yield surface the second one is numerical it consists in replacing the algebraic equation expressing a mechanical equilibrium between components by a partial differential equation with a short relaxation time the numerical method was tested in 1d case wilkins flying plate problem 2d plane case impact of a projectile on a plate and axisymmetrical case taylor test problem impact with penetration effects etc numerical examples show the ability of the model to deal with real physical phenomena

solid fluid diffuse interface model in cases of extreme deformations diffuse interface methods have been recently proposed and successfully used for accurate compressible multi fluid computations abgrall 1 kapila et al 20 saurel et al 30 these methods deal with extended systems of hyperbolic equations involving a non conservative volume fraction equation and relaxation terms following the same theoretical frame we derive here an eulerian diffuse interface model for elastic solid compressible fluid interactions in situations involving extreme deformations elastic effects are included following the eulerian conservative formulation proposed in godunov 16 miller and colella 23 godunov and romenskii 17 plohr and plohr 27 and gavrilyuk et al 14 we apply first the hamilton principle of stationary action to derive the conservative part of the model the relaxation terms are then added which are compatible with the entropy inequality in the limit of vanishing volume fractions the euler equations of compressible fluids and a conservative hyperelastic model are recovered it is solved by a unique hyperbolic solver valid at each mesh point pure fluid pure solid and mixture cell capabilities of the model and methods are illustrated on various tests of impacts of solids moving in an ambient compressible fluid

gauss jacobi type quadrature rules for fractional directional integrals fractional directional integrals are the extensions of the riemann liouville fractional integrals from one to multi dimensional spaces and play an important role in extending the fractional differentiation to diverse applications in numerical evaluation of these integrals the weakly singular kernels often fail the conventional quadrature rules such as newton cotes and gauss legendre rules it is noted that these kernels after simple transforms can be taken as the jacobi weight functions which are related to the weight factors of gauss jacobi and gauss jacobi lobatto rules these rules can evaluate the fractional integrals at high accuracy comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the gauss jacobi type rules in handling weakly singular kernels of different strengths potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section

detectability of discrete event systems in this note we investigate the detectability problem in discrete event systems we assume that we do not know initially which state the system is in the problem is to determine the current and subsequent states of the system based on a sequence of observations the observation includes partial event observation and or partial state observation which leads to four possible cases we further define four types of detectabilities strong detectability weak detectability strong periodic detectability and weak periodic detectability we derive necessary and sufficient conditions for these detectabilities these conditions can be checked by constructing an observer which models the estimation of states under different observations the theory developed in this note can be used in feedback control and diagnosis if the system is detectable then the observer can be used as a diagnoser to diagnose the failure states of the system

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

high order numerical approximation formulas for riemann liouville riesz tempered fractional derivatives construction and application i abstract in this paper we develop a new numerical algorithm for solving the riesz tempered space fractional diffusion equation the stability and convergence of the numerical scheme are discussed via the technique of matrix analysis finally numerical experiments are performed to confirm the effectiveness of our numerical algorithm

computing fractional laplacians on complex geometry domains algorithms and simulations we consider a fractional laplacian defined in bounded domains by the eigen decomposition of the integer order laplacian and demonstrate how to compute very accurately using the spectral element method the eigenspectrum and corresponding eigenfunctions in two dimensional prototype complex geometry domains we then employ these eigenfunctions as trial and test bases to first solve the fractional diffusion equation and subsequently to simulate two phase flow based on the navier stokes equations combined with a fractional allen cahn mass preserving model a key point to the effectiveness of an exponential convergence of this approach is the use of a weighted gram schmidt orthonormalization of the eigenfunctions that guarantees accurate projection and recovery of spectral accuracy for smooth solutions we demonstrate that even when only part of the eigenspectrum is computed accurately we can still obtain exponential convergence if we employ the complete set of the eigenvectors of the discrete laplacian

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

modelling anomalous diffusion using fractional bloch torrey equations on approximate irregular domains abstract diffusion weighted imaging is an in vivo non invasive medical diagnosis technique that uses the brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment recent work highlights that the diffusion induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay to investigate the underlying mechanism of this deviated signal decay diffusion is re examined through the bloch torrey equation by using fractional calculus with respect to both time and space in this study we explore the influence of the complex geometrical structure on the diffusion process an effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two dimensional time space riesz fractional partial differential equation with dirichlet boundary conditions this scheme is proved to be unconditionally stable and convergent numerical examples are given to support our analysis we then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time space fractional bloch torrey equations on irregular domains