application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

error dynamics the propagation of a signal in a continuous medium and the associated evolution of error is of prime importance in many applications of applied physics there have been many efforts in analyzing error dynamics using method attributed to von neumann 1 2 that is readily applied for linear equations and in quasilinearized form for non linear equations the main assumption for linear problems is that the error and the signal follow the same dynamics while this appears intuitively correct the main aim behind this work is to show that this is not correct for discrete computing due to dispersion or phase error or when the numerical method is not strictly neutrally stable we demonstrate the above with the help of the linear advection equation for the analysis of space time discretization schemes the linear advection equation as a model that represents many flows and wave phenomena is used

dispersion and group velocity in numerical schemes for three dimensional hydrodynamic equations we investigate dispersion and group velocity relations of numerical schemes for the three dimensional hydrodynamic equations by using the transform function method the numerical methods are developed for two of arakawa s spatial grid types namely b grid and c grid using a spectral method in the vertical direction one of our results is that if the horizontal scale length is greater than the rossby radius of deformation then the b grid is more appropriate than the commonly used c grid

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a compact finite difference scheme on a non equidistant mesh abstract a fourth order compact finite difference scheme is presented for second order partial differential equations the scheme is derived for a one dimensional non equidistant mesh which makes it particularly useful in problems with sharp boundary layers the inclusion of general boundary conditions does not reduce the order of the scheme for the boundary points the scheme is tested on a representative model equation we shortly discuss its stability in the simplest time dependent heat flow problem and its use in more dimensions

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

numerical wave propagation on the hexagonal c grid inertio gravity mode and rossby mode dispersion properties are examined for discretizations of the linearized rotating shallow water equations on a regular hexagonal c grid in planar geometry it is shown that spurious non zero rossby mode frequencies found by previous authors in the f plane case can be avoided by an appropriate discretization of the coriolis terms three generalizations of this discretization that conserve energy even for non constant coriolis parameter are presented a quasigeostrophic b plane analysis is carried out to investigate the rossby mode dispersion properties of these three schemes the rossby mode dispersion relation is found to have two branches the primary branch modes are good approximations in terms of both structure and frequency to corresponding modes of the continuous governing equations and offer some improvements over a quadrilateral c grid scheme the secondary branch modes have vorticity structures approximating those of small scale modes of the continuous governing equations suggesting that the hexagonal c grid might have an advantage in terms of resolving extra rossby modes however the frequencies of the secondary branch rossby modes are much smaller than those of the corresponding continuous modes so this potential advantage is not fully realized

a new velocity vorticity formulation for direct numerical simulation of 3d transitional and turbulent flows accuracy of velocity vorticity v formulations over other formulations in solving navier stokes equation has been established in recent times however the issue of non satisfaction of solenoidality conditions on vorticity is not addressed in the literature which can possibly lead to non physical solution in this respect here we have developed and reported conservative rotational form of the v formulation which preserves the solenoidality condition on vorticity in a much simpler way compared to other formulations superiority of rotational form over the conventional laplacian form of v formulation is also shown by comparing the results for flows inside cubical lid driven cavity ldc for solving the 3d navier stokes equation using a staggered grid we use optimized compact schemes for a interpolation and b evaluation of first and second derivatives as illustrations we have solved problems of i flow inside a 3d lid driven cavity ldc whose solutions are compared with experimental results reported by koseff and street 19 and ii 3d transitional flow of an equilibrium zero pressure gradient zpg boundary layer over a flat plate as demonstration of the effectiveness of the rotational form of v formulation

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

a vorticity velocity method for the numerical solution of 3d incompressible flows a new method for the numerical solution of the 3d navier stokes equations written in terms of vorticity velocity is presented the advantages of this formulation with respect to primitive variables and vorticity vector potential ones are discussed in view of physical as well as engineering applications a suitable form of the continuum equations the most appropriate discretization scheme and variable location in order to guarantee the solenoidality of the velocity and vorticity fields are introduced and justified a 3d lid driven cavity problem for 400 re 3200 is chosen as a test case for comparison and validation purposes a parallel implementation of the method as been performed on a shared memory architecture mainframe speedup results and efficiency considerations are given and discussed

direct numerical simulation of flow past elliptic cylinders flow over elliptic cylinders can be considered prototypical of flow over a range of bluff bodies since the geometry allows one to study the effect of both thickness and angle of attack on the flow field therefore a careful study of this flow should provide valuable insight into the phenomenon of unsteady separation and the structure of bluff body wakes with this in mind a spectral collocation technique has been developed to simulate the full three dimensional incompressible flow over elliptic cylinders and unlike spectral element and spectral multidomain techniques here the flow is solved in a single domain the equations are discretized on a body fitted elliptic cylindrical grid and properties of the metric associated with this coordinate system are used to solve the governing equations in an efficient manner key issues including the inflow and outflow boundary conditions and time discretization are discussed in detail with the hope that this will facilitate future simulations of similar flows finally we present results of two and three dimensional simulations for a range of flow and geometric parameters the results are compared with available experimental data and it is found that important quantities like strouhal numbers and drag coefficients match well with established values

effective fluctuation and response theory the so called fluctuation theorems pushed the study of systems far beyond equilibrium whose response to thermodynamic forces affinities is characterized by the reciprocal and the fluctuation dissipation relations all these results rely on the assumption that the observer has complete information about the system no hidden leakage to the environment exact evaluation of the thermodynamic cost of processes will an observer who has marginal information be able to perform an effective thermodynamic analysis given that such observer will only be able to establish local equilibrium by perturbing the stalling currents will he she observe equilibrium like fluctuations within the formalism of markov jump processes on finite networks we provide a broad theory of the statistical behavior of some out of many currents that flow across a system in particular 1 there exist effective affinities for which an integral fluctuation relation holds 2 at stalling i e where the marginal currents vanish a symmetrized fluctuation dissipation relation holds 3 under reasonable assumptions on the parametrization of the rates effective affinities can be operationally defined by a procedure of tuning to stalling 4 there exists a notion of marginally time reversed process which restores the full fledged fluctuation relation and reciprocity 5 there exist fluctuation relations across different levels in the hierarchy of more and more complete theories the above results apply to configuration space currents and to their phenomenological linear combinations provided certain symmetries of the effective affinities are respected a condition whose range of validity we deem the most interesting question left open to future inquiry

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

a conservative phase field method for solving incompressible two phase flows in this paper a conservative phase field method based on the work of sun and beckermann y sun c beckermann sharp interface tracking using the phase field equation j comput phys 220 2007 626 653 for solving the two and three dimensional two phase incompressible navier stokes equations is proposed the present method can preserve the total mass as the cahn hilliard equation but the calculation and implementation are much simpler than that the dispersion relation preserving schemes are utilized for the advection terms while the helmholtz smoother is applied to compute the surface tension force term to verify the proposed method several benchmarks are examined and shown to have good agreements with previous results it also shows that the satisfactions of mass conservations are guaranteed

a compact difference scheme for the navier stokes equations in vorticity velocity formulation this paper presents a new numerical method for solving the incompressible unsteady navier stokes equations in vorticity velocity formulation the method is applicable to spatial simulations of transitional and turbulent boundary layer flows it is based on a compact difference discretization of the streamwise and wall normal derivatives in cartesian coordinates a fourier collocation approach is used for the spanwise derivatives important new features of the numerical method are the use of nonequidistant differences in the wall normal direction the use of split compact differences in the streamwise direction a new fast iteration for a semi implicit time integration of the wall normal diffusion terms and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary results of test calculations are presented to verify the improvements obtained by the use of these new techniques

a two dimensional dispersion analysis of selected methods for solving the linearized shallow water equations abstract the accuracy and cost of three finite element methods for solving the linearized two dimensional shallow water equations are compared with a traditional explicit finite difference technique accuracy is determined by comparing numerical and analytic plane wave solutions cost is measured as the number of computations per unit of real time and per unit of model area two of the finite element methods are shown to be cost competitive and as accurate as the chosen explicit finite difference technique though not comprehensive the finite element analyses also suggest that meshes composed of equilateral triangles most accurately represent phase and group velocity

fast methods for computing centroidal laguerre tessellations for prescribed volume fractions with applications to microstructure generation of polycrystalline materials abstract ideas from the mathematical theory of optimal transport have recently been transferred to the micromechanics of polycrystalline materials leading to fast methods for generating polycrystalline microstructures with grains of prescribed volume fraction in terms of centroidal laguerre tessellations in this work we improve the state of the art solvers for a given set of seeds and corresponding volume fractions summing to unity there is a set of laguerre weights such that the corresponding laguerre tessellation realizes the prescribed volume fractions exactly furthermore the laguerre weights are unique up to a constant and can be determined by solving a convex optimization problem however whenever the optimization algorithm encounters a weight vector leading to an empty cell the optimization problem is no longer locally strictly convex to account for the latter backtracking strategies are typically employed we show that modern gradient based optimization algorithms devoid of backtracking like the malitsky mishchenko method and the barzilai borwein scheme easily overcome the described difficulty leading to a significant speed up compared to more traditional solvers furthermore for computing centroidal laguerre tessellations of prescribed volume fraction we propose an anderson accelerated version of lloyd s algorithm and show by numerical experiments that it consistently reduces the run time we demonstrate the capabilities of our proposed methods for generating microstructures of polycrystalline materials with prescribed grain size distribution

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

weighted ultrafast diffusion equations from well posedness to long time behaviour in this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures these equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic wasserstein distance exploiting this structure in particular through the so called jko scheme we introduce a notion of weak solutions prove existence uniqueness bv and h1 estimates l1 weighted contractivity harnack inequalities and exponential convergence to a steady state

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

a new combined stable and dispersion relation preserving compact scheme for non periodic problems a new compact scheme is presented for computing wave propagation problems and navier stokes equation a combined compact difference scheme is developed for non periodic problems called nccd henceforth that simultaneously evaluates first and second derivatives improving an existing combined compact difference ccd scheme following the methodologies in sengupta et al t k sengupta s k sircar a dipankar high accuracy schemes for dns and acoustics j sci comput 26 2 2006 151 193 stability and dispersion relation preservation drp property analysis is performed here for general ccd schemes for the first time emphasizing their utility in uni and bi directional wave propagation problems that is relevant to acoustic wave propagation problems we highlight a specific points in parameter space those give rise to least phase and dispersion errors for non periodic wave problems b the solution error of ccd nccd schemes in solving stommel ocean model an elliptic p d e and c the effectiveness of the nccd scheme in solving navier stokes equation for the benchmark lid driven cavity problem at high reynolds numbers showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature

further improvement and analysis of ccd scheme dissipation discretization and de aliasing properties in this paper we further analyze a combined compact difference ccd scheme proposed recently t k sengupta v lakshmanan v v s n vijay a new combined stable and dispersion relation preserving compact scheme for non periodic problems j comput phys 228 8 2009 3048 3071 for its dissipation discretization properties to show that its superiority also helps in controlling aliasing error for a benchmark internal flow however application of the same ccd method to study the receptivity of a boundary layer experiencing adverse pressure gradient is not successful this is traced to the nature of the equilibrium flow where the better dissipation property is not helpful in the inviscid part of the flow where the aliasing problems continue to persist a further modification is proposed to the ccd method here to solve complex physical problems requiring information on higher order disturbance quantities as in problems of flow receptivity and instability

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

hexagonal patterns in a simplified model for block copolymers in this paper we study a new model for patterns in two dimensions inspired by diblock copolymer melts with a dominant phase the model is simple enough to be amenable not only to numerics but also to analysis yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments starting from a sharp interface continuum model a nonlocal energy functional involving a wasserstein cost we derive the new model using gamma convergence in a limit where the volume fraction of one phase tends to zero the limit energy is defined on atomic measures in three dimensions the atoms represent small spherical blobs of the minority phase and in two dimensions they represent thin cylinders of the minority phase we then study local minimizers of the limit energy numerical minimization is performed in two dimensions by recasting the problem as a computational geometry problem involving power diagrams the numerical results suggest that the sma

analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods fourier analysis is used to quantitatively assess the resolution and in particular the isotropy of wave solution using finite difference spatial discretization schemes along with fourth order runge kutta temporal scheme aspect ratio of the grid in two dimension along with the angle of wave propagation are the parameters varied to qualitatively and quantitatively assess the anisotropy of the solutions for a a skewed one dimensional wave convecting in two dimensions following the standard convection equation and b a wave propagating following the two dimensional linearized rotating shallow water equations results show the effect of changing the aspect ratio and the propagation angle on the directional nature of the solution as obtained by different methods for the above non dispersive and dispersive wave system

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

a new compact scheme for parallel computing using domain decomposition direct numerical simulation dns of complex flows require solving the problem on parallel machines using high accuracy schemes compact schemes provide very high spectral resolution while satisfying the physical dispersion relation numerically however as shown here compact schemes also display bias in the direction of convection often producing numerical instability near the inflow and severely damping the solution always near the outflow this does not allow its use for parallel computing using domain decomposition and solving the problem in parallel in different sub domains to avoid this in all reported parallel computations with compact schemes the full domain is treated integrally while using parallel thomas algorithm pta or parallel diagonal dominant pdd algorithm in different processors with resultant latencies and inefficiencies for domain decomposition methods using compact scheme in each sub domain independently a new class of compact schemes is proposed and specific strategies are developed to remove remaining problems of parallel computing this is calibrated here for parallel computing by solving one dimensional wave equation by domain decomposition method we also provide the error norm with respect to the wavelength of the propagated wave packet next the advantage of the new compact scheme on a parallel framework has been shown by solving three dimensional unsteady navier stokes equations for flow past a cone cylinder configuration at a mach number of 4 additionally a test case is conducted on the advection of a vortex for a subsonic case to provide an estimate for the error and parallel efficiency of the method using the proposed compact scheme in multiple processors

high accuracy solution of bi directional wave propagation in continuum mechanics solution of partial differential equations by numerical method is strongly affected due to numerical errors which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation to quantify and control such errors and obtain high accuracy solutions we consider a class of problems which involve second derivative of unknowns with respect to time here we analyse numerical metrics such as the numerical group velocity numerical phase speed and the numerical amplification factor for different methods in solving the model bi directional wave equation bdwe such equations can be solved directly for example by runge kutta nystrom rkn method alternatively the governing equation can be converted to a set of first order in time equations and then using four stage fourth order runge kutta rk4 method for time integration spatial discretisation considered are the classical second and fourth order central difference schemes along with lele s central compact scheme for evaluating second derivatives in another version we have used lele s scheme for evaluating first derivatives twice to obtain the second derivative as bdwe represents non dissipative non dispersive dynamics we also consider the canonical problem of linearised rotating shallow water equation lrswe in a new formulation involving second order derivative in time which represents dispersive waves along with a stationary mode the computations of lrswe with rk4 and rkn methods for temporal discretisation and lele s compact schemes for spatial discretisation are compared with computations performed with rk4 method for time discretisation and staggered compact scheme scs for spatial discretisation by treating it as a set of three equations as reported in rajpoot et al 2012 1

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

fluctuation theorem and large deviation function for a solvable model of a molecular motor we study a discrete stochastic model of a molecular motor this discrete model can be viewed as a minimal ratchet model we extend our previous work on this model by further investigating the constraints imposed by the fluctuation theorem on the operation of a molecular motor far from equilibrium in this work we show the connections between different formulations of the fluctuation theorem one formulation concerns the generating function of the currents while another one concerns the corresponding large deviation function which we have calculated exactly for this model a third formulation concerns the ratio of the probability of observing a velocity v to the same probability of observing a velocity ensuremath v finally we show that all the formulations of the fluctuation theorem can be understood from the notion of entropy production

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

a new compact difference scheme for second derivative in non uniform grid expressed in self adjoint form a single parameter family of self adjoint compact difference sacd schemes is developed for discretizing the laplacian operator in self adjoint form developed implicit scheme is formally second order accurate with respect to truncation error with a free parameter a which helps control the numerical properties in the spectral plane the sacd scheme is analyzed in the spectral plane for its resolution properties for both periodic and non periodic problems using the matrix spectral analysis t k sengupta g ganeriwal s de analysis of central and upwind schemes j comput phys 192 2 2003 677 694 the major objective here is to identify the advantages of the new scheme over the traditional explicit second order cd2 scheme in discretizing the laplacian operator in self adjoint form this appears in navier stokes equation and in other transport equations and boundary value problems bvp expressed in orthogonal co ordinate systems either in physical or in transformed plane we also compare the developed method with the higher order compact schemes for non uniform grids to demonstrate the accuracy of sacd scheme we have tested it for i bi directional wave propagation problem given by the second order wave equation and ii an elliptic bvp as in the stommel ocean model for the stream function these examples help infer the properties of sacd scheme when solving different types of partial differential equations most importantly the effects of grid stretching and choice of value of the free parameter a are investigated here we also compare the present implicit compact method with explicit compact method known as the higher order compact hoc method also the practical applications of the sacd scheme are explored by solving the navier stokes equation for the cases of a a flow inside a lid driven cavity and b the receptivity and instability of an external adverse pressure gradient flow over a flat plate in the former unsteadiness of the flow is captured and in the latter the receptivity of the flow is studied in causing flow instability by triggering tollmien schlichting waves the new scheme shows a marked improvement over the explicit scheme for low reynolds number steady flow in lid driven cavity whereas for the flow in the same geometry at higher reynolds numbers efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer

global spectral analysis of multi level time integration schemes an analysis is reported here for three time level integration methods following the global spectral analysis gsa described in high accuracy computing methods t k sengupta cambridge univ press usa the focus is on the second order adamsbashforth ab2 and the extrapolation in time ext2 methods careful distinction is made for the first time step at t 0 by either euler forward or four stage fourth order rungekutta rk4 time schemes the latter is used to solve a benchmark aeroacoustic problem several one dimensional wave propagation models are analyzed pure advection and advection diffusion equations various spatial discretizations are discussed including fourier spectral method attention is paid to the presence of physical and numerical modes as noted in the quadratic equation obtained from the difference equation for the model 1d convection equation it is shown that ab2 method is less stable and accurate than ext2 method with respect to numerical dissipation and dispersion this is true for the methods in which the physical mode dominates over the numerical mode presented analysis provides useful guide to analyze any three time level methods

a simplified ratchet model of molecular motors a limiting case of one of the ratchet models of ajdari prost et al is analyzed an explicit solution is obtained for the probability distribution as a function of the time for any initial distribution with all the transients included in the long time limit the drift velocity and diffusion coefficient are obtained in terms of the microscopic transition rates that are the parameters in the model in spite of its extreme simplicity with realistic values of its kinetic parameters the model yields values of the drift velocity and effective force that are of the right magnitude for a molecular motor the model proves to be a simple special case of derrida s periodic one dimensional hopping model for which he found a solution in the long time limit

a gallavotti cohen evans morriss like symmetry for a class of markov jump processes we investigate a new symmetry of the large deviation function of certain time integrated currents in non equilibrium systems the symmetry is similar to the well known gallavotti cohen evans morriss symmetry for the entropy production but it concerns a different functional of the stochastic trajectory the symmetry can be found in a restricted class of markov jump processes where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints we provide three physical examples where time integrated observables display such a symmetry moreover we argue that the origin of the symmetry can be traced back to time reversal if stochastic trajectories are grouped appropriately

macroscopic constraints for the minimum entropy production principle in an essential and quite general setup based on networks we identify schnakenberg s observables as the constraints that prevent a system from relaxing to equilibrium showing that in the linear regime steady states satisfy a minimum entropy production principle the result is applied to master equation systems opening a new path to a well known version of the principle regarding invariant states moreover with the aid of a simple example the principle is shown to conform to prigogine s original formulation finally we discuss analogies and differences with a recently proposed maximum entropy production principle

a two level time stepping method for layered ocean circulation models this paper describes a numerical method with two time levels for solving partial differential equations that represent layered models of ocean circulation the method is designed to be used with a barotropic baroclinic splitting that separates the fast and slow motions into subsystems that are solved by different techniques with this method some of the dependent variables are predicted and then corrected after the initial prediction steps all steps involve centered differencing and averaging about the midpoint of the time interval in question the coriolis and pressure terms are evaluated at the same time levels to avoid a first order truncation error in the geostrophic balance between those terms compared to the three level leapfrog method that is widely used in geophysical fluid dynamics the present method does not admit a computational mode and the maximum permissible time step is at least twice as large in addition with the two level method it is possible to use a nearly nonoscillatory advection algorithm to solve the equations for mass and momentum in a simple test problem the present method gives less phase error than the leapfrog method and two other methods and it does not give any amplitude error the differences are especially large when comparing the present method to the leapfrog method the two level method also gives good results in a nonlinear test problem involving vanishing layers at the top and bottom of the fluid domain and an interface moving along sloping bottom topography

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

a three point sixth order nonuniform combined compact difference scheme a three point nonuniform combined compact difference nccd scheme with sixth order accuracy is proposed for numerical models the nccd scheme is a generalization of the previously proposed combined compact difference ccd scheme with a global hermitan polynomial spline and has major improved features such as error and computational cpu time reduction for nonperiodic boundaries additional sixth or fifth order nonuniform boundary conditions are proposed the nccd scheme with either sixth or fifth order additional boundary conditions can increase the accuracy and decrease the cpu time about 1 2 orders of magnitude compared to the ccd scheme

a novel velocity vorticity formulation of the navier stokes equations with applications to boundary layer disturbance evolution a novel velocity vorticity formulation of the unsteady three dimensional navier stokes equations is presented the formulation is particularly suitable for simulating the evolution of three dimensional disturbances in boundary layers a key advantage is that there are only three governing equations for three primary dependent variables another advantage is that no wall boundary conditions are needed for the vorticity instead the conditions placed on the velocity are linked to the vorticity field through integral constraints based on the definition of vorticity numerical methods are presented in the context of application to the three dimensional boundary layer over a rotating disc the discretization scheme uses spectral expansions in the wall normal and azimuthal or spanwise directions and compact finite differences in the radial or streamwise direction the scheme is implemented so that the advantages of spectral convergence can be combined with the use of an efficient line iteration solution procedure the linearized form of the new velocity vorticity method is validated for the case of convective instabilities evolving over both rigid and compliant discs

a compact high order alternating direction implicit method for three dimensional acoustic wave equation with variable coefficient abstract efficient and accurate numerical simulation of seismic wave propagation is important in various geophysical applications such as seismic full waveform inversion fwi problem however due to the large size of the physical domain and requirement on low numerical dispersion many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media despite the great efforts that have been devoted during the past decades it still remains a challenging task in the development of efficient and accurate finite difference method for multi dimensional acoustic wave equation with variable velocity in this paper we proposed a pade approximation based finite difference scheme for solving the acoustic wave equation in three dimensional heterogeneous media the new method is obtained by combining the pade approximation and a novel algebraic manipulation the efficiency of the new algorithm is further improved through the alternative directional implicit adi method the stability of the new algorithm has been theoretically proved by energy method the new method is conditionally stable with a better courant friedrichs lewy condition cfl condition which has been verified numerically extensive numerical examples have been solved which demonstrated that the new method is accurate efficient and stable

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

solution of linearized rotating shallow water equations by compact schemes with different grid staggering strategies high accuracy solution of pdes requires proper error analysis previous analysis for a non dispersive system t k sengupta a dipankar p sagaut error dynamics beyond von neumann analysis j comput phys 226 2007 1211 1218 identified sources of error correctly here the aim is to extend the spectral analysis for the model linearized rotating shallow water equations lrswe as an example of dispersive system we perform the analysis when high accuracy compact schemes are used to solve the lrswe relevant to geophysical fluid dynamics using different grid arrangements proposed in mesinger and arakawa f mesinger a arakawa numerical methods used in atmospheric models garp publ ser no 17 vol 1 wmo geneva 1976 pp 43 64 and randall d a randall geostrophic adjustment and the finite difference shallow water equations mon wea rev 122 1994 1371 1377 compact schemes are used for fluid dynamical problem as these afford near spectral accuracy in solving non periodic problems however higher accuracy methods also suffer from errors those are often filtered by low order methods for example dispersion error is present in all numerical methods and extreme form of it leads to q waves which appear at higher wavenumbers for compact schemes as compared to lower order method we also evaluate a compact scheme specifically designed for use with staggered grids here two and four time level temporal discretization methods have been compared for solving lrswe by considering classical fourth order four stage runge kutta rk 4 two time level forward backward fb and four time level generalized fb temporal integration schemes

gallavotti cohen type symmetry related to cycle decompositions for markov chains and biochemical applications we slightly extend the fluctuation theorem obtained in lebowitz and spohn in j stat phys 95 333 365 1999 for sums of generators considering continuous time markov chains on a finite state space whose underlying graph has multiple edges and no loop this extended frame is suited when analyzing chemical systems as simple corollary we derive by a different method the fluctuation theorem of d andrieux and p gaspard for the fluxes along the chords associated to a fundamental set of oriented cycles andrieux and gaspard in j stat phys 127 107 131 2007

centroidal power diagrams lloyd s algorithm and applications to optimal location problems in this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems the target energy can be written either in terms of atomic measures and the wasserstein distance or in terms of weighted points and power diagrams generalized voronoi diagrams the latter formulation is more suitable for computation we show that critical points of the energy are centroidal power diagrams which are generalizations of centroidal voronoi tessellations and that they can be approximated by a generalization of lloyd s algorithm lloyd s algorithm is a common method for finding centroidal voronoi tessellations we prove that the algorithm is energy decreasing and prove a convergence theorem numerical experiments suggest that the algorithm converges linearly we illustrate the algorithm in two and three dimensions using simple models of optimal location and crystallization see online supplementary material

a staggered grid high order accurate method for the incompressible navier stokes equations a high order accurate finite difference method for the numerical solution of the incompressible navier stokes equations is presented fourth order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite difference formulas high order accuracy in time is obtained by marching the solution with runge kutta methods the incompressibility constraint is enforced for each runge kutta stage iteratively either by local pressure correction or by a poisson equation based global pressure correction method local pressure correction is carried out on cell by cell basis using a local fourth order accurate discrete analog of the continuity equation the global pressure correction is based on the numerical solution of a poisson type equation which is discretized to fourth order accuracy and solved using gmres in both cases the updated pressure is used to recompute the velocities in order to satisfy the incompressibility constraint to fourth order accuracy the accuracy and efficiency of the proposed method is demonstrated in test problems

optimal time advancing dispersion relation preserving schemes in this paper we examine the constrained optimization of explicit runge kutta rk schemes coupled with central spatial discretization schemes to solve the one dimensional convection equation the constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von neumann analysis developed in sengupta et al t k sengupta a dipankar p sagaut error dynamics beyond von neumann analysis j comput phys 226 2007 1211 1218 the efficiency of these optimal schemes is demonstrated for the one dimensional convection problem and also by solving the navier stokes equations for a two dimensional lid driven cavity ldc problem for the ldc problem results for re 1000 are compared with results using spectral methods in botella and peyret o botella r peyret benchmark spectral results on the lid driven cavity flow comput fluids 27 1998 421 433 to calibrate the method in solving the steady state problem we also report the results of the same flow at re 10 000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems finally we also compare our results for a wave packet propagation problem with another method developed for computational aeroacoustics

effective vorticity velocity formulations for three dimensional incompressible viscous flows the vorticity velocity formulation for incompressible viscous flows is studied the main concern is effective numerical solution of the kinematic cauchy riemann equations for velocity especially when the divergence free condition of vorticity is violated due to numerical error the mathematical formulation of a differential and an integral approaches are revisited a novel projection of vorticity onto the divergence free space and its application to the two approaches are studied as an illustration of the differential approach the 3d lid driven cavity problem is solved the projection scheme is justified through numerical tests by comparing it to a new fully divergence free scheme numerical tests indicate that keeping vorticity divergence free is important for obtaining correct and accurate solutions the theory and methods apply to other divergence free fields of physical interest as well such as that in electro magnetic dynamics

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

accuracy study of the ido scheme by fourier analysis the numerical accuracy of the interpolated differential operator ido scheme is studied with fourier analysis for the solutions of partial differential equations pdes advection diffusion and poisson equations the ido scheme solves governing equations not only for physical variable but also for first order spatial derivative spatial discretizations are based on hermite interpolation functions with both of them in the fourier analysis for the ido scheme the fourier coefficients of the physical variable and the first order derivative are coupled by the equations derived from the governing equations the analysis shows the ido scheme resolves all the wavenumbers with higher accuracy than the fourth order finite difference fd and compact difference cd schemes for advection equation in particular for high wavenumbers the accuracy is superior to that of the sixth order combined compact difference ccd scheme the diffusion and poisson equations are also more accurately solved in comparison with the fd and cd schemes these results show that the ido scheme guarantees highly resolved solutions for all the terms of fluid flow equations

fluctuation theorem for currents and schnakenberg network theory a fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state by using schnakenberg network theory the theorem can be applied in particular in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution