cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

periodic solutions of some infinite dimensional hamiltonian systems associated with non linear partial difference equations ii this is our second paper devoted to the study of some non linear schrodinger equations with random potential we study the non linear eigenvalue problems corresponding to these equations we exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails contrary to our results in the first paper the results established in the present paper hold for an arbitrary strength of the non linear cubic term in the non linear schrodinger equation

effective hamiltonians and phase diagrams for tight binding models we present rigorous results for several variants of the hubbard model in the strong coupling regime we establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are in fact leading order terms of well defined volume independent unitarily equivalent interactions in addition in the very asymmetric falicov kimball regime we are able to apply recently developed phase diagram technology quantum pirogov sinai theory to conclude that the zero temperature phase diagrams obtained for the leading classical part remain valid except for thin excluded regions and small deformations for the full fledged quantum interaction at zero or low temperature moreover the phase diagram is stable against addition of arbitrary but sufficiently small further quantum terms that do not break the ground state symmetries this generalizes and unifies a number of previous results on the subject in particular published results on the zero temperature phase diagram of the falikov kimball model with and without magnetic flux are extended to small temperatures and or small ionic hopping we give explicit expressions for the first few orders in the hopping amplitude of equivalent interactions and we describe the resulting phase diagram our approach yields algorithms to compute equivalent interactions to arbitrarily high order in the hopping amplitude

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined

variational multiscale element free galerkin vmefg and local discontinuous galerkin ldg methods for solving two dimensional brusselator reaction diffusion system with and without cross diffusion abstract the finite element method fem is one of the basic methods for solving deterministic and stochastic partial differential equations this method is proposed in the 19 decade and after years several modifications for this well known technique such as mortal fem discontinuous galerkin fem extended fem least squares fem spectral fem mixed fem immersed fem adaptive fem etc have been proposed also for improving the accuracy and for considering complex domain shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique the name of this improved method is element free galerkin efg method which is classified in the category of meshless methods the efg method has been applied for solving many problems in engineering in the current paper we select two numerical procedures that are extracted from fem the discontinuous galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity it should be noted that many enriched ideas have been explained for improving accuracy of efg method in the current paper the local discontinuous galerkin and variational multiscale efg methods are applied for obtaining numerical solution of two dimensional brusselator system with and without cross diffusion the brusselator system is a theoretical model for a type of autocatalytic reaction up to best of authors knowledge the accuracy of the obtained numerical solutions using local discontinuous galerkin method is very related to the used time discrete scheme therefore fourth order exponential time differencing method has been employed for discretizing the time variable also to achieve a full discretization scheme the local discontinuous galerkin finite element method is used moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

remediation of contaminated groundwater by meshless local weak forms in this paper four meshless local weak form methods such as direct meshless local petrovgalerkin method dmlpg meshless local petrovgalerkin method mlpg local weak radial point interpolation method lwrpim and local weak moving kriging method lwmk are applied to find the numerical solution of coupled non linear advectiondiffusionreaction system arising in the prevention of groundwater contamination problem a comparison between these methods is done from the perspective of accuracy and computational efficiency an efficient fourth order exponential time differencing rungekutta method is utilized for the time discretization the computational efficiency is the most significant advantage of the dmlpg method in comparison with the other meshless local weak form methods because dmlpg reduces the computational costs significantly this is due to the fact that dmlpg shifts the numerical integrations over low degree polynomials rather than over complicated shape functions the main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non linear partial differential equations especially coupled non linear advectiondiffusionreaction system the numerical results confirm the good efficiency of the proposed methods for solving our model

a real distinct poles exponential time differencing scheme for reaction diffusion systems a second order exponential time differencing etd method for reaction diffusion systems which uses a real distinct poles discretization method for the underlying matrix exponentials is developed the method is established to be stable and second order convergent it is demonstrated to be robust for problems involving non smooth initial and boundary conditions and steep solution gradients we discuss several advantages over competing second order etd schemes

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

high order time stepping fourier spectral method for multi dimensional space fractional reaction diffusion equations abstract this paper introduces a high order time stepping scheme that is based on using fourier spectral in space and a fourth order diagonal pade approximation to the matrix exponential function for solving multi dimensional space fractional reaction diffusion equations the resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non linear system at each time step while maintaining the stability of the scheme the non locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices this scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator it also attains spectral convergence for multiple spatial dimensions the stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions which provides an indication of the stability of the method the convergence analysis is performed empirically to show that the scheme is fourth order accurate in time as expected numerical experiments on reaction diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space fractional reaction diffusion equations and to validate the effectiveness of the scheme

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

higher order exponential time differencing scheme for system of coupled nonlinear schrodinger equations the coupled nonlinear schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics like propagation of pulses efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system in this paper we introduce a new version of the cox and matthews third order exponential time differencing runge kutta etd3rk scheme based on the 1 2 pade approximation to the exponential function in addition we present its local extrapolation form to improve temporal accuracy to the fourth order the developed scheme and its extrapolation are seen to be strongly stable which have ability to damp spurious oscillations caused by high frequency components in the solution a computationally efficient algorithm of the new scheme based on a partial fraction splitting technique is presented in order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear schrodinger equations and performed several numerical experiments on them the numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long range solitary solutions given by coupled nonlinear schrodinger equations and conserved the conserved quantities mass and energy exactly to at least five decimal places

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

serk2v3 solving mildly stiff nonlinear partial differential equations abstract multidimensional nonlinear parabolic partial differential equations pdes appear in a large variety of disciplines usually the scientific literature advises against the use of explicit ode solvers for the integration of stiff problems however in this manuscript we would like to show how some explicit methods can be very efficient for specific problems stabilized explicit runge kutta methods serk are a class of explicit methods with extended stability domains along the negative real axis it is necessary to evaluate the function s times but the stability region is o s 2 hence the computational cost is o s times lower than for traditional explicit algorithms in this way moderately stiff problems can be integrated by the use of simple explicit evaluations serk schemes can easily be applied to many different classes of problems with large dimensions and additionally they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes and therefore are especially well suited for the method of lines mol discretizations of parabolic multi dimensional pdes additionally the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way that is with minimal number of additional stages hence adaptation of the length step is possible at practically no extra cost in this work we study several strategies to change the time step for non smooth functions and finally propose a new technique this procedure improves the numerical results especially when the data is non smooth we also derive two algorithms to estimate the spectral radius in the new code a nonlinear power method and another procedure based on the gershgorin theorem although serk algorithms are only second order schemes in this manuscript we propose to utilize them after higher order discretizations in space to obtain higher accurate solutions efficiently and usually faster since these higher order schemes allow the use of a smaller number of nodes some numerical experiments shown in the paper support these conclusions

uniqueness of the ground state in weak perturbations of non interacting gapped quantum lattice systems we consider a general weak perturbation of a non interacting quantum lattice system with a non degenerate gapped ground state we prove that in a finite volume the dependence of the ground state on the boundary condition exponentially decays with the distance to the boundary which implies in particular that the infinite volume ground state is unique also equivalent forms of boundary conditions for ground states of general finite quantum systems are discussed

quantum lattice models at intermediate temperature we analyze the free energy and construct the gibbs kms states for a class of quantum lattice systems at low temperature and when the interactions are almost diagonal in a suitable basis the models we study may have continuous symmetries our results however apply to intermediate temperatures where discrete symmetries are broken but continuous symmetries are not our results are based on quantum pirogov sinai theory and a combination of high and low temperature expansions

phase diagram of the two dimensional t t falicov kimball model the ground state phase diagram of the two dimensional falicov kimball model with nearest neighbour and next nearest neighbour hoppings has been studied in the perturbative regime where hoppings are small compared with the on site coulomb interaction the phase diagram at fourth order exhibits a richer structure than the one of the ordinary falicov kimball model

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

exponential fitting bdf algorithms explicit and implicit 0 stable methods we present bdf type formulas capable of the exact integration with only round off errors of differential equations the solutions of which belong to the space generated by the linear combinations of 1 eax xeax plots of their 0 stability regions in terms of are provided we will see that the explicit method is 0 stable with many parameters h plots of their regions absolute stability that include all the negative real axis are provided numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems with the explicit method

time discretization and stability regions for dissipative dispersive kuramoto sivashinsky equation arising in turbulent gas flow over laminar liquid abstract in this study spatio temporal discretization for semilinear dissipative partial differential equations type is introduced analyzed and implemented the model studied here is the dispersively kuramoto sivashinsky equation with an additional term representing the dispersive term arising in turbulent gas flow over laminar liquid tseluiko and kalliadasis 2011 this additional term is multiplied by a parameter that represents the influence of the turbulent gas flow our objective is to examine the effect of this additional term on the dynamics of the kuramoto sivashinsky equation characterized by its chaotic behavior this is achieved by combining the exponential time differencing crank nicolson etd cn scheme derived by kleefed et al 2012 and fourier pseudospectral schemes for temporal and spatial stepping respectively the method is known to be stable and second order convergent in addition a theoretical study and a plot of stability regions of etd cn were performed showing its effectiveness for the stiff problem studied here

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

numerical study of three dimensional turing patterns using a meshless method based on moving kriging element free galerkin efg approach in this paper a numerical procedure is presented for solving a class of three dimensional turing system first we discrete the spatial direction using element free galerkin efg method based on the shape functions of moving kriging interpolation then to achieve a high order accuracy we use the fourth order exponential time differencing runge kutta etdrk4 method using this discretization for the temporal dimension we obtain an explicit scheme and do not need to solve nonlinear system of equations the efg method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the efg method test and trial functions are moving least squares approximation mls shape functions since the shape functions of moving least squares mls approximation do not have kronecker delta property we cannot implement the essential boundary condition directly also building shape functions of mls approximation is a time consuming procedure because of the mentioned reasons we employ the shape functions of moving kriging interpolation technique which have the mentioned property and less cpu time is required for building them for testing this method on three dimensional pdes we select some equations and system of pdes such as allen cahn gray scott ginzburg landau brusselator models predator prey model with additional food supply to predator several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme

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

a variable θ method for parabolic problems of nonsmooth data abstract parabolic initial boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution for those problems specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution this article investigates characteristics of the method and introduces a variable method as a synergistic combination of the crank nicolson cn method 1 2 and the implicit method 1 it suppresses spurious oscillations by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness and maintains as a similar accuracy as the cn method with smooth data an effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations the wobble set the resulting variable method is analyzed for its accuracy and stability various numerical examples are given to verify its effectiveness

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

geometric aspects of quantum spin states a number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system s equilibrium states methods of general applicability are introduced in the context of the su 2s 1 invariant quantum spin s chains with the interaction p o wherep o is the projection onto the singlet state of a pair of nearest neighbor spins the phenomena discussed here include the absence of neel order the possibility of dimerization conditions for the existence of a spectral gap and a dichotomy analogous to one found by affleck and lieb stating that the systems exhibit either slow decay of correlations or translation symmetry breaking our representation elucidates the relation evidence for which was found earlier of the p o spin s systems with the potts and the fortuin kasteleyn random cluster models in one more dimension the method reveals the geometric aspects of the listed phenomena and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin e g within such structure the dichotomy is implied by a topological argument and the alternatives correspond to whether or not the clusters are of finite mean length

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

estimations of the free energy for the hubbard model a series of upper bounds and two lower bounds for the partition function of the hubbard model have been derived these bounds are expressible by certain properties of the falicov kimball model the upper bounds have been derived mainly by the use of the golden thompson inequality and its generalizations and the holder inequality the lower bounds are based on the bogoliubov peierls inequality the numerical values of bounds have also been calculated for small systems

unitary dressing transformations and exponential decay below threshold for quantum spin systems parts i and ii this is the continuation of a series of articles concerning a class of quantum spin systems with hamiltonian operators of the form h lambda sum limits x in lambda s x sum limits yo subset lambda lambda left yo right c 1 t yo where is a graph is a small parameter andsx has a gap 1 for all x e i in the singular seti the gap ofsx can be arbitrarily small part iii is devoted to the proof of a preliminary result while in part iv we consider the case in whichi is a subset of finite density of this completes the first iteration step of the deterministic part of the proof of localization in the ground state of the random field quantumxy model

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

segregation in the asymmetric hubbard model we study the asymmetric hubbard model where hoppings of electrons depend on their spin for strong interactions and sufficiently asymmetric hoppings it is proved that the ground state displays phase separation away from half filling this extends a recent result obtained with freericks and lieb for the falicov kimball model it is based on estimates for the sum of lowest eigenvalues of the discrete laplacian in arbitrary domains

eserk5 a fifth order extrapolated stabilized explicit runge kutta method abstract a new algorithm is developed and analyzed for multi dimensional non linear parabolic partial differential equations pdes which are semi discretized in the spatial variables leading to a system of ordinary differential equations odes it is based on fifth order extrapolated stabilized explicit runge kutta schemes eserk they are explicit methods and therefore it is not necessary to employ complicated software for linear or non linear system of equations additionally they have extended stability regions along the negative real semi axis hence they can be considered to solve stiff problems coming from very common diffusion or reaction diffusion problems previously only lower order codes up to fourth order have been constructed and made available in the scientific literature however at the same time higher order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe and where functions change very fast the new schemes allow changing the step length very easily and with a very small computational cost thus a variable step length with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well known ode solvers

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

spectral variational integrators for semi discrete hamiltonian wave equations abstract in this paper we present a highly accurate hamiltonian structure preserving numerical method for simulating hamiltonian wave equations this method is obtained by applying spectral variational integrators svi to the system of hamiltonian odes which are derived from the spatial semi discretization of the hamiltonian pde the spatial variable is discretized by using high order symmetric finite differences an efficient implementation of svi for high dimensional systems of odes is presented

law of large numbers and central limit theorem for randomly forced pde s we consider a class of dissipative pde s perturbed by an external random force under the condition that the distribution of perturbation is sufficiently non degenerate a strong law of large numbers slln and a central limit theorem clt for solutions are established and the corresponding rates of convergence are estimated it is also shown that the estimates obtained are close to being optimal the proofs are based on the property of exponential mixing for the problem in question and some abstract slln and clt for mixing type markov processes

efficient l stable method for parabolic problems with application to pricing american options under stochastic volatility efficient l stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve heston s stochastic volatility model based pde for pricing american options under stochastic volatility the proposed new method is also used to solve two asset american options pricing problem cox and matthews s m cox p c matthews exponential time differencing for stiff systems journal of computational physics 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic problems kassam and trefethen a k kassam l n trefethen fourth order time stepping for stiff pdes siam journal on scientific computing 26 4 2005 1214 1233 showed that while computing certain functions involved in the cox matthews schemes severe cancelation errors can occur which affect the accuracy and stability of the schemes kassam and trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors but this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems we modify the etdrk schemes using positivity preserving pade approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique as a result of this approach it is required only to solve several backward euler linear problems in serial or parallel

mott transition and sign problem for a model of lattice fermions we consider a model of spinless fermions on the square latticez2 with an interaction potential of strengthu 0 at distance one and strengthj at distance two in the largeu limit t j u wheret is the hopping amplitude as the chemical potential is varied ift t 0 we find three different phases corresponding to full half and zero filling fractions we study the system at low temperaturet 0 by a method involving a canonical transformation and a functional integral representation ift 0 we locate the phase boundaries of the mott metal insulator transition for all j u with upper and lower bounds show that mean field theory is valid ifj 0 we have only one sided bounds for the phase boundaries and we can t validate mean field theory in casej 0 we introduce a new resummation scheme for low temperature expansions which yields finite and convergent perturbation series and permits us to study issues like the sign problem our algorithm gives an optimal canonical transformation for the functional integral such that the expectation of the sign observables is exp c cdot t cdot ve frac beta 2 wherev is the volume and t 1

a fast numerical method to price american options under the bates model we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the price of an american option can be obtained as the solution of a linear complementarity problem governed by a partial integro differential equation in this paper a numerical method for solving such a problem is proposed in particular first of all using a bermudan approximation and a richardson extrapolation technique the linear complementarity problem is reduced to a set of standard linear partial differential problems see for example ballestra and sgarra 2010 chang et al 2007 2012 then these problems are solved using an ad hoc pseudospectral method which efficiently combines the chebyshev polynomial approximation an implicit explicit time stepping and an operator splitting technique numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing american options under the bates model

a hybrid fem for solving the allen cahn equation abstract we present an unconditionally stable hybrid finite element method for solving the allen cahn equation which describes the temporal evolution of a non conserved phase field during the antiphase domain coarsening in a binary mixture its various modified forms have been applied to image analysis motion by mean curvature crystal growth topology optimization and two phase fluid flows the hybrid method is based on the operator splitting method the equation is split into a heat equation and a nonlinear equation an implicit finite element method is applied to solve the diffusion equation and then the nonlinear equation is solved analytically various numerical experiments are presented to confirm the accuracy and efficiency of the method our simulation results are consistent with previous theoretical and numerical results

ground states in relatively bounded quantum perturbations of classical lattice systems we consider ground states in relatively bounded quantum perturbations of classical lattice models we prove general results about such perturbations existence of the spectral gap exponential decay of truncated correlations analyticity of the ground state and also prove that in particular the aklt model belongs to this class if viewed on a large enough length scale this immediately implies a general perturbation theory about this model

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

multiplicative ergodic theorem for a non irreducible random dynamical system we study the asymptotic properties of the trajectories of a discrete time random dynamical system in an infinite dimensional hilbert space under some natural assumptions on the model we establish a multiplica tive ergodic theorem with an exponential rate of convergence the assumptions are satisfied for a large class of parabolic pdes including the 2d navier stokes and complex ginzburg landau equations perturbed by a non degenerate bounded random kick force as a consequence of this er godic theorem we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system in particular we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible the proof relies on a refined version of the uniform feller property combined with some contraction and bootstrap arguments

influence of anisotropic next nearest neighbour hopping on diagonal charge striped phases we consider the model of a strongly correlated system of electrons on a two dimensional square lattice described by an extended falicov kimball hamiltonian where the stability of some axial and diagonal striped phases was proved introducing a next nearest neighbour hopping small enough not to destroy the striped structure we examine rigorously how the presence of the next nearest neighbour hopping anisotropy reduces the 2 rotation degeneracy of the diagonal striped phase the effect appears to be similar to that in the case of anisotropy of the nearest neighbour hopping diagonal stripes are oriented in the direction of the weaker next nearest neighbour hopping

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

on smoothing of the crank nicolson scheme and higher order schemes for pricing barrier options most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities in pricing barrier options certain aspects are triggered if the asset price becomes too high or too low standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied moreover these unwanted oscillations become worse when estimating the hedging parameters e g delta and gamma we have an improved smoothing strategy for the crank nicolson method which is unique in achieving optimal order convergence for barrier option problems numerical experiments are discussed for one asset and two asset problems time evolution graphs are obtained for one asset problems to show how option prices change with respect to time this smoothing strategy is then extended to higher order methods using diagonal m m pade main schemes under a smoothing strategy of using as damping schemes the 0 2m 1 subdiagonal pade schemes

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

stabilized explicit runge kutta methods for multi asset american options american derivatives have become very popular instruments in financial markets however they are more complicated to price than european options since at each time level we have to determine not only the option value but also whether or not it should be exercised several procedures have been proposed to dissolve these difficulties but they usually involve the solution of nonlinear partial differential equations pdes in the case of multi dimensional problems solving these equations is a very challenging task in this paper we propose stabilized explicit runge kutta serk methods to solve this kind of problems they can easily be applied to many different classes of problems with large dimensions and they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes as for example lapack and blas packages or multigrid or iterative methods applied together with newton type algorithms and are especially well suited for the method of lines mol discretizations of parabolic pdes

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

low temperature phase diagrams of quantum lattice systems i stability for quantum perturbations of classical systems with finitely many ground states starting from classical lattice systems ind 2 dimensions with a regular zerotemperature phase diagram involving a finite number of periodic ground states we prove that adding a small quantum perturbation and or increasing the temperature produce only smooth deformations of their phase diagrams the quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds for fermions the interactions must be given by monomials of even degree in creation and annihilation operators our methods can be applied to some anyonic systems as well our analysis is based on an extension of pirogov sinai theory to contour expansions ind 1 dimensions obtained by iteration of the duhamel formula

low temperature expansions for the gibbs states of quantum ising lattice systems by conditioning on a subalgebra of observables we develop low temperature expansions for certain quantum ising models which include transverse ising models and anisotropic xy models these expansions are majorized by the series for functions solving certain nonlinear partial differential equations convergence for the majorizing series follows from cauchy kowalevski theory applied to the partial differential equations the low temperature expansions imply long range order