numerical methods for a multicomponent two phase interface model with geometric mean influence parameters in this paper we consider an interface model for multicomponent two phase fluids with geometric mean influence parameters which is popularly used to model and predict surface tension in practical applications for this model there are two major challenges in theoretical analysis and numerical simulation the first one is that the influence parameter matrix is not positive definite the second one is the complicated structure of the energy function which requires us to find out a physically consistent treatment to overcome these two challenging problems we reduce the formulation of the energy function by employing a linear transformation and a weighted molar density and furthermore we propose a local minimum grand potential energy condition to establish the relation between the weighted molar density and mixture compositions from this we prove the existence of the solution under proper conditions and prove the maximum principle of the weighted molar density for numerical simulation we propose a

statistical properties of skew endomorphisms with bernoulli base abstract we analyse numerically the statistical properties of a class of mappings of the 2 torus t 2 onto itself whose investigation is suggested by some models of modulated diffusion these transformations can be written as a skew product of the endomorphism bp x px mod 1 p z 1 0 1 on the 1 torus t 1 0 1 and a translation on t 1 under suitable assumptions the skew product can be proven to be mixing w r t the lebesgue measure central limit cl and functional central limit fcl properties are numerically checked for analytic observables the result is remarkable because the mappings show no hyperbolic or quasi hyperbolic structure crucial for the proof of central limit theorem and donsker s invariance principle in all of the dynamical systems where these properties have been established up to now moreover cl and fcl behaviours seem to hold also in the case of purely ergodic endomorphisms and even for observables whose correlations do not decay at infinity

nonlocal boundary conditions for finite difference parabolic equation solvers theoretical numerical models of underwater sound propagation usually incorporate a downgoing radiation condition on the transverse component of the acoustic field most one way or parabolic equation pe solvers approximate this radiation condition by appending an absorbing layer to the computational mesh and setting the field to zero at the base of this layer papadakis et al j acoust soc am 92 2030 2038 1992 replaced such approximate treatments with a nonlocal boundary condition nlbc that exactly transforms the semi infinite pe problem to an equivalent one in a bounded domain papadakis approach requires the evaluation of a spectral wave number integral whose integrand is inversely proportional to the impedance of the subbottom medium in this paper an alternate procedure is analyzed for obtaining nlbcs directly from the z space crank nicolson solvers for both the tappert and claerbout pes formulas for the field at range r r are derived in terms of the known field at the previously c

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

weak galerkin finite element methods for darcy flow anisotropy and heterogeneity this paper presents a family of weak galerkin finite element methods wgfems for darcy flow computation the wgfems are new numerical methods that rely on the novel concept of discrete weak gradients the wgfems solve for pressure unknowns both in element interiors and on the mesh skeleton the numerical velocity is then obtained from the discrete weak gradient of the numerical pressure the new methods are quite different than many existing numerical methods in that they are locally conservative by design the resulting discrete linear systems are symmetric and positive definite and there is no need for tuning problem dependent penalty factors we test the wgfems on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in darcy flow

covolume upwind finite volume approximations for linear elliptic partial differential equations in this paper we study covolume upwind finite volume methods on rectangular meshes for solving linear elliptic partial differential equations with mixed boundary conditions to avoid non physical numerical oscillations for convection dominated problems nonstandard control volumes covolumes are generated based on local peclet s numbers and the upwind principle for finite volume approximations two types of discretization schemes with mass lumping are developed with use of bilinear or biquadratic basis functions as the trial space respectively some stability analyses of the schemes are presented for the model problem with constant coefficients various examples are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

impedance matched absorbers for finite difference parabolic equation algorithms in this paper a perfectly matched layer pml absorber recently introduced into the electromagnetic propagation literature by berenger j comput phys 114 185 200 1994 is adapted for use with both paraxial and wide angle acoustic parabolic equations pes our procedure incorporates an imaginary component into the transverse coordinate that mimics the introduction of a fictitious absorber on the edge of the computational grid use of such an impedance matched layer can significantly reduce spurious reflections compared to physical absorbing layer methods and thus allows a smaller number of boundary points to be employed in pe calculations numerical results obtained with several higher order propagator approximations confirm that such impedance matched absorbers efficiently eliminate reflections

stabilized mixed continuous enriched galerkin formulations for locally mass conservative poromechanics abstract local element wise mass conservation is often highly desired for numerical solutions to coupled poromechanical problems as an efficient numerical method featuring this property mixed continuous galerkin cg enriched galerkin eg finite elements have recently been proposed in which piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed cg cg elements while this enrichment of the pressure space provides local mass conservation it unavoidably alters the stability condition for mixed finite elements because no stabilization method has been available for the new stability condition high order displacement interpolation has been required for mixed cg eg elements if undrained condition is expected to circumvent this requirement here we develop stabilized formulations for the mixed cg eg elements that permit equal order interpolation functions even in the undrained limit we begin by identifying the inf sup condition for mixed cg eg elements by phrasing an enriched poromechanical problem as a twofold saddle point problem we then derive two types of stabilized formulations one based on the polynomial pressure projection ppp method and the other based on the fluid pressure laplacian fpl method a key finding of this work is that both methods lead to stabilization terms that should be augmented only to the cg part of the pore pressure field not to the enrichment part the two stabilized formulations are verified and investigated through numerical examples that involve various conditions spanning 1d to 3d isotropy to anisotropy and homogeneous to heterogeneous domains the methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions

wide angle parabolic equation solutions to two range dependent benchmark problems numerical solutions are presented to two benchmark problems involving acoustic propagation in range dependent media the two problems deal with 1 upslope propagation in a wedge shaped channel with a penetrable bottom and 2 propagation in a plane parallel waveguide with a range varying sound speed profile the solutions are based on a pair of wide angle variable density parabolic equations one of which is solved using a finite difference algorithm while the other is solved using a split step algorithm

a locking free solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of lagrangian elements abstract this paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear lagrangian elements it solves the primal variable displacement in the strain div formulation and can handle both displacement and traction boundary conditions it is a locking free solver based on conforming finite elements the solver has second order accuracy in displacement and first order accuracy in stress and dilation divergence of displacement as validated by theoretical analysis and illustrated by numerical experiments on benchmarks deal ii implementation is also discussed

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

dual scale galerkin methods for darcy flow abstract the discontinuous galerkin dg method has found widespread application in elliptic problems with rough coefficients of which the darcy flow equations are a prototypical example one of the long standing issues of dg approximations is the overall computational cost and many different strategies have been proposed such as the variational multiscale dg method the hybridizable dg method the multiscale dg method the embedded dg method and the enriched galerkin method in this work we propose a mixed dual scale galerkin method in which the degrees of freedom of a less computationally expensive coarse scale approximation are linked to the degrees of freedom of a base dg approximation we show that the proposed approach has always similar or improved accuracy with respect to the base dg method with a considerable reduction in computational cost for the specific definition of the coarse scale space we consider raviart thomas finite elements for the mass flux and piecewise linear continuous finite elements for the pressure we provide a complete analysis of stability and convergence of the proposed method in addition to a study on its conservation and consistency properties we also present a battery of numerical tests to verify the results of the analysis and evaluate a number of possible variations such as using piecewise linear continuous finite elements for the coarse scale mass fluxes

postprocessing of non conservative flux for compatibility with transport in heterogeneous media a conservative flux postprocessing algorithm is presented for both steady state and dynamic flow models the postprocessed flux is shown to have the same convergence order as the original flux an

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

a locally conservative enriched galerkin approximation and efficient solver for elliptic and parabolic problems we present and analyze an enriched galerkin finite element method eg to solve elliptic and parabolic equations with jump coefficients eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions which can be considered as a penalty stabilization the method is shown to be locally and globally conservative while keeping fewer degrees of freedom in comparison with discontinuous galerkin finite element methods dg moreover we present and analyze a fast effective eg solver whose cost is roughly that of cg and which can handle an arbitrary order of approximations a number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of eg when coupled with transport

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

a new method for a realistic treatment of the sea bottom in the parabolic approximation a new method for realistic treatment of the bottom in the parabolic approximation in underwater acoustics is presented the method utilizes a finite difference scheme for the numerical implementation of the parabolic equation and the boundary condition is introduced to the problem in the form of an impedance condition the impedance condition for the fluid half space bottom can be implemented analytically while total numerical implementation is necessary for a general type bottom a characteristic type of bottoms that can be handled by this method is the homogeneous elastic half space the reliability of the method is confirmed by examples where the results obtained by the method are compared with well known results of reference problems

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

machine learning and transport simulations for groundwater anomaly detection abstract this paper presents studies on modeling and algorithms for groundwater anomaly detection specifically conductivity along with four other surrogates are used for identifying anomaly in groundwater the one class support vector machine 1 svm technique is utilized for model training and real data from colorado water watch is used for testing the model and algorithms design of code modules in python is briefly discussed since groundwater contamination rarely happens in reality we also use synthetic data from numerical simulations of flow and transport in porous media to test this model for anomaly detection

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

decay of correlations for certain quadratic maps we prove exponential decay of correlations for f wheref belongs in a positive measure set of quadratic maps of the interval and is its absolutely continuous invariant measure these results generalize to other interval maps

a locally conservative finite element method based on piecewise constant enrichment of the continuous galerkin method this paper presents a locally conservative finite element method based on enriching the approximation space of the continuous galerkin method with elementwise constant functions the proposed method has a smaller number of degrees of freedom than the discontinuous galerkin method numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method we also present a theoretical analysis of the method and establish optimal convergence of numerical solutions

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

discrete transparent boundary conditions for the numerical solution of fresnel s equation the paper presents a construction scheme of deriving transparent i e reflection free boundary conditions for the numerical solution of fresnel s equation being formally equivalent to schrodinger s equation in contrast to previous suggestions the method advocated here treasts the discrete problem after discretization of the time like variable i e in a rothe method which leads to a sequence of coupled boundary value problems the thus obtained boundary conditions appear to be of a nonlocal cauchy type as it turns out each kind of linear implicit discretization induces its own discrete transparent boundary conditions numerical experiments on technologically relevant examples from integrated optics are included

towards a complete determination of the spectrum of a transfer operator associated with intermittency the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point for an example of such an intermittent map the farey map we give a simple proof that the transfer operator is self adjoint on a suitably defined hilbert space and characterize its spectrum using a suitable first return map we present a highly efficient numerical method for the determination of all the eigenvalues including those embedded in the continuous spectrum

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

unconditionally stable methods for simulating multi component two phase interface models with peng robinson equation of state and various boundary conditions in this paper we consider multi component dynamic two phase interface models which are formulated by the cahn hilliard system with peng robinson equation of state and various boundary conditions these models can be derived from the minimum problems of helmholtz free energy or grand potential in the realistic thermodynamic systems the resulted cahn hilliard systems with various boundary conditions are fully coupled and strongly nonlinear a linear transformation is introduced to decouple the relations between different components and as a result the models are simplified from this we further propose a semi implicit unconditionally stable time discretization scheme which allows us to solve the cahn hilliard system by a decoupled way and thus our method can significantly reduce the computational cost and memory requirements the mixed finite element methods are employed for the spatial discretization and the approximate errors are also analyzed for both space and time numerical examples are tested to demonstrate the efficiency of our proposed methods

penalty factor free discontinuous galerkin methods for 2 dim stokes problems this paper presents two new finite element methods for two dimensional stokes problems these methods are developed by relaxing the constraints of the crouzeix raviart nonconforming p 1 finite elements penalty terms are introduced to compensate for lack of continuity or the divergence free property however there is no need for choosing penalty factors and the formulations are symmetric these new methods are easy to implement and avoid solving saddle point linear systems numerical experiments are presented to illustrate the proved optimal error estimates

adaptive enriched galerkin methods for miscible displacement problems with entropy residual stabilization we present a novel approach to the simulation of miscible displacement by employing adaptive enriched galerkin finite element methods eg coupled with entropy residual stabilization for transport in particular numerical simulations of viscous fingering instabilities in heterogeneous porous media and hele shaw cells are illustrated eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions the method provides locally and globally conservative fluxes which are crucial for coupled flow and transport problems moreover eg has fewer degrees of freedom in comparison with discontinuous galerkin dg and an efficient flow solver has been derived which allows for higher order schemes dynamic adaptive mesh refinement is applied in order to reduce computational costs for large scale three dimensional applications in addition entropy residual based stabilization for high order eg transport systems prevents spurious oscillations numerical tests are presented to show the capabilities of eg applied to flow and transport enriched galerkin approximations for coupled flow and transport system entropy residual stabilization for enriched galerkin to stabilize higher order transport scheme dynamic mesh adaptivity for viscous fingering in two and three dimensions

application of a conservative generalized multiscale finite element method to flow models in this paper we propose a method for the construction of locally conservative flux fields from generalized multiscale finite element method gmsfem pressure solutions the flux values are obtained from an element based postprocessing procedure in which an independent set of 4x4 linear systems need to be solved to test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the postprocessed flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model a number of numerical examples are offered to validate the performance of the method

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

a comparative study on the weak galerkin discontinuous galerkin and mixed finite element methods this paper presents a comparative study on the newly introduced weak galerkin finite element methods wgfems with the widely accepted discontinuous galerkin finite element methods dgfems and the classical mixed finite element methods mfems for solving second order elliptic boundary value problems we examine the differences similarities and connection among these methods in scheme formulations implementation strategies accuracy and computational cost the comparison and numerical experiments demonstrate that wgfems are viable alternatives to mfems and hold some advantages over dgfems due to their properties of local conservation normal flux continuity no need for penalty factor and definiteness of discrete linear systems

enriched galerkin methods for two phase flow in porous media with capillary pressure abstract in this paper we propose an enriched galerkin eg approximation for a two phase pressure saturation system with capillary pressure in heterogeneous porous media the eg methods are locally conservative have fewer degrees of freedom compared to discontinuous galerkin dg and have an efficient pressure solver to avoid non physical oscillations an entropy viscosity stabilization method is employed for high order saturation approximations entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains the iterative and sequential implicit pressure and explicit saturation impes algorithms are treated in time numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of eg

enriched galerkin finite element approximation for elastic wave propagation in fractured media abstract finite element methods fem are becoming increasingly popular in modeling seismic wave propagation these methods provide higher order accuracy geometrical flexibility and adaptive gridding capabilities that are not easy to incorporate in traditional finite difference methods employed for generation of synthetic seismograms moreover several studies have shown that discontinuous galerkin fem dgm is a promising approach for modeling wave propagation in fractured media here we propose an enriched galerkin fem egm for elastic wave propagation egm uses the same bilinear form as dgm and the continuous galerkin finite element spaces enriched by discontinuous piecewise constants or bilinear functions egm satisfies local equilibrium while reducing the degrees of freedom in dgm formulations in this paper we consider elastic wave propagation and derive optimal a priori error estimates for dgm and egm we present numerical examples in two spatial dimensions that confirm these theoretical results in addition we provide numerical comparisons with the spectral element method in previous work dgm has been shown to be effective in modeling elastic wave propagation in fractured media using the linear slip model we now extend these results to egm with reduced computational costs over dgm

efficient numerical methods for simulating surface tension of multi component mixtures with the gradient theory of fluid interfaces abstract surface tension significantly impacts subsurface flow and transport and it is the main cause of capillary effect a major immiscible two phase flow mechanism for systems with a strong wettability preference in this paper we consider the numerical simulation of the surface tension of multi component mixtures with the gradient theory of fluid interfaces major numerical challenges include that the system of the euler lagrange equations is solved on the infinite interval and the coefficient matrix is not positive definite we construct a linear transformation to reduce the euler lagrange equations and naturally introduce a path function which is proven to be a monotonic function of the spatial coordinate variable by using the linear transformation and the path function we overcome the above difficulties and develop the efficient methods for calculating the interface and its interior compositions moreover the computation of the surface tension is also simplified the proposed methods do not need to solve the differential equation system and they are easy to be implemented in practical applications numerical examples are tested to verify the efficiency of the proposed methods

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

cluster approximation for the farey fraction spin chain we consider the farey fraction spin chain in an external field h utilising ideas from dynamical systems the free energy of the model is derived by means of an effective cluster energy approximation this approximation is valid for divergent cluster sizes and hence appropriate for the discussion of the magnetizing transition we calculate the phase boundaries and the scaling of the free energy at h 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy for h 0 our results are largely consistent with those found previously using mean field theory and renormalization group arguments