discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

stable cell centered finite volume discretization for biot equations in this paper we discuss a new discretization for the biot equations the discretization treats the coupled system of deformation and flow directly as opposed to combining discretizations for the two separate subproblems the coupled discretization has the following key properties the combination of which is novel 1 the variables for the pressure and displacement are co located and are as sparse as possible e g one displacement vector and one scalar pressure per cell center 2 with locally computable restrictions on grid types the discretization is stable with respect to the limits of incompressible fluid and small time steps 3 no artificial stabilization term has been introduced furthermore due to the finite volume structure embedded in the discretization explicit local expressions for both momentum balancing forces and mass conservative fluid fluxes are available we prove stability of the proposed method with respect to all relevant limits together with consistency this proves conve

optimal gradient estimates and asymptotic behaviour for the vlasov poisson system with small initial data the vlasov poisson system describes interacting systems of collisionless particles for solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t 3 at late times in this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay so that in n dimensions for n geqq 3 the derivative of the density of order k decays as t n k an asymptotic formula for the solution at late times is also obtained

stochastic diffusion processes on cartesian meshes diffusion of molecules is simulated stochastically by letting them jump between voxels in a cartesian mesh the jump coefficients are first derived using finite difference finite element and finite volume approximations of the laplacian on the mesh an alternative is to let the first exit time for a molecule in random walk in a voxel define the jump coefficient such coefficients have the advantage of always being non negative these four different ways of obtaining the diffusion propensities are compared theoretically and in numerical experiments a finite difference and a finite volume approximation generate the most accurate coefficients

monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions we suggest here a least change correction to available finite element fe solution this postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the fe solution although our approach is presented for fes it admits natural extension to other numerical schemes such as finite differences and finite volumes for the postprocessing a priori information about the monotonicity is assumed to be available either for the whole domain or for a subdomain where the lost monotonicity is to be recovered the obvious requirement is that such information is to be obtained without involving the exact solution e g from expected symmetries of this solution the postprocessing is based on solving a monotonic regression problem with some extra constraints one of them is a linear equality type constraint that models the conservativity requirement the other ones are box type constraints and they originate from the discrete maximum principle the resulting postprocessing problem is a large scale quadratic optimization problem it is proved that the postprocessed fe solution preserves the accuracy of the discrete fe approximation we introduce an algorithm for solving the postprocessing problem it can be viewed as a dual ascent method based on the lagrangian relaxation of the equality constraint we justify theoretically its correctness its efficiency is demonstrated by the presented results of numerical experiments

a small stencil and extremum preserving scheme for anisotropic diffusion problems on arbitrary 2d and 3d meshes in this paper a nonlinear extremum preserving scheme for the heterogeneous and anisotropic diffusion problems is proposed on general 2d and 3d meshes through a certain linearity preserving approach the so called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns this new scheme is locally conservative has only cell centered unknowns and possesses a small stencil which is five point on the structured quadrilateral meshes and seven point on the structured hexahedral meshes the stability result in h 1 norm is obtained under quite general assumptions numerical results show that our scheme is robust and extremum preserving and the optimal convergence rates are verified on general distorted meshes in case that the diffusion tensor is taken to be anisotropic at times discontinuous

on the design of general purpose flux limiters for finite element schemes i scalar convection the algebraic flux correction afc paradigm is extended to finite element discretizations with a consistent mass matrix it is shown how to render an implicit galerkin scheme positivity preserving and remove excessive artificial diffusion in regions where the solution is sufficiently smooth to this end the original discrete operators are modified in a mass conserving fashion so as to enforce the algebraic constraints to be satisfied by the numerical solution a node oriented limiting strategy is employed to control the raw antidiffusive fluxes which consist of a convective part and a contribution of the consistent mass matrix the former offsets the artificial diffusion due to upwinding of the spatial differential operator and lends itself to an upwind biased flux limiting the latter eliminates the error induced by mass lumping and calls for the use of a symmetric flux limiter the concept of a target flux and a new definition of upper lower bounds make it possible to combine the advantages of algebraic fct and tvd schemes introduced previously by the author and his coworkers unlike other high resolution schemes for unstructured meshes the new algorithm reduces to a consistent high order galerkin scheme in smooth regions and is designed to provide an optimal treatment of both stationary and time dependent problems its performance is illustrated by application to the linear advection equation for a number of 1d and 2d configurations

a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes abstract we construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes where the strict convexity restriction on the meshes is removed the main contributions of this paper include three aspects firstly the classical finite volume element c fve method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation in fact the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme secondly a new monotonicity correction is suggested based on which we obtain a monotone finite volume element m fve method the resulting m fve method still keeps the local conservation and is easy for implementation finally we analyze theoretically the truncation error and the monotonicity for this scheme besides the existence of a solution to this nonlinear scheme is proved by applying the brouwer s fixed point theorem numerical results demonstrate that the m fve method has the approximate second order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

a hybrid finite volume finite element method for bulk surface coupled problems abstract the paper develops a hybrid method for solving a system of advection diffusion equations in a bulk domain coupled to advection diffusion equations on an embedded surface a monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface in our approach the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way moreover a triangulation of the surface into regular shaped elements is not required the background mesh is an octree grid with cubic cells as an example of an application we consider the modeling of contaminant transport in fractured porous media one standard model leads to a coupled system of advection diffusion equations in a bulk matrix and along a surface fracture a series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach the method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

convergent finite element discretization of the multi fluid nonstationary incompressible magnetohydrodynamics equations we propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density viscosity and electric conductivity the discretization satisfies a discrete energy law and a discrete maximum principle for the positive density and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters a simple fixed point scheme together with an appropriate stopping criterion is proposed which decouples the computation of density velocity and magnetic field and inherits the above properties provided a mild mesh constraint holds computational studies are provided

an improvement of the two point flux approximation scheme on polygonal meshes abstract a new diffusion scheme satisfying the maximum principle on two dimensional 2d polygonal meshes is proposed being different from the existing schemes which are based on the nonlinear combination of the two one side fluxes the scheme in this paper is constructed by a modification of the conventional linear two point flux approximation tpfa scheme in which the two side flux is constructed to improve its accuracy on 2d skewed meshes the tpfa scheme is rewritten by adding a tangential part i e the tangential component of the flux multiplied with a nonlinear factor a new method for the discretization of the tangential part is presented so that both the conservation condition and the maximum principle are satisfied while the tpfa part remains unchanged moreover by virtue of the harmonic averaging points for the calculation of the tangential part the scheme has a minimal stencil similar to the tpfa scheme finally some numerical examples are presented to verify that it is second order accurate on 2d polygonal meshes for anisotropic diffusion problems

stable self similar blow up dynamics for the three dimensional relativistic gravitational vlasov poisson system r f x v dv describes the mechanical state of a stellar system subject to its own gravity smooth initial data yield unique global in time solutions from a celebrated result by pfaffelmoser 37 there exists a hierarchy of physical models which aim at taking into account further relativistic effects the simplest one is the three dimensional relativistic gravitational vlasov poisson system tf v 1 v xf ef vf 0 which we study here a striking feature as observed by glassey schaeffer 18 is that this system now admits finite blow up solutions nevertheless the existence argument is purely obstructive and provides no insight into the description of the singularity formation we exhibit in this paper a family of finite time blow up self similar solutions and prove that their blow up dynamic is stable with respect to radially symmetric perturbations our analysis applies to the four dimensional gravitational vlasov poisson system as well whose study began in 26

the finite volume scheme preserving maximum principle for diffusion equations with discontinuous coefficient abstract we construct a new scheme whose primary unknowns include both cell centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient first two linear fluxes are given on both sides of the cell edge respectively in order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient in addition to cell centered unknowns we also introduce cell edge unknowns on discontinuous line as basic unknowns second the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes for the cell edge unknowns on discontinuous line we add an equation by using the continuity of normal flux compared to the classical cell centered nonlinear finite volume scheme the introduction of cell edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient then we prove that the scheme satisfies the discrete maximum principle based on this the existence of a solution for the scheme is also obtained numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes preserves discrete maximum principle and is superior to the existing scheme in sheng and yuan 2011 preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes

convergence of finite volume schemes for semilinear convection diffusion equations the topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method dirichlet boundary conditions are considered here a discrete poincare inequality is used and estimates on the approximate solutions are proven the convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions

a positivity preserving finite element method for chemotaxis problems in 3d we present an implicit finite element method for a class of chemotaxis models in three spatial dimensions the proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density to enforce the discrete maximum principle the standard galerkin discretization is constrained using a local extremum diminishing flux limiter to demonstrate the efficiency and robustness of this approach we solve blow up problems in a 3d chemostat domain to give a flavor of more complex and realistic chemotactic applications we investigate the pattern dynamics and aggregating behavior of the bacteria escherichia coli and salmonella typhimurium the obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature

an extremum preserving finite volume scheme for convection diffusion equation on general meshes abstract we present an extremum preserving finite volume scheme for the convection diffusion equation on general meshes in this article the harmonic averaging point locating at the interface of heterogeneity are utilized to define the auxiliary unknowns the second order upwind method with a slope limiter is used for the discretization of convection flux this scheme has only cell centered unknowns and possesses a small stencil the extremum preserving property of this scheme is proved by standard assumption numerical results demonstrate that the extremum preserving scheme is an efficient method in solving the convection diffusion equation on distorted meshes

on almost global existence for nonrelativistic wave equations in 3d almost global solutions are constructed to three dimensional quadratically nonlinear wave equations the proof relies on generalized energy estimates and a new decay estimate the method applies to equations that are only classically invariant such as the nonlinear system of hyperelasticity 1996 john wiley sons inc

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

finite volume hydromechanical simulation in porous media cell centered finite volume methods are prevailing in numerical simulation of flow in porous media however due to the lack of cell centered finite volume methods for mechanics coupled flow and deformation is usually treated either by coupled finite volume finite element discretizations or within a finite element setting the former approach is unfavorable as it introduces two separate grid structures while the latter approach loses the advantages of finite volume methods for the flow equation recently we proposed a cell centered finite volume method for elasticity herein we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media we detail in particular the issue of coupling terms and show how this is naturally handled furthermore we observe how the cell centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions we support the discussion with a set of numerical examples the convergence properties of the coupled scheme are first investigated second we illustrate the practical applicability of the method both for fractured and heterogeneous media

a maximum principle preserving finite element method for scalar conservation equations this paper introduces a first order viscosity method for the explicit approximation of scalar conservation equations with lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension provided the lumped mass matrix is positive definite the method is shown to satisfy the local maximum principle under a usual cfl condition the method is independent of the cell type for instance the mesh can be a combination of tetrahedra hexahedra and prisms in three space dimensions

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

an analysis of time discretization in the finite element solution of hyperbolic problems abstract the problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined a modified equation analysis reveals that the classical second order accurate time stepping algorithms i e the lax wendroff leap frog and crank nicolson methods properly combine with piecewise linear finite elements in advection problems only for small values of the time step on the contrary as the courant number increases the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit cfl property are not achieved these fundamental numerical properties can however be recovered while still remaining in the standard galerkin finite element setting by increasing the order of accuracy of the time discretization this is accomplished by exploiting the taylor series expansion in the time increment up to the third order before performing the galerkin spatial discretization using piecewise linear interpolations as a result it appears that the proper finite element equivalents of second order finite difference schemes are implicit methods of incremental type having third and fourth order global accuracy on uniform meshes taylor galerkin methods numerical results for several linear examples are presented to illustrate the properties of the taylor galerkin schemes in one and two dimensional calculations

blending low order stabilised finite element methods a positivity preserving local projection method for the convection diffusion equation in this work we propose a nonlinear blending of two low order stabilisation mechanisms for the convection diffusion equation the motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions the approach is to blend a first order artificial diffusion method which will be active only in the vicinity of layers and extrema with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution we prove existence of discrete solutions as well as convergence under appropriate assumptions on the nonlinear terms and on the exact solution numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle while the layers are not smeared significantly

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

the newtonian limit for asymptotically flat solutions of the vlasov einstein system it is shown that there exist families of asymptotically flat solutions of the einstein equations coupled to the vlasov equation describing a collisionless gas which have a newtonian limit these are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition a central role in the proof is played by energy estimates in unweighted sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation

a robust linearization scheme for finite volume based discretizations for simulation of two phase flow in porous media in this work we consider a mathematical model for two phase flow in porous media the fluids are assumed immiscible and incompressible and the solid matrix non deformable the mathematical model for the two phase flow is written in terms of the global pressure and a complementary pressure obtained by using the kirchhoff transformation as primary unknowns for the spatial discretization finite volumes have been used more precisely the multi point flux approximation method and in time the backward euler method has been employed we present here a new linearization scheme for the nonlinear system arising after the temporal and spatial discretization we show that the scheme is linearly convergent numerical experiments are presented that sustain the theoretical results

a vertex centered and positivity preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids abstract we suggest a new positivity preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids the scheme has vertex centered edge midpoint and cell centered unknowns the vertex centered unknowns are primary and have finite volume equations associated with them the edge midpoint and cell centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns which makes the final scheme a pure vertex centered one unlike most existing positivity preserving schemes the construction of the scheme is based on a special nonlinear two point flux approximation that has a fixed stencil and does not require the convex decomposition of the co normal in order to solve efficiently the nonlinear systems resulting from the nonlinear scheme picard method and its anderson acceleration are discussed numerical experiments demonstrate the second order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the anderson acceleration is also shown on reduction of the number of nonlinear iterations moreover the proposed scheme does not have the so called numerical heat barrier issue suffered by most existing cell centered and hybrid schemes however further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

enforcing positivity with conservation for nine point scheme of nonlinear diffusion equations abstract the nine point scheme nps is a finite volume scheme for solving nonlinear diffusion equations on quadrilateral grids it s well known that the solution of the scheme on distorted grids cannot preserve positivity when all given data are nonnegative here for the nps of nonlinear diffusion problems we propose a conservative enforcing positivity algorithm cepa which is posterior correction of the discrete solution and allows the new solution preserves positivity as well as local conservation at each step of nonlinear iteration numerical results are presented to demonstrate that the accuracy of our algorithm of enforcing positivity with conservation is higher than some common procedures of positivity correction

physics based preconditioners for flow in fractured porous media discrete fracture models are an attractive alternative to upscaled models for flow in fractured media as they provide a more accurate representation of the flow characteristics a major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large scale simulations in this work two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations first the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there this assures good scalability of the preconditioners in terms of problem size and permeability contrast for fracture dominated problems numerical examples show that such geometric preconditioners are comparable or preferable when compared to state of the art algebraic multigrid preconditioners the robustness of the physics based preconditioner for less favorable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy second the preconditioners are physics preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions this facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates and without sacrificing basic physical modeling principles with the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

monotone corrections for generic cell centered finite volume approximations of anisotropic diffusion equations we present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems which provides a discrete maximum principle we point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties we then study two specific corrections proving under numerical assumptions that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme

explicit and implicit fem fct algorithms with flux linearization a new approach to the design of flux corrected transport fct algorithms for continuous linear multilinear finite element approximations of convection dominated transport problems is pursued the algebraic flux correction paradigm is revisited and a family of nonlinear high resolution schemes based on zalesak s fully multidimensional flux limiter is considered in order to reduce the cost of flux correction the raw antidiffusive fluxes are linearized about an auxiliary solution computed by a high or low order scheme by virtue of this linearization the costly computation of solution dependent correction factors is to be performed just once per time step and there is no need for iterative defect correction if the governing equation is linear a predictor corrector algorithm is proposed as an alternative to the hybridization of high and low order fluxes three fem fct schemes based on the runge kutta crank nicolson and backward euler time stepping are introduced a detailed comparative study is performed for linear convection diffusion equations

accelerated non linear finite volume method for diffusion a non linear finite volume method with monotone matrix for the diffusion equation is presented it does not extrapolate the primary variable to neumann boundaries as this was previously done in similar methods this change results in faster convergence computation time is significantly shortened further using the reduced rank extrapolation method rre and imposing an upper limit on the number of linear iterations per non linear step second order accuracy and performance improvement are demonstrated by numerical examples

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

goal oriented mesh adaptation for flux limited approximations to steady hyperbolic problems the development of adaptive numerical schemes for steady transport equations is addressed a goal oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation the error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems the galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints the localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions a nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible the developed simulation tools are applied to a linear convection problem in two space dimensions

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

a vector field method for massless relativistic transport equations and applications abstract in this article we present a vector field method for the study of solutions to massless relativistic transport equations compared to 10 we remove the lorentz boosts of the commutation vector fields and we prove a new functional inequality in order to derive pointwise decay estimates on the solutions it makes our method more suitable for the study of systems coupling a massless vlasov equation with an equation which has a different speed of propagation we also believe that our approach can be adapted more easily to the context of curved backgrounds in the second part of this paper we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless vlasov poisson system no compact support assumption is required on the data and in particular the initial decay in the velocity variable is optimal in order to compensate the slow decay rate of the particle density near the light cone we take advantage of the null structure of the equations

coprimeness preserving non integrable extension to the two dimensional discrete toda lattice equation we introduce a so called coprimeness preserving non integrable extension to the two dimensional toda lattice equation we believe that this equation is the first example of such discrete equations defined over a three dimensional lattice we prove that all the iterates of the equation are irreducible laurent polynomials of the initial data and that every pair of two iterates is co prime which indicate confined singularities of the equation by reducing the equation to two or one dimensional lattices we obtain coprimeness preserving non integrable extensions to the one dimensional toda lattice equation and the somos 4 recurrence

on the design of algebraic flux correction schemes for quadratic finite elements abstract a fully algebraic approach to the design of nonlinear high resolution schemes is revisited and extended to quadratic finite elements the matrices resulting from a standard galerkin discretization are modified so as to satisfy sufficient conditions of the discrete maximum principle for nodal values in order to provide mass conservation the perturbation terms are assembled from skew symmetric internodal fluxes which are redefined as a combination of first and second order divided differences the new approach to the construction of artificial diffusion operators is combined with a node oriented limiting strategy the resulting algorithm is applied to p 1 and p 2 approximations of stationary convection diffusion equations in 1d 2d

simulation of stochastic diffusion via first exit times in molecular biology it is of interest to simulate diffusion stochastically in the mesoscopic model we partition a biological cell into unstructured subvolumes in each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion the jump rates can be computed by discretizing the diffusion equation on that unstructured mesh if the mesh is of poor quality due to a complicated cell geometry standard discretization methods can generate negative jump coefficients which no longer allows the interpretation as the probability to jump between the subvolumes we propose a method based on the mean first exit time of a molecule from a subvolume which guarantees positive jump coefficients two approaches to exit times a global and a local one are presented and tested in simulations on meshes of different quality in two and three dimensions

a monotone finite volume method for advection diffusion equations on unstructured polygonal meshes we present a new second order accurate monotone finite volume fv method for the steady state advection diffusion equation the method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non negativity the interpolation free approximation of the diffusive flux uses the nonlinear two point stencil proposed in lipnikov 23 approximation of the advective flux is based on the second order upwind method with a specially designed minimal nonlinear correction the second order convergence rate and monotonicity are verified with numerical experiments

edge based nonlinear diffusion for finite element approximations of convection diffusion equations and its relation to algebraic flux correction schemes for the case of approximation of convection diffusion equations using piecewise affine continuous finite elements a new edge based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle the diffusion operator is shown to be lipschitz continuous and linearity preserving using these properties we provide a full stability and error analysis which in the diffusion dominated regime shows existence uniqueness and optimal convergence then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters the performance of the method is illustrated on some numerical test cases in two space dimensions

on discrete maximum principles for discontinuous galerkin methods the aim of this work is to propose a monotonicity preserving method for discontinuous galerkin dg approximations of convection diffusion problems to do so a novel definition of discrete maximum principle dmp is proposed using the discrete variational setting of the problem and we show that the fulfilment of this dmp implies that the minimum maximum depending on the sign of the forcing term is on the boundary for multidimensional problems then an artificial viscosity av technique is designed for convection dominant problems that satisfies the above mentioned dmp the noncomplete stabilized interior penalty dg method is proved to fulfil the dmp property for the one dimensional linear case when adding such av with certain parameters the benchmarks for the constant values to satisfy the dmp are calculated and tested in the numerical experiments section finally the method is applied to different test problems in one and two dimensions to show its performance

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

non negative mixed finite element formulations for a tensorial diffusion equation we consider the tensorial diffusion equation and address the discrete maximum minimum principle of mixed finite element formulations in particular we address non negative solutions which is a special case of the maximum minimum principle of mixed finite element formulations it is well known that the classical finite element formulations like the single field galerkin formulation and raviart thomas variational multiscale and galerkin least squares mixed formulations do not produce non negative solutions that is they do not satisfy the discrete maximum minimum principle on arbitrary meshes and for strongly anisotropic diffusivity coefficients in this paper we present two non negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques these proposed mixed formulations produce non negative numerical solutions on arbitrary meshes for low order i e linear bilinear and trilinear finite elements the first formulation is based on the raviart thomas spaces and the second non negative formulation is based on the variational multiscale formulation for the former formulation we comment on the effect of adding the non negative constraint on the local mass balance property of the raviart thomas formulation we perform numerical convergence analysis of the proposed optimization based non negative mixed formulations we also study the performance of the active set strategy for solving the resulting constrained optimization problems the overall performance of the proposed formulation is illustrated on three canonical test problems

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes abstract in the present work we deal with the convergence of cell centered nonlinear finite volume schemes for anisotropic and heterogeneous diffusion operators a general framework for the convergence study of finite volume methods is provided and used to establish the convergence of the new methods thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with linear finite volume schemes is provided

flux corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements abstract the flux corrected transport fct algorithm is applied to the unsteady and steady state particle transport equation the proposed fct method employs the following 1 a low order positivity preserving scheme based on the application of m matrix properties 2 a high order scheme based on the entropy viscosity method introduced by guermond 1 and 3 local discrete solution bounds derived from the integral transport equation the resulting scheme is second order accurate in space enforces an entropy inequality mitigates the formation of spurious oscillations and guarantees the absence of negativities space discretization is achieved using continuous finite elements time discretizations for unsteady problems include theta schemes such as explicit and implicit euler and strong stability preserving runge kutta ssprk methods the developed fct scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady state and implicit time discretizations

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems the maximum principle is one of the most important properties of solutions of partial differential equations its numerical analog the discrete maximum principle dmp is one of the most difficult properties to achieve in numerical methods especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic violation of the dmp may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one in this work we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference mfd method on two and three dimensional meshes these conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element efficient solutions are devised for meshes consisting of simplexes parallelograms and parallelepipeds and orthogonal locally refined elements as those used in the amr methodology on simplicial meshes it turns out that the mfd method coincides with the mixed hybrid finite element methods based on the low order raviart thomas vector space thus in this case we recover the well established conventional angle conditions of such approximations instead in the other cases a suitable design of the mfd method allows us to formulate a monotone discretization for which the existence of a dmp can be theoretically proved moreover on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the multi point flux approximation mpfa methods numerical experiments confirm the effectiveness of the considered monotonicity conditions

nonlinear forms of coprimeness preserving extensions to the somos 4 recurrence and the two dimensional toda lattice equation investigation into their extended laurent properties coprimeness property was introduced to study the singularity structure of discrete dynamical systems in this paper we shall extend the coprimeness property and the laurent property to further investigate discrete equations with complicated pattern of singularities as examples we study extensions to the somos 4 recurrence and the 2d discrete toda equation by considering their non autonomous polynomial forms we prove that their tau function analogues possess the extended laurent property with respect to their initial variables and some extra factors related to the non autonomous terms using this laurent property we prove that these equations satisfy the extended coprimeness property this coprimeness property reflects the singularities that trivially arise from the equations

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

a constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems nonlinear constrained finite element approximations to anisotropic diffusion problems are considered starting with a standard linear or bilinear galerkin discretization the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle dmp an algebraic splitting is employed to separate the contributions of negative and positive off diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes respectively in order to prevent the formation of spurious undershoots and overshoots a symmetric slope limiter is designed for the antidiffusive part the corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes the recovery of nodal gradients is performed by means of a lumped mass l 2 projection the proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix symmetry zero row and column sums a positivity preserving defect correction scheme is devised for the nonlinear algebraic system to be solved numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions

a new nonlinear finite volume scheme preserving positivity for diffusion equations in this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations the main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided two nonnegative parameters are introduced to define a new nonlinear two point flux in which one point is the cell center and the other is the midpoint of cell edge the final flux on the edge is obtained by the continuity of normal flux numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

two dimensional exactly and completely integrable dynamical systems monopoles instantons dual models relativistic strings lund regge model generalized toda lattice etc an investigation of two dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary lie algebra mathfrak g whose grading is consistent with an arbitrary integral embedding of 3d subalgebra in mathfrak g has been carried out we have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a goursat problem a method for the construction of a wide class of infinite dimensional lie algebras of finite growth has been proposed

space time least squares finite element method for convection reaction system with transformed variables we present a method to solve a convection reaction system based on a least squares finite element method lsfem for steady state computations issues related to recirculation flow are stated and demonstrated with a simple example the method can compute concentration profiles in open flow even when the generation term is small this is the case for estimating hemolysis in blood time dependent flows are computed with the space time lsfem discretization we observe that the computed hemoglobin concentration can become negative in certain regions of the flow it is a physically unacceptable result to prevent this we propose a quadratic transformation of variables the transformed governing equation can be solved in a straightforward way by lsfem with no sign of unphysical behavior the effect of localized high shear on blood damage is shown in a circular couette flow with blade configuration and a physiological condition is tested in an arterial graft flow

fast computation of multiphase flow in porous media by implicit discontinuous galerkin schemes with optimal ordering of elements we present a family of implicit discontinuous galerkin schemes for purely advective multiphase flow in porous media in the absence of gravity and capillary forces to advance the solution one time step one must solve a discrete system of nonlinear equations by reordering the grid cells the nonlinear system can be shown to have a lower triangular block structure where each block corresponds to the degrees of freedom in a single or a small number of cells to reorder the system we view the grid cells and the fluxes over cell interfaces as vertices and edges in a directed graph and use a standard topological sorting algorithm then the global system can be computed by processing the blocks sequentially using a standard newton raphson algorithm for the degrees of freedom in each block decoupling the system offers greater control over the nonlinear solution procedure and reduces the computational costs memory requirements and complexity of the scheme significantly in particular the first order version of the method may be at least as efficient as modern streamline methods when accuracy requirements or the dynamics of the flow allow for large implicit time steps

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100