optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

pinning of a rough interface by an external potential abstract an analysis is given of the behavior of an interface between two phases in the presence of an external pinning potential in the solid on solid limit of the two dimensional ising model it is found that the potential turns a rough interface into a smooth one except in the case of a boundary potential where a minimum potential strength is required the connection with the roughening transition found by abraham is discussed the interface width is calculated as a function of the potential parameters in the limit of a weak pining potential

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

diffusion on a hypercubic lattice with pinning potential exact results for the error catastrophe problem in biological evolution in the theoretical biology framework one fundamental problem is the so called error catastrophe in darwinian evolution models we re examine eigen s fundamental equations by mapping them into a polymer depinning transition problem in a genotype space represented by a unitary hypercubic lattice the exact solution of the model shows that error catastrophe arises as a direct consequence of the equations involved and confirms some previous qualitative results

two a posteriori error estimates for one dimensional scalar conservation laws in this paper we propose a posteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws we first consider the schemes for which a consistent in cell entropy inequality can be derived then we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities as an illustration we show several numerical tests on the burgers equation and we propose an adaptive algorithm for the selection of the mesh

entropy formulation for parabolic degenerate equations with general dirichlet boundary conditions and application to the convergence of fv methods this paper is devoted to the analysis and the approximation of parabolic hyperbolic degenerate problems defined on bounded domains with nonhomogeneous boundary conditions it consists of two parts the first part is devoted to the definition of an original notion of entropy solutions to the continuous problem which can be adapted to define a notion of measure valued solutions or entropy process solutions the uniqueness of such solutions is established in the second part the convergence of the finite volume method is proved this result relies on weak estimates and on the theorem of uniqueness of the first part it also entails the existence of a solution to the continuous problem

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

error estimates for finite element methods for scalar conservation laws in this paper new a posteriors error estimates for the shock capturing streamline diffusion scsd method and the shock capturing discontinuous galerkin scdg method for scalar conservation laws are obtained these estimates are then used to prove that the scsd method and the scdg method converge to the entropy solution with a rate of at least h 1 8 and h 1 8 respectively in the l infty l 1 norm the triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k the result is independent of the dimension of the space

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh we study here the discretisation of the nonlinear hyperbolic equationu t div vf u 0 in 3 with given initial conditionu 0 u 0 in 2 wherev is a function from 2 to 2 such that divv 0 andf is a given nondecreasing function from to an explicit euler scheme is used for the time discretisation of the equation and a triangular mesh for the spatial discretisation under a usual stability condition we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation

probabilistic analysis of the upwind scheme for transport equations we provide a probabilistic analysis of the upwind scheme for d dimensional transport equations we associate a markov chain with the numerical scheme and then obtain a backward representation formula of kolmogorov type for the numerical solution we then understand that the error induced by the scheme is governed by the fluctuations of the markov chain around the characteristics of the flow we show in various situations that the fluctuations are of diffusive type as a by product we recover recent results due to merlet and vovelle numer math 106 129 155 2007 and merlet siam j numer anal 46 1 124 150 2007 we prove that the scheme is of order 1 2 in l infty 0 t l 1 mathbb r d for an integrable initial datum of bounded variation and of order 1 2 e for all e 0 in l infty 0 t times mathbb r d for an initial datum of lipschitz regularity our analysis provides a new interpretation of the numerical diffusion phenomenon

a supraconvergent scheme for nonlinear hyperbolic systems abstract supraconvergent difference schemes are methods for which the local truncation error degrades when applied to an irregular mesh yet the observed error in the solution is as good as if the mesh were uniform for a version of the lax wendroff scheme requiring the computation of the jacobian we show that superconvergence holds on an arbitrary grid one dimensional for the standard version we present a computation which shows that second order accuracy is not always maintained but the convergence is better than first order

modular equations and the genus zero property of moonshine functions and let n be the normalizer of 0 n in sl 2 r we shall call a discrete subgroup of sl 2 r a congruence group if it contains n for some n necessarily the index of n in is finite and acts on the extended upper half plane h h q i by fractional linear transformations the quotient h has the structure of a compact riemann surface and will be denoted by x conway and norton cn conjectured relationships between certain congruence groups and m known as the moonshine conjectures these have been proved by borcherds

l 1 framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations we develop a general l 1 framework for deriving continuous de pendence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the chen perthame kinetic approach 9 we ap ply our l 1 framework to establish an explicit estimate for continuous depen dence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method without imposition of bounded variation of the approximate solutions finally as an example of a direct application of this framework to numerical methods we focus on a linear convection diusi on model equation and derive an l 1 error estimate for an upwind central finite dierence scheme

well posedness and long time behavior of lipschitz solutions to generalized extremal surface equations we show that in one space dimension lipschitz solutions of generalized extremal surface equations are equivalent to entropy solutions in l r of a non strictly hyperbolic system of conservation laws we obtain an explicit representation formula and the uniqueness of the entropy solutions to the cauchy problem of the system by using this formula we also obtain the convergence and convergence rates as t of the entropy solutions to explicit traveling waves in the l1 r norm moreover when initial data are constants outside of a finite space interval the entropy solutions become the explicit traveling waves after a finite time finally we prove l1 stabilities of the entropy solutions

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

a supraconvergent scheme for the korteweg de vries equation a finite difference method for the integration of the korteweg de vries equation on irregular grids is analyzed under periodic boundary conditions the method is shown to be supraconvergent in the sense that though being inconsistent it is second order convergent however such a convergence only takes place on grids with an odd number of points per period when a grid with an even number of points is used the inconsistency of the method leads to divergence numerical results backing the analysis are presented

invariant domains and second order continuous finite element approximation for scalar conservation equations we propose a technique for approximating nonlinear scalar conservation equations that uses continuous finite elements and is formally at least second order accurate in space and maximum principle preserving the method is explicit in time uses unstructured continuous finite elements in space and works in any space dimension the stability and accuracy of the method are achieved by adapting the artificial viscosity

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

a geometric approach to error estimates for conservation laws posed on a spacetime abstract we consider a hyperbolic conservation law posed on an n 1 dimensional spacetime whose flux is a field of differential forms of degree n generalizing the classical kuznetsov s method we derive an l 1 error estimate which applies to a large class of approximate solutions in particular we apply our main theorem and deal with two entropy solutions associated with distinct flux fields as well as with an entropy solution and an approximate solution our framework encompasses for instance equations posed on a globally hyperbolic lorentzian manifold

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

a note on entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids recently cockburn coquel and lefloch proved convergence and error estimates for higher order finite volume schemes their result is based on entropy inequalities which are derived under restrictive assumptions on either the flux function or the numerical fluxes moreover they assume that the spatial grid satisfies a standard regularity assumption using instead entropy inequalities derived in previous work by kroner noelle and rokyta and a weaker condition on the grid we can generalize and simplify the error estimates

error control for a class of runge kutta discontinuous galerkin methods for nonlinear conservation laws we propose an a posteriori error estimate for the runge kutta discontinuous galerkin method rk dg of arbitrary order in arbitrary space dimensions for stabilization of the scheme a general framework of projections is introduced finally it is demonstrated numerically how the a posteriori error estimate is used to design both an efficient grid adaption and gradient limiting strategy numerical experiments show the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids

an implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes we are interested here in the numerical approximation of a family of probability measures solution of the chapman kolmogorov equation associated to some non diffusion markov process with uncountable state space such an equation contains a transport term and another term which implies redistribution of the probability mass on the whole space an implicit finite volume scheme is proposed which is intermediate between an upstream weighting scheme and a modified lax friedrichs one due to the seemingly unusual probability framework a new weak bounded variation inequality had to be developed in order to prove the convergence of the discretised transport term such an inequality may be used in other contexts such as for the study of finite volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in l 1 also due to the redistribution term the tightness of the family of approximate probability measures had to be proven numerical examples are provided showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

on nonlinear stochastic balance laws we are concerned with multidimensional stochastic balance laws we identify a class of nonlinear balance laws for which uniform spatial bv bound for vanishing viscosity approximations can be achieved moreover we establish temporal equicontinuity in l1 of the approximations uniformly in the viscosity coefficient using these estimates we supply a multidimensional existence theory of stochastic entropy solutions in addition we establish an error estimate for the stochastic viscosity method as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions various further generalizations of the results are discussed

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

level and index in the modular group it is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup this allows a number of existence theorems about non congruence subgroups the level of a subgroup of the modular group can be defined in terms of the action on q we define a similar action to get information on congruence subgroups in fact we get a more powerful result but this appears to be the most useful version

continuous dependence on the nonlinearities of solutions of degenerate parabolic equations explicit estimates for the continuous dependence inl 0 t l1 rd of solutions of the equation lv v v in 0 rdwith initial conditionv 0 h with respect to the nonlinear continuously differential functions and are established

stability of solutions of quasilinear parabolic equations abstract we bound the difference between solutions u and v of u t a u div x f h and v t b v div x g k with initial data and respectively by u t v t l p e a e t l r n 2 p b t a b x f x g f u g u h k p e p here all functions a f and h are smooth and bounded and may depend on u x r n and t the functions a and h may in addition depend on u identical assumptions hold for the functions that determine the solutions v furthermore e r n is assumed to be a bounded set and p and p are fractions that depend on n and p the diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth

an explicit a priori estimate for a finite volume approximation of linear advection on non cartesian grids we propose an elementary proof of strong convergence for a finite volume approximation of nonstationary linear advection on arbitrary grids in two space dimensions this proof is elementary in the sense that the basic a priori estimate uses only some discrete integrations by parts and the cauchy schwarz inequality numerical results show that the estimate is probably nonoptimal

validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws nonlinear geometric optics with various frequencies for entropy so lutions only in l 1 of multidimensional scalar conservation laws is analyzed a new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling compactness homogenization and l 1 stability new multidimensional features are recognized especially including nonlinear propagations of oscillations with high frequencies the validity of nonlinear geometric optics for entropy solutions in l 1 of multidimensional scalar con servation laws is justified

the initial boundary value problem for general non local scalar conservation laws in one space dimension we prove global well posedness results for weak entropy solutions of bounded variation bv of scalar conservation laws with non local flux on bounded domains under suitable regularity assumptions on the flux function in particular existence is obtained by proving the convergence of an adapted lax friedrichs algorithm lipschitz continuos dependence from initial and boundary data is derived applying kru hkov s doubling of variable technique

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

error estimate for godunov approximation of locally constrained conservation laws we consider a model of traffic flow with unilateral constraint on the flux introduced by colombo and goatin j differential equations 234 2007 pp 654 675 for which the convergence of numerical approximation using monotone finite volume schemes has been performed by andreianov goatin and seguin numer math 115 2010 pp 609 645 we derive for this problem a new rm bv estimate and make use of it to provide an error estimate for the godunov approximation of the problem of order h 1 3 that is improved into the optimal order h 1 2 under a reasonable assumption numerical experiments are then provided to illustrate the optimality of the result

finite volume relaxation schemes for multidimensional conservation laws we consider finite volume relaxation schemes for multidimensional scalar conservation laws these schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1 4 in l 0 t l loc 1 r d

on convergence of minmod type schemes a class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered this class of methods contains the classical lax friedrichs and the second order nessyahu tadmor schemes in the case of linear flux new l2 stability results and error estimates for the methods are proved numerical experiments confirm that these methods are one sided l2 stable for convex flux instead of the usual lip stability

groups commensurable with the modular group abstract let a and b be maximal subgroups of psl 2 r that commensurable with psl 2 z s and t are commensurable with each other if s t is of finite index in both s and t in this article we determine the index a a b and the level of a b appendix a

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

convergence of a staggered lax friedrichs scheme for nonlinear conservation laws on unstructured two dimensional grids based on nessyahu and tadmor s nonoscillatory central difference schemes for one dimensional hyperbolic conservation laws 16 for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented the experiments show the wide applicability of these multidimensional schemes the theoretical arguments which support this are some maximum principles and a convergence proof in the scalar linear case a general proof of convergence as obtained for the original one dimensional nt schemes does not exist for any of the extensions to multidimensional nonlinear problems for the finite volume extension on two dimensional unstructured grids introduced by arminjon and viallon 3 4 we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law

a finite difference formula for the discretization of d 3 dx 3 on nonuniform grids we analyze the use of a five point difference formula for the dis cretization of the third derivative operator on nonuniform grids the formula was derived so as to coincide with the standard five point formula on regular grids and to lead to skew symmetric schemes it is shown that under periodic boundary conditions the formula is supraconvergent in the sense that in spite of being inconsistent it gives rise to schemes with second order convergence however such a supraconvergence only takes place when the number of points in the grid is odd for grids with an even number of points the inconsistency of the formula results in lack of convergence both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments

convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws we prove the convergence of flux vector splitting schemes associated to hyperbolic systems of conservation laws with a single compatible entropy c we prove estimate on the l2 norm of the gradient of the numerical approximation in the inverse square root of the space increment x this estimate is related to the notion of strictly c dissipativity on f f and id f f where f f is the flux decomposition the second tool of the proof is a kinetic formulation of the flux splitting scheme with three velocities then we get a control for all entropies and apply the compensated compactness theory

stochastic nonlocal conservation laws on whole space abstract after introducing an entropy formulation for stochastic nonlocal conservation laws via an integral formula we develop a uniqueness theory by using a doubling variables technique and an existence theory by using a vanishing viscosity method we further consider the continuous dependence of the entropy solution on initial data

weakly nonoscillatory schemes for scalar conservation laws a new class of godunov type numerical methods called here weakly nonoscillatory or wno for solving nonlinear scalar conservation laws in one space dimension is introduced this new class generalizes the classical nonoscillatory schemes in particular it contains modified versions of min mod and uno under certain conditions convergence and error estimates for wno methods are proved

continuous dependence estimate for conservation laws with levy noise abstract we are concerned with multidimensional stochastic balance laws driven by levy processes using bounded variation bv estimates for vanishing viscosity approximations we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that levy noise only depends on the solution this result is used to show the error estimate for the stochastic vanishing viscosity method in addition we establish fractional bv estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable

a priori error estimates for numerical methods for scalar conservation laws part i the general approach we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original kuznetsov approximation theory as a first application of this general technique we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution thus we obtain optimal error estimates for the engquist osher scheme without using the fact i that the solution is uniformly bounded ii that the scheme is total variation diminishing and iii that the discrete semigroup associated with the scheme has the l 1 contraction property which guarantees an upper bound for the modulus of continuity in time of the approximate solution

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

how the quasispecies evolution depends on the topology of the genome space abstract we compared the properties of the error threshold transition in quasispecies evolution for three different topologies of the genome space they are a hypercube b rugged landscape modelled by an ultrametric space and c holey landscape modelled by bethe lattice in all studied topologies the phase transition exists we calculated the critical exponents in all the cases for the critical exponent corresponding to appropriately defined susceptibility we found super universal value

modified equation for adaptive monotone difference schemes and its convergent analysis a modified parabolic equation for adaptive monotone difference schemes based on equal arclength mesh applied to the linear convection equation is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous piecewise smooth solution of the linear convection equation at order one rate in the l 1 norm it is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half order rate in the l 1 norm therefore the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy numerical experiments also confirm the theoretical conclusions