some remarks on the blow up rate for the 3d magnetic zakharov system we continue our study 8 on the cauchy problem for the three dimensional magnetic zakharov system zsm iet e ne i e b e 0 nt v vt n e 2 b i e e b 0 with initial data zsm 1 e 0 x e0 x n 0 x n0 x v 0 x v0 x let e n v c 0 t h1 r3 l2 r3 l2 r3 be a blow up solution to the cauchy problem zsm zsm 1 and let t be its blow up time then e n satisfies the space time integral estimate e 1 0t r3 n t x qdx 1q r3 e t x 2qdx 1q dt where 1e e 0 14 and q 32 1 e 32 2 the estimate e 1 implies that for a 1 supt 0 t t t ae n t lq r3 e t l2q r3 2 in particular if e n v is a radially symmetric solution to zsm zsm 1 then supt 0 t t t ae n t lq d e t l2q d 2 where a 0 13 and d x r3 x 1

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

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

combinatorial characterization of sub hyperbolic rational maps in 1980 s thurston established a combinatorial characterization for post critically finite rational maps among post critically finite branched coverings of the two sphere to itself a completed proof was written by douady and hubbard in their paper a douady j h hubbard a proof of thurston s topological characterization of rational functions acta math 171 1993 263 297 this criterion was then extended by cui jiang and sullivan to sub hyperbolic rational maps among sub hyperbolic semi rational branched coverings of the two sphere to itself the goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub hyperbolic rational maps by adapting some arguments in the proof in douady and hubbard s paper

limit behavior of saturated approximations of nonlinear schrodinger equation we consider the solutionu t of the saturated nonlinear schrodinger equation i partial u partial t delta u left u right 4 n u varepsilon left u right q 1 uandu 0 varphi 1 where n geqslant 2 varepsilon 0 1 4 n 0 such that mathop lim limits t to t left u t right h 1 infty whereu t is solution of i partial u partial t delta u left u right 4 n uandu 0 varphi 1 for 0 fixed u t is defined for all time we are interested in the limit behavior as 0 ofu t fort t in the case where there is no loss of mass inu at infinity in a sense to be made precise we describe the behavior ofu as goes to zero and we derive an existence result for a solution of 1 after the blow up timet in a certain sense nonlinear schrodinger equation with supercritical exponents are also considered

blow up solutions of inhomogeneous nonlinear schrodinger equations on torus abstract in this paper we study blow up solutions to the cauchy problem of the inhomogeneous nonlinear schrodinger equation t u i f x u f x u k x u 2 u on t 2 we present the l 2 concentration property for general initial data and investigate the l 2 minimality

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

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

supersonic limit for the zakharov rubenchik system abstract we study the asymptotic behavior of the solutions of zakharov rubenchik system when appropriate parameters go to zero namely we state weak and strong convergence results of these solutions to solutions of zakharov system the proof of the weak limit is a classical argument in the theory of compactness whose main ingredient is the aubin lions theorem and the ascoli theorem strong limits are conveniently treated by decomposing the nonlinearities and using the strichartz estimates associated with the group of the schrodinger equation and the wave group

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

on uniqueness and continuation properties after blow up time of self similar solutions of nonlinear schrodinger equation with critical exponent and critical mass we consider the nonlinear schrodinger equation with critical power where u 0 t n c and o h1 o x o l2 with this nonlinear term the equation 1 1 has a conformal invariance thus one yields explicit self similar solutions which have the following property they have minimum l2 norm among blow up solutions in this paper we first focus on uniqueness properties of these self similar solutions in a certain class we then look at the possible continuations of these solutions after the blow up time

the electrostatic limit for the 3d zakharov system abstract we consider the vectorial zakharov system describing langmuir waves in a weakly magnetized plasma in its original derivation zakharov 1972 the evolution for the electric field envelope is governed by a schrodinger type equation with a singular parameter which is usually large in physical applications motivated by this we study the rigorous limit as this parameter goes to infinity by using some strichartz type estimates to control separately the fast and slow dynamics in the problem we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vector fields

optimal biquadratic finite volume element methods on quadrilateral meshes in this paper an optimal biquadratic finite volume element scheme is established and analyzed for elliptic equations on quadrilateral meshes it is proved that the new scheme has optimal o h 2 convergence rate in h 1 norm and o h 3 convergence rate in l 2 norm with the assumption that each element is an h 2 parallelogram

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

a characterization of hyperbolic rational maps we give a topological characterization of rational maps with disconnected julia sets our results extend thurston s characterization of postcritically finite rational maps in place of iteration on teichmuller space we use quasiconformal surgery and thurston s original result

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

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

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

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

some new discretization and adaptation and multigrid methods for 2 d 3 t diffusion equations in the simulation of laser driven implosion of a fuel capsule in inertial confinement fusion experiments a system of two dimensional diffusion equations coupled with electron iron and photon temperature are widely used to approximately describe the process of energy across multiple materials and the exchange of energy among electrons irons and photons the numerical solution of such equations is always challenging because of its strong nonlinear phenomena and strong discontinuous interfaces in this article we design a symmetric finite volume method develop the corresponding preconditioning technique and propose a mesh adaptation algorithm based on hessian matrix and a two grid method using these new methods we demonstrate that the energy conservation error and computation efficiency of the integrated algorithm are much better than classical method

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

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

l2 concentration of blow up solutions for the nonlinear schrodinger equation with critical power nonlinearity abstract we consider the solution of the nonlinear schrodinger equation i u t u f u and u 0 where u is defined on 0 t rn and f is a nonlinear complex valued function we consider the case of the critical power that is f z z 4 n z as z in a sense which will be made precise in addition we suppose that u t blows up in the norm h1 at time t we prove that u t has no limit in l2 as t t in the particular case where has a spherical symmetry we further show a phenomenon of l2 concentration at the origin

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 decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

on the lowest eigenvalue of the laplacian for the intersection of two domains ifa andb are two bounded domains in n and a b are the lowest eigenvalues of with dirichlet boundary conditions then there is some translate b x ofb such that a b x a b a similar inequality holds for lambda p a inf parallel nabla f parallel p p parallel f parallel p p f in w 0 1 p a there are two corollaries of this theorem i a lower bound for sup x volume a b x in terms of a whenb is a ball ii a compactness lemma for certain sequences inw 1 p n

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

dynamics of stochastic zakharov equations the asymptotic behavior of zakharov equations driven by random force is studied the force which is smooth enough and homogeneous in space and white noise in time acts on both equations of the zakharov system independently by some a priori estimates we prove the existence of a solution in energy spaces e1 and e2 via the galerkin approximation this solution is defined on the given probability space rather than a martingale solution then a global random attractor is constructed in energy space e2 equipped with weak topology further the existence of a stationary measure is proved in energy space e2 with usual topology

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

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

virial type blow up solutions for the zakharov system with magnetic field in a cold plasma abstract in this paper we study blow up solutions of virial type to the zakharov system with magnetic field in a cold plasma in r n n 2 3 after obtaining some a priori estimates on those terms generated by the magnetic field we obtain a virial type blow up result to the system under consideration the result suggests that the magnetic field in a cold plasma doesn t affect the virial type blow up character of the zakharov system

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

optimal lower bound estimates for the blow up rate for the zakharov system in a nonhomogeneous medium abstract in this paper we obtain lower bound estimates for the blow up rate of finite time blow up solutions to the cauchy problem for the zakharov system in a nonhomogeneous medium in two space dimensions by introducing suitable scale transformations of space and time and the use of compactness arguments we derive an optimal lower bound estimate in the energy space h 2 r 2 l 2 r 2 h 1 r 2 for the blow up rate for t near the finite blow up time t also we give an application to the virial identity for the zakharov system under study

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

blow up and nonlinear instability for the magnetic zakharov system abstract this study deals with the generalized zakharov system with magnetic field first of all we construct a kind of blow up solutions and establish the existence of blow up solutions to the system through considering an elliptic system next we show the nonlinear instability for a kind of periodic solutions in addition we consider the concentration properties of blow up solutions for the system under study at the end of this paper we establish the global existence of weak solutions to the cauchy problem of the system under consideration

error estimates for a finite volume element method for elliptic pdes in nonconvex polygonal domains we consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain based on sharp shift estimates we derive error estimations in h1 l2 and l infty norms taking into consideration the regularity of the data numerical experiments and counterexamples illustrate the theoretical results

blow up results of viriel type for zakharov equations we consider the zakharov equations in n for n 2 n 3 we first establish a viriel identity for such equations and then prove a blow up result for solutions with a negative energy

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

a perturbative analysis of the time envelope approximation in strong langmuir turbulence abstract we investigate a nonlinear set of coupled wave equations describing the inertial regime of the strong langmuir turbulence 1 2 2 e t 2 2i e t e ne 1 c 2 2 n t 2 n e 2 which differs from the usual zakharov equations by the inclusion in the first equation for e of a second time derivative multiplied by the parameter 1 2 that vanishes under the so called time envelope approximation 2 a from these perturbed zakharov equations it is shown that the latter limit is not compatible with a strongly dominant ion inertia corresponding to the formal case c 2 a 0 in the opposite case i e as c2 remains of order unity the local in time cauchy problem attached to the above equations is solved and the limit 2 a is detailed for a fixed value of c2 under some specific initial data the solution e is proved to blow up at least in an infinite time provided that lies below a threshold value when this condition is not fulfilled the global existence of the solution set e n is finally restored in a one dimensional space

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

geometrically finite and semi rational branched coverings of the two sphere in 1982 thurston gave a necessary and sufficient condition for a critically finite branched covering of the two sphere to itself to be combinatorially equivalent to a rational map we discuss extending this result to geometrically finite rational maps we give an example to show that thurston s original condition is not sufficient this example is topologically pathological near accumulation points of the postcritical set we give two conditions forbidding such pathology show that they are equivalent and in a sequel to the present paper will show that thurston s condition together with this tameness is both necessary and sufficient to characterize geometrically finite

equations of langmuir turbulence and nonlinear schrodinger equation smoothness and approximation we consider the following family of systems parametrized by e 0 which describe langmuir s turbulence in the case where initial data are sufficiently small we study the asymptotic behaviour of the solutions ee ne when e goes to zero for d 1 2 3 namely we state convergence results of ee ne to the couple e e 2 where e is the solution of the nonlinear schrodinger equation moreover in three dimensions we show that the smooth solution ee ne is defined on 0 tmax e with tmax e going to infinity as e goes to 0

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

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

a moving mesh finite difference method for equilibrium radiation diffusion equations an efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions the method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations the mesh adaptation is controlled through a hessian based monitor function and the so called equidistribution and alignment principles several challenging issues in the numerical solution are addressed particularly the radiation diffusion coefficient depends on the energy density highly nonlinearly this nonlinearity is treated using a predictor corrector and lagged diffusion strategy moreover the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations numerical examples with multi material multiple spot concentration situations are presented numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy it is also shown that a two level mesh movement strategy can significantly improve the efficiency of the computation

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes we construct a monotone finite volume scheme on distorted meshes for multimaterial nonequilibrium radiation diffusion problems which are described by the coupled radiation diffusion and material conduction equations moreover we prove theoretically that the scheme is monotone numerical results are presented to show that our scheme preserves positivity of solution on various distorted meshes and the contours of numerical solution obtained by our scheme on distorted meshes accord with that on rectangular meshes moreover numerical tests indicate that our monotone scheme is more computationally efficient than the nine point scheme these results show that our nonlinear monotone finite volume scheme is a practical and attractive method for solving nonlinear diffusion equations on distorted meshes

concentration properties of blow up solutions and instability results for zakharov equation in dimension two part ii we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions