adaptive numerical simulation of traffic flow density in this paper we describe a one dimensional adaptive moving mesh method and apply it to the traffic flow model introduced in 2006 by sopasakis and katsoulakis and its modified model proposed in 2009 by kurganov and polizzi these models can be written as a scalar conservation law with a global flux the proposed scheme is an extension of the moving non oscillatory central scheme which belongs to a class of moving finite volume methods we also modify the model given by kurganov and polizzi to account for the driver s reaction i e delayed response finally the moving finite volume method is extended accordingly

adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

the approximation for the boundary optimal control problem of burgers fisher equation with constraints this paper deals with the numerical approximation with meshless method for the boundary optimal control problem with some control and state constraints governed by the burgers fisher equation which is a nonlinear evolution equation and is the prototype model for the reaction convection and diffusion phenomena arising in many spatial temporal processes by making use of the element free galerkin efg method the original optimal control problem is discretized spatially to a semi discrete optimal control problem governed by a system of nonlinear ordinary differential equations then by using the control parameterization method the original problem can be reduced to an optimal parameter selection problem governed by a lumped parameter system which can be solved as a nonlinear optimization problems by using the sequential quadratic programming sqp algorithm the numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method

variational barrier method of adaptive grid generation in hyperbolic problems of gas dynamics application of the harmonic mapping using a variational approach to generate moving adaptive grids in the hyperbolic problems of gas dynamics is considered using the example of a three point model of adaptation the possibility to generate an unfolded mesh with strong grid lines condensing in the vicinity of discontinuities of the control monitor function is demonstrated the algorithm of redistributing the boundary nodes is suggested and consists of using constrained minimization of the discrete harmonic functional when constraints define the boundary of the domain in real computations due to mesh adaptation it is possible to reduce the errors caused by shock waves smearing over the cells by many factors of ten modeling of the two dimensional 2 d supersonic gas flow in the channel has shown that the same accuracy on the adaptive grid with the same structure as the quasi uniform mesh can be acheived while requiring less cpu memory by a factor of 25 and less running time by a factor of 50 to 60 computational tests of the steady transonic and supersonic flow over an airfoil demonstrate the ability of the method to control mesh sizes across shocks

adaptive moving mesh computations for reaction diffusion systems in this paper we describe an adaptive moving mesh technique and its application to reaction diffusion models from chemistry the method is based on a coordinate transformation between physical and computational coordinates the transformation can be viewed as a solution of adaptive mesh partial differential equations pdes which is derived from the minimization of a mesh energy integral for an efficient implementation we have used an approach in which the numerical solution of the physical pdes and the adaptive pdes are decoupled further to avoid solving large nonlinear systems a second order implicit explicit time integration method in combination with the iterative method bi cgstab is applied in the method of lines procedure numerical examples are given in one and two space dimensions

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

numerical simulation of the generalized huxley equation by he s variational iteration method abstract by means of variational iteration method the solution of generalized huxley equation are obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

differentiability of solutions with respect to parameters in neutral differential equations with state dependent delays in this paper we consider a class of nonlinear neutral differential equations with state dependent delays in both the neutral and the retarded terms we study well posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the c norm

cover time for random walks on arbitrary complex networks we present an analytical method for computing the mean cover time of a discrete time random walk process on arbitrary complex networks the cover time is defined as the time a random walker requires to visit every node in the network at least once this quantity is particularly important for random search processes and target localization on network structures based on the global mean first passage time of target nodes we derive a method for computing the cumulative distribution function of the cover time based on first passage time statistics our method is viable for networks on which random walks equilibrate quickly we show that it can be applied successfully to various model and real world networks our results reveal an intimate link between first passage and cover time statistics and offer a computationally efficient way for estimating cover times in network related applications

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

moving mesh discontinuous galerkin methods for pdes with traveling waves abstract in this paper a moving mesh discontinuous galerkin dg method is developed for nonlinear partial differential equations pdes with traveling wave solutions the moving mesh strategy for one dimensional pdes is based on the rezoning approach which decouples the solution of the pde from the moving mesh equation we show that the dg moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal arc length and curvature monitor functions numerical results reveal the efficiency of the proposed moving mesh dg method for solving burgers burgers fisher and schlogl nagumo equations

fourier pade approximations and filtering for spectral simulations of an incompressible boussinesq convection problem in this paper we present rational approximations based on fourier series representation for periodic piecewise analytic functions the well known gibbs phenomenon hampers the convergence of the standard fourier method here for a given set of the fourier coefficients from a periodic piecewise analytic function we define fourier pade galerkin and fourier pade collocation methods by expressing the coefficients for the rational approximations using the fourier data we show that those methods converge exponentially in the smooth region and successfully reduce the gibbs oscillations as the degrees of the denominators and the numerators of the pade approximants increase numerical results are demonstrated in several examples the collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one dimensional inviscid burgers equation and the two dimensional incompressible inviscid boussinesq convection flow

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

effective distances for epidemics spreading on complex networks we show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network based measures derived from random walks theory using the daily air traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path the comparison with direct simulations reveals a higher correlation compared to the shortest path approach used previously in addition our method allows to connect fundamental observables in epidemic spreading with the cumulant generating function of the hitting time for a markov chain our results provides a general and computationally efficient approach using only algebraic methods

a self adaptive moving mesh method for the camassa holm equation a self adaptive moving mesh method is proposed for the numerical simulations of the camassa holm equation it is an integrable scheme in the sense that it possesses the exact n soliton solution it is named a self adaptive moving mesh method because the non uniform mesh is driven and adapted automatically by the solution once the non uniform mesh is evolved the solution is determined by solving a tridiagonal linear system due to these two superior features of the method several test problems give very satisfactory results even if by using a small number of grid points

adomian decomposition method for burger s huxley and burger s fisher equations the approximate solutions for the burger s huxley and burger s fisher equations are obtained by using the adomian decomposition method solving frontier problems of physics the decomposition method kluwer boston 1994 the algorithm is illustrated by studying an initial value problem the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

stability of newton tvd runge kutta scheme for one dimensional euler equations with adaptive mesh abstract in this paper we propose a moving mesh method with a newton total variation diminishing tvd runge kutta scheme for the euler equations our scheme improves time discretization in the moving mesh algorithms by analyzing the semi discrete euler equations with the discrete moving mesh equations as constraints the stability of the newton tvd runge kutta scheme is proved thus we can conclude that the proposed algorithm can generate a weak solution to the euler equations finally numerical examples are presented to verify the theoretical results and demonstrate the accuracy of the proposed scheme

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

adaptive mesh methods for one and two dimensional hyperbolic conservation laws we develop efficient moving mesh algorithms for one and two dimensional hyperbolic systems of conservation laws the algorithms are formed by two independent parts pde evolution and mesh redistribution the first part can be any appropriate high resolution scheme and the second part is based on an iterative procedure in each iteration meshes are first redistributed by an equidistribution principle and then on the resulting new grids the underlying numerical solutions are updated by a conservative interpolation formula proposed in this work the iteration for the mesh redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state the main idea of the proposed method is to keep the mass conservation of the underlying numerical solution at each redistribution step in one dimension we can show that the underlying numerical approximation obtained in the mesh redistribution part satisfies the desired tvd property which guarantees that the numerical solution at any time level is tvd provided that the pde solver in the first part satisfies such a property several test problems in one and two dimensions are computed using the proposed moving mesh algorithm the computations demonstrate that our methods are efficient for solving problems with shock discontinuities obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

a high resolution mapped grid algorithm for compressible multiphase flow problems we describe a simple mapped grid approach for the efficient numerical simulation of compressible multiphase flow in general multi dimensional geometries the algorithm uses a curvilinear coordinate formulation of the equations that is derived for the euler equations with the stiffened gas equation of state to ensure the correct fluid mixing when approximating the equations numerically with material interfaces a c based and a a based model have been described that is an easy extension of the cartesian coordinates counterpart devised previously by the author 30 a standard high resolution mapped grid method in wave propagation form is employed to solve the proposed multiphase models giving the natural generalization of the previous one from single phase to multiphase flow problems we validate our algorithm by performing numerical tests in two and three dimensions that show second order accurate results for smooth flow problems and also free of spurious oscillations in the pressure for problems with interfaces this includes also some tests where our quadrilateral grid results in two dimensions are in direct comparisons with those obtained using a wave propagation based cartesian grid embedded boundary method

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

analytic study on burgers fisher huxley equations and combined forms of these equations abstract in this work we formally derive travelling wave solutions for the burgers fisher huxley equations and combined forms of these equations the tanh coth method is used to determine these sets of travelling wave solutions kinks and periodic wave solutions are constructed the proposed scheme can be used to a wide class of nonlinear evolution equations

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

remapping free ale type kinetic method for flow computations based on the integral form of the fluid dynamic equations a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed different from the earlier unified moving mesh gas kinetic method c q jin k xu an unified moving grid gas kinetic method in eulerian space for viscous flow computation j comput phys 222 2007 155 175 the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2d case since a purely lagrangian method is always associated with mesh entangling in order to avoid computational collapsing in multidimensional flow simulation the mesh velocity is constructed by considering both fluid velocity lagrangian methodology and diffusive velocity regenerating eulerian mesh function therefore we obtain a generalized arbitrary lagrangian eulerian ale method by properly designing a mesh velocity instead of re generating a new mesh after distortion as a result the remapping step to interpolate flow variables from old mesh to new mesh is avoided the current method provides a general framework which can be considered as a remapping free ale type method since there is great freedom in choosing mesh velocity in order to improve the accuracy and robustness of the method the adaptive moving mesh method h z tang t tang adaptive mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

reaction diffusion on random spatial networks with scale free jumping rates via effective medium theory we study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites an average transport network is found using a recently developed variant of the effective medium approximation ema that is capable of dealing with these long range connections using a feynman kac argument in the effective medium we derive an estimate on the size of the infected domain and reproduce the known result of its exponential growth in time we hereby demonstrate the applicability of long range ema to dynamical processes on networks more intricate than simple diffusion

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

a numerical simulation and explicit solutions of the generalized burgers fisher equation we consider solitary wave solutions of the generalized burgers fisher bf equation u t pu ru x u x x qu 1 u r 0 in this paper by considering the decomposition scheme we first obtain the exact solutions of the generalized bf equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values r 2 of the generalized bf equation

an adaptive grid method and its application to steady euler flow calculations an adaptive remeshing procedure based on a cell volume deformation method is presented starting with an initial grid this method offers direct cell volume control through the specification of the transformation jacobian grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points the grid velocities are determined by solving a scalar poisson equation this method is applied to solving the compressible euler equations computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves

moving mesh methods for one dimensional hyperbolic problems using clawpack we develop a one dimensional moving mesh method for hyperbolic systems of conser vation laws this method is based on the high resolution finite volume wave propagation method implemented in the clawpack software package a modified system of conservation laws is solved on a fixed uniform computational grid with a grid mapping function computed simultaneously in such a way that in physical space certain features are tracked by cell interfaces the method is tested on a shock tube problem with multiple reflections where the contact discontinuity is tracked and also on two multifluid problems where the interface between two distinct gases is tracked one is a standard test problem and the other also involves a moving piston whose motion is also tracked by the moving mesh 2003 elsevier science ltd all rights reserved

adaptive numerical method for burgers type nonlinear equations abstract in this paper a moving mesh partial differential equation mmpde approach will be applied to some types of nonlinear partial differential equations the method is described in detail and a number of computational examples are presented using different monitor functions applied to the burgers type equations

travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations in this work exact travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations are obtained the analysis rests mainly on the standard tanh method the work emphasizes the need for a transformation formula for the case where the parameter m is non integer the approach can be used in a variety of many types of nonlinearity

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

an adaptive moving mesh method with static rezoning for partial differential equations abstract adaptive mesh methods are valuable tools in improving the accuracy and efficiency of the numerical solution of evolutionary systems of partial differential equations if the mesh moves to track fronts and large gradients in the solution then larger time steps can be taken than if it were to remain stationary we derive explicit differential equations for moving the mesh so that the time variation of the solution at the mesh points is minimized moving the mesh based on this approach allows for larger time steps but does not guarantee that the solution is well resolved in space we maintain spatial accuracy when there are new emerging layers or wave fronts by adaptively rezoning the mesh points to equidistribute an error estimate when using a multistep integration method the past solution values are also interpolated so that the same multistep method can be used after rezoning the resulting algorithm has very few problem dependent numerical parameters and is appropriate for a large class of one dimensional partial differential equations we illustrate the performance of the algorithm by examples and demonstrate that the proposed algorithm is efficient and accurate when compared with other adaptive mesh strategies

a robust moving mesh finite volume method applied to 1d hyperbolic conservation laws from magnetohydrodynamics in this paper we describe a one dimensional adaptive moving mesh method and its application to hyperbolic conservation laws from magnetohydrodynamics mhd the method is robust because it employs automatic control of mesh adaptation when a new model is considered without manually set parameters adaptive meshes are a common tool for increasing the accuracy and reducing computational costs when solving time dependent partial differential equations pdes mesh points are moved towards locations where they are needed the most to obtain a time dependent adaptive mesh monitor functions are used to automatically monitor the importance of the various parts of the domain by assigning a weight value to each location based on the equidistribution principle all mesh points are distributed according to their assigned weights we use a sophisticated monitor function that tracks both small local phenomena as well as large shocks in the same solution the combination of the moving mesh method and a high resolution finite volume solver for hyperbolic pdes yields a serious gain in accuracy at relatively no extra costs the results of several numerical experiments including comparisons with h refinement are presented which cover many intriguing aspects typifying nonlinear magneto fluid dynamics with higher accuracy than often seen in similar publications

adaptive moving mesh upwind scheme for the two species chemotaxis model chemotaxis systems are used to model the propagation aggregation and pattern formation of bacteria cells in response to an external stimulus usually a chemical one a common property of all chemotaxis systems is their ability to model a concentration phenomenon rapid growth of the cell density in small neighborhoods of concentration points curves more precisely the solution may develop singular spiky structures or even blow up in finite time therefore the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task we study the two species patlak keller segel type chemotaxis system in which the two species do not compete but have different chemotactic sensitivities which may lead to a significantly difference in cell density growth rates this phenomenon was numerically investigated in kurganov and lukacova medvid ova discrete contin dyn syst ser b 19 2014 pp 131 152 and chertock et al adv comput math 44 2018 pp 327 350 where second and higher order methods on uniform cartesian grids were developed however in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected in this work we consider an alternative approach relying on mesh adaptation which helps to improve the approximation of the singular structures evolved by chemotaxis models we develop in particular an adaptive moving mesh amm finite volume semi discrete upwind method for the two species chemotaxis system the proposed amm technique allows one to increase the density of mesh nodes at the blowup regions this helps to substantially improve the resolution while using a relatively small number of finite volume cells

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

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