application of the θ method to a telegraphic model of fluid flow in a dual porosity medium abstract this work focuses mainly on the study of numerical solutions which are obtained using the method of a generalized warren and root model that includes a second order wave like equation in its formulation the solutions approximately describe the single phase hydraulic head in fractures by considering the finite velocity of propagation by means of a cattaneo like equation the corresponding discretized model is obtained by utilizing a non uniform grid and a non uniform time step a simple relationship is proposed to give the time step distribution convergence is analyzed by comparing results from explicit fully implicit and crank nicolson schemes with exact solutions a telegraphic model of fluid flow in a single porosity reservoir with relaxation dynamics the warren and root model and our studied model which is solved with the inverse laplace transform we find that the flux and the hydraulic head have spurious oscillations that most often appear in small time solutions but are attenuated as the solution time progresses furthermore we show that the finite difference method is unable to reproduce the exact flux at time zero obtaining results for oilfield production times which are in the order of months in real units is only feasible using parallel implicit schemes in addition we propose simple parallel algorithms for the memory flux and for the explicit scheme

convergence analysis of moving finite element methods for space fractional differential equations most existent papers have focused on the fixed mesh methods for solving space fractional differential equations however since some classes of space fractional differential equations may have singular or even finite time blowup solutions it is highly needed to develop adaptive mesh methods to solve these problems in this paper the moving finite element methods are studied for a class of time dependent space fractional differential equations the convergence theories of the methods are derived with l 2 norm and numerical examples are provided to support the theoretical results to simplify the analysis a fractional ritz projection operator is introduced and the error estimation of the projection is derived under the moving mesh setting

a numerical investigation of blow up in reaction diffusion problems with traveling heat sources this paper studies the numerical solution of a reaction diffusion differential equation with traveling heat sources according to the fact that the locations of heat sources are known we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem several examples are provided to demonstrate the efficiency of the new moving mesh method especially in the case of two or three traveling heat sources moreover numerical results illustrate that the speed of the movement of the heat source is critical for blow up when there is only one traveling heat source for the case of two traveling heat sources blow up depends not only on the speed but also on the distance between the two traveling heat sources

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

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

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

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 special purpose processor for the monte carlo simulation of ising spin systems a description is given of a special purpose computer for the study of static and dynamic properties of stochastic ising like systems of up to 4 106 spins elementary monte carlo steps are executed in the machine at a rate of 1 5 mhz the first results are given obtained as part of a testing procedure of the machine

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

moving mesh methods for blowup in reaction diffusion equations with traveling heat source in this paper moving mesh methods are used to simulate the blowup in a reaction diffusion equation with traveling heat source the finite time blowup occurs if the speed of the movement of the heat source remains sufficiently low and the blowup procedure is not fixed at one point not like that for stationary heat source as time goes to the blowup time the blowup profile converges to a stationary state in the simulation a new moving mesh algorithm is designed to deal with the difficulty caused by the delta function in the traveling heat source the convergence rates are verified and new blowup figures are generated from the numerical experiments

adaptive reconnection based arbitrary lagrangian eulerian method ew present a new adaptive arbitrary lagrangian eulerian ale method this method is based on the reconnection based ale reale methodology of refs 35 34 6 the main elements in a standard reale method are an explicit lagrangian phase on an arbitrary polygonal in 2d mesh in which the solution and positions of grid nodes are updated a rezoning phase in which a new grid is defined by changing the connectivity using voronoi tessellation but not the number of cells and a remapping phase in which the lagrangian solution is transferred onto the new grid in the standard reale method the rezoned mesh is smoothed by using one or several steps toward centroidal voronoi tessellation but it is not adapted to the solution in any way in the current paper we present a new adaptive reale method a reale that is based on the following design principles first a monitor function or error indicator based on the hessian of some flow parameter s is utilized second an equi distribution principle for the monitor function is used as a criterion for adapting the mesh third a centroidal voronoi tessellation is used to adapt the mesh fourth we scale the monitor function to avoid very small and large cells and then smooth it to permit the use of theoretical results related to weighted centroidal voronoi tessellation in the a reale method both number of cells and their locations are allowed to change at the rezone stage on each time step the number of generators at each time step is chosen to guarantee the required spatial resolution in regions where monitor function reaches its maximum value we present all details required for implementation of new adaptive a reale method and demonstrate its performance in comparison with standard reale method on series of numerical examples

numerical verification of delta shock waves for pressureless gas dynamics the subject of this paper is theoretical analysis and numerical verification of delta shock wave existence for pressureless gas dynamic system the existence of overcompressive delta shock wave solution in the framework of colombeau generalized functions is proved this result is verified numerically by specially designed procedure that is based on wave propagation method implemented in clawpack the method is coupled with dynamic refinement mesh we also consider a strictly hyperbolic system obtained from the original one by perturbation and change of variables the same numerical procedure is applied to the perturbed problem the obtained numerical results in both cases confirm theoretical expectations

a third order accurate direct eulerian grp scheme for the euler equations in gas dynamics the paper proposes and implements a third order accurate direct eulerian generalized riemann problem grp scheme for one and two dimensional 1d 2d euler equations in gas dynamics it is an extension of the second order accurate grp scheme proposed in ben artzi et al 2006 5 the approximate states in numerical fluxes of the third order accurate grp scheme are derived by using the higher order weno reconstruction of the initial data the limiting values of the time derivatives of the solutions at the singularity point and the jacobian matrix besides the limiting values of the first order time derivatives of fluid variables the second order time derivatives are also needed in developing the present grp scheme and obtained by directly and analytically resolving the local grp in the eulerian formulation via two main ingredients i e the riemann invariants and rankine hugoniot jump conditions unfortunately for the sonic case that the transonic rarefaction wave appears in the grp the jacobian matrix is singular on the sonic line to this end those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation the 2d grp scheme is implemented by using the third order accurate time splitting method several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed grp scheme in comparison to the second order accurate grp scheme

a numerical study of air vapor heat transport through textile materials with a moving interface this paper focuses on the numerical study of heat and moisture transfer in clothing assemblies which is described by a multi component and multiphase air vapor heat flow with a moving interface a splitting semi implicit finite volume method is applied for the system of nonlinear parabolic equations and an implicit euler scheme is used for the interface equation in terms of classical dirichlet to neumann map the implicit system can be solved directly and no iteration is needed two types of clothing assemblies are investigated and the comparison with experimental measurements is also presented

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

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

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

the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

an hr adaptive method for the cubic nonlinear schrodinger equation abstract the nonlinear schrodinger equation nlse is one of the most important equations in quantum mechanics and appears in a wide range of applications including optical fibre communications plasma physics and biomolecule dynamics it is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients adaptive moving mesh methods r adaptive attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features this approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest adaptive mesh refinement h adaptive where the mesh is locally coarsened or refined is an alternative adaptive strategy which is popular for time independent problems in this paper we consider the effectiveness of a combined method h r adaptive to solve the nlse in one space dimension simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches the method is also shown to control the spatial error based on the user s input error tolerance evidence is also presented indicating second order spatial convergence using a novel monitor function to generate the adaptive moving mesh

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

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

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 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

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

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

an adaptive moving finite volume scheme for modeling flood inundation over dry and complex topography 1 a new geometrical conservative interpolation on unstructured meshes is developed for preserving still water equilibrium and positivity of water depth at each iteration of mesh movement leading to an adaptive moving finite volume amfv scheme for modeling flood inundation over dry and complex topography unlike traditional schemes involving position fixed meshes the iteration process of the afmv scheme moves a fewer number of the meshes adaptively in response to flow variables calculated in prior solutions and then simulates their posterior values on the new meshes at each time step of the simulation the amfv scheme consists of three parts an adaptive mesh movement to shift the vertices position a geometrical conservative interpolation to remap the flow variables by summing the total mass over old meshes to avoid the generation of spurious waves and a partial differential equations pdes discretization to update the flow variables for a new time step five different test cases are presented to verify the computational advantages of the proposed scheme over nonadaptive methods the results reveal three attractive features i the amfv scheme could preserve still water equilibrium and positivity of water depth within both mesh movement and pde discretization steps ii it improved the shock capturing capability for handling topographic source terms and wet dry interfaces by moving triangular meshes to approximate the spatial distribution of time variant flood processes iii it was able to solve the shallow water equations with a relatively higher accuracy and spatial resolution with a lower computational cost

an adaptive grid method for two dimensional viscous flows this paper extends the gas kinetic bgk ns scheme to an adaptive grid for the viscous flow simulations the grid movement and adaptation is controlled by a monitor function which may depend on velocity gradient or other flow variables such as density or pressure for the viscous flow computation the use of adaptive mesh much improves the efficiency and accuracy of the method in comparison with the methods with static mesh points the current method is an accurate and efficient method for the viscous flow computation where the grid points can be easily moved and concentrated on the regions with large velocity and density gradients such as the boundary layer and multi material interface many numerical examples validate the current approach for the viscous flow simulations

an adaptive numerical method to handle blow up in a parabolic system we study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval coupled at the boundary in a nonlinear way in certain cases the system develops a blow up singularity in finite time fixed mesh methods are not well suited to approximate the problem near the singularity as an alternative to reproduce the behaviour of the continuous solution we present an adaptive in space procedure the scheme recovers the conditions for blow up and non simultaneous blow up it also gives the correct non simultaneous blow up rate and set moreover the numerical simultaneous blow up rates coincide with the continuous ones in the cases when the latter are known finally we present numerical experiments that illustrate the behaviour of the adaptive method

well balanced residual distribution for the ale spherical shallow water equations on moving adaptive meshes we consider the numerical approximation of the shallow water equations swes in spherical geometry for oceanographic applications to provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere to this end we restate all arbitrary lagrangian eulerian ale transport formulas as well as the volume transformation laws for a 2d manifold using these results we write the set of ale swes on the sphere we then propose a residual distribution discrete approximation of the governing equations classical properties as the dgcl and the c property well balancedness are reformulated in this more general context an adaptive mesh movement strategy is proposed the discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement

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

resolving small scale structures in boussinesq convection by adaptive grid methods inviscid boussinesq convection is a challenging problem both analytically and numerically due to the complex dynamic development of small scales and the rapid loss of solution regularity the boussinesq convection pushes any numerical strategy to the limit in e and shu phys fluids 6 1994 49 a detailed numerical study of the boussinesq convection in the absence of viscous effects is carried out using filtered pseudospectral method and a high order accurate eno schemes in their computations very fine grids have to be used in order to resolve the small structures of the boussinesq fluid in this work we will develop an efficient adaptive grid method for solving the inviscid incompressible flows which can be useful in resolving extremely small structures with reasonably small number of grid points to demonstrate the effectiveness of the proposed method the boussinesq convection problem will be computed using the adaptive grid method

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

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

generalized nonlinear diagonal dominance and applications to asynchronous iterative methods abstract we introduce a concept of generalized diagonal dominance for nonlinear functions as in the linear case this brings together several apparently different classes of nonlinear functions such as strictly diagonally dominant functions and certain m functions with our concept we easily obtain a quite far reaching result on the global convergence of asynchronous iterative methods for finding zeros of nonlinear functions special cases include some known and several new convergence results for special iterative methods such as the nonlinear jor sor and ssor method

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

simulating finger phenomena in porous media with a moving finite element method the non equilibrium richards equation is solved using a moving finite element method in this paper the governing equation is discretized spatially with a standard finite element method and temporally with second order runge kutta schemes a strategy of the mesh movement is based on the work by li et al r li t tang p w zhang a moving mesh finite element algorithm for singular problems in two and three space dimensions journal of computational physics 177 2002 365 393 a beckett and mackenzie type monitor function is adopted to obtain high quality meshes around the wetting front a smoothing method which is based on the diffusive mechanism is used with the moving mesh technique high mesh quality and high numerical accuracy are obtained successfully the numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments

a second order accurate conservative front tracking method in one dimension this paper presents a conservative front tracking method for shocks and contact discontinuities that is second order accurate it is based on a volume of fluid method that treats the moving front with concepts similar to those of an embedded boundary method special care is taken in the centering of the data to ensure the right order of accuracy at the front and a redistribution step guarantees conservation a suite of test problems for tracking both shocks and contact discontinuities is presented that confirms that the method is second order accurate

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

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

an l stability analysis for the finite difference solution of one dimensional linear convection diffusion equations on moving meshes the stability of three moving mesh finite difference schemes is studied in the l norm for one dimensional linear convection diffusion equations these schemes use central finite differences for spatial discretization and the q method for temporal discretization and they are based on conservative and non conservative forms of transformed partial differential equations the stability conditions obtained consist of the cfl condition and the mesh speed related conditions the cfl condition is independent of the mesh speed and has the same form as that for fixed meshes the mesh speed related conditions restrict how fast the mesh can move the conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine illustrative numerical results are presented

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

a direct eulerian grp scheme for relativistic hydrodynamics two dimensional case this paper develops a direct eulerian generalized riemann problem grp scheme for two dimensional 2d relativistic hydrodynamics rhd it is an extension of the grp scheme for one dimensional 1d rhds z c yang p he h z tang j comput phys 230 2011 7964 7987 and the grp scheme for the non relativistic hydrodynamics m ben artzi j q li g warnecke j comput phys 218 2006 19 43 in order to derive the direct eulerian grp scheme the local grp of the split 2d rhd equations in the eulerian formulation has to be directly resolved by using corresponding riemann invariants and rankine hugoniot jump conditions so that the crucial and delicate lagrangian treatment in the original grp scheme m ben artzi j falcovitz j comput phys 55 1984 1 32 may be avoided an important difference of resolving the grp of the split 2d rhd equations from the grp of the 1d rhd equations or the non relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the grp of the split 2d rhd equations are nonlinearly coupled through the lorentz factor which is also built in terms of the tangential velocities it is a purely multi dimensional relativistic feature several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2d grp scheme

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

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

a study of moving mesh pde methods for numerical simulation of blowup in reaction diffusion equations a new concept called the dominance of equidistribution is introduced for analyzing moving mesh partial differential equations for numerical simulation of blowup in reaction diffusion equations theoretical and numerical results show that a moving mesh method works successfully when the employed moving mesh equation has the dominance of equidistribution the property can be verified using dimensional analysis in several aspects the current work generalizes previous work where a moving mesh equation is shown to have this dominance of equidistribution if it preserves the scaling invariance of the underlying physical partial differential equation and uses a small constant value for t a parameter used for adjusting response time of the mesh movement to the change in the physical solution also cases with both constant and variable t are considered here

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

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

microcanonical monte carlo study of a two dimensional blume capel model a microcanonical monte carlo algorithm has been developed to calculate the density of states of a two dimensional blume capel model in zero field the full density of states is calculated for a 16 16 lattice as a function of the two spin summations which appear in the hamiltonian this permits the partition function and related thermodynamic functions to be evaluated for any temperature and any single site anisotropy from a single data set results are presented for the pseudo critical transition temperature and the maximum in the specific heat as a function of the single site anisotropy parameter the technique will allow the region near the tricritical point to be explored in detail and will permit the future determination of the zeros of the partition function it should be noted that the microcanonical sampling method of lee 1990 was found to be unusably slow for the s 1 problem in regions were the density of states had a significant slope and the microcanonical sampling described in this work is a significant improvement on his sampling method

adaptivity and blow up detection for nonlinear evolution problems this work is concerned with the development of a space time adaptive numerical method based on a rigorous a posteriori error bound for a semilinear convection diffusion problem which may exhibit blow up in finite time more specifically a posteriori error bounds are derived in the l infty l 2 l 2 h 1 type norm for a first order in time implicit explicit interior penalty discontinuous galerkin in space discretization of the problem although the theory presented is directly applicable to the case of conforming finite element approximations in space the choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for odes that exhibit finite time blow up the new adaptive algorithm is shown to accurately estimate the blow up time of a number of problems including one which exhibits regional blow up

a method of constructing adaptive hexahedral moving grids a variational method of constructing an adaptive moving grid composed of hexahedral cells is considered the method is based on minimizing a functional defined on the riemannian manifold in space of variables including the physical space coordinates and components of a vector valued monitor function the mesh is generated in the manifold and then it is projected down to the physical region to provide the adaptive grid the integrand of the functional is the non dimensional ratio of the invariants of the metric the functional depends on metric elements of two metrics the curvilinear mesh generated in the manifold induces the monitor metric another control metric defined in a special way is responsible for additional cell shape control the issues of non degeneracy conditions for a hexahedral mesh are considered examples of adaptive grids are presented

numerical simulation of blowup in nonlocal reaction diffusion equations using a moving mesh method in this paper we implement the moving mesh pde method for simulating the blowup in reaction diffusion equations with temporal and spacial nonlinear nonlocal terms by a time dependent transformation the physical equation is written into a lagrangian form with respect to the computational variables the time dependent transformation function satisfies a parabolic partial differential equation usually called moving mesh pde mmpde the transformed physical equation and mmpde are solved alternately by central finite difference method combined with a backward time stepping scheme the integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach the monitor function in mmpdes plays a key role in the performance of the moving mesh pde method the dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle a variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests

a simple moving mesh method for one and two dimensional phase field equations a simple moving mesh method is proposed for solving phase field equations the numerical strategy is based on the approach proposed in li et al j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution the phase field equations are discretized by a finite volume method and the mesh moving part is realized by solving the conventional euler lagrange equations with the standard gradient based monitors numerical results demonstrate the accuracy and effectiveness of the proposed algorithm

second order godunov type scheme for reactive flow calculations on moving meshes the method of calculating the system of gas dynamics equations coupled with the chemical reaction equation is considered the flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ode part the numerical algorithm is based on the godunov s scheme on deforming meshes with some modification to increase the scheme order in time and space the variational approach is applied to generate the moving adaptive mesh at every time step the functional of smoothness written on the graph of the control function is minimized the grid lines are condensed in the vicinity of the main solution singularities e g precursor shock fire zones intensive transverse shocks and slip lines which allows resolving a fine structure of the reaction domain the numerical examples relating to the chapman jouguet detonation and unstable overdriven detonation are considered in both one and two space dimensions

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

moving mesh finite element methods for the incompressible navier stokes equations this work presents the first effort in designing a moving mesh algorithm to solve the incompressible navier stokes equations in the primitive variables formulation the main difficulty in developing this moving mesh scheme is how to keep it divergence free for the velocity field at each time level the proposed numerical scheme extends a recent moving grid method based on harmonic mapping r li t tang and p w zhang j comput phys 170 2001 pp 562 588 which decouples the pde solver and the mesh moving algorithm this approach requires interpolating the solution on the newly generated mesh designing a divergence free preserving interpolation algorithm is the first goal of this work selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations which is the second goal of this study the performance of the moving mesh scheme is tested on the standard periodic double shear layer problem no spurious vorticity patterns appear when even fairly coarse grids are used

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach

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

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

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

a concept of nonlinear block diagonal dominance we introduce a concept of block diagonal dominance for nonlinear functions and discuss the relations to strictly diagonally dominant functions and m functions some sufficient conditions for the new kind of functions are given the global convergence of block asynchronous sor methods for finding zeros of block diagonal dominant nonlinear functions is proved

the efficient implementation of a finite element multi resolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flows adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant solution developed in the context of spectral methods by eitan tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy we incorporate this idea into the finite element framework by using hierarchical functions as a multi frequency basis the result is a new finite element method for solving hyperbolic conservation laws for this method convergence for a one dimensional scalar conservation law has previously been proved here the method is described in detail several issues connected with its efficient implementation are considered and numerical results for several examples involving one and two dimensional hyperbolic conservation laws are provided several advantageous features of the method are discussed including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space

resolving the shock induced combustion by an adaptive mesh redistribution method an adaptive mesh method is presented for numerical simulation of shock induced combustion the method is composed of two independent ingredients a flow solver and a mesh redistribution algorithm the flow solver is a finite volume based second order upwind tvd scheme together with a lower upper symmetric gauss seidel relaxation scheme for solving the multispecies navier stokes equations with finite rate chemistry the adaptive mesh is determined by a grid generation method based on solving poisson equations with the monitor function carefully designed to resolve both sharp fronts and the induction zone between them numerically it is found that the resolution of the induction zone is particularly critical to the combustion problems and an under resolved numerical method may cause excessive energy release and spurious runaway reactions the monitor function proposed in this paper which is based on the relative rate of change of mass fraction covered this issue satisfactorily numerical simulations of supersonic flows past an axisymmetric projectile in a premixed hydrogen oxygen or air mixture are carried out the results show that the spurious runaway chemical reactions appearing on coarse grids can be eliminated by using adaptive meshes without invoking any ad hoc treatment for reaction rates the adaptive mesh approach is more effective than the fixed mesh one in obtaining grid independent results finally discrepancies between the numerical and benchmark experimental results and difficulties in simulating the detonation waves are delineated which appeal for further investigation

an adaptive grp scheme for compressible fluid flows this paper presents a second order accurate adaptive generalized riemann problem grp scheme for one and two dimensional compressible fluid flows the current scheme consists of two independent parts mesh redistribution and pde evolution the first part is an iterative procedure in each iteration mesh points are first redistributed and then a conservative interpolation formula is used to calculate the cell averages and the slopes of conservative variables on the resulting new mesh the second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the eulerian grp scheme which is directly extended to two dimensional arbitrary quadrilateral meshes several numerical examples show that the current adaptive grp scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points but also reduces possible errors or oscillations effectively

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

a new efficient monte carlo technique a new efficient monte carlo technique is developed and tested using the 2d ising model for which exact solutions are known the new technique calculates continuous thermodynamic functions of continuous thermodynamic variables from independent samples taken within an interval of typically a few hundredths of a monte carlo step per spin the new technique is not only efficient and accurate but also furnishes some new information concerning the relationship between canonical and microcanonical averages for finite systems it also furnishes primary thermodynamic functions such as free energy directly from the monte carlo data a feature not available in the conventional monte carlo method

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

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

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

the moving boundary node method a level set based finite volume algorithm with applications to cell motility eukaryotic cell crawling is a highly complex biophysical and biochemical process where deformation and motion of a cell are driven by internal biochemical regulation of a poroelastic cytoskeleton one challenge to built quantitative models that describe crawling cells is solving the reaction diffusion advection dynamics for the biochemical and cytoskeletal components of the cell inside its moving and deforming geometry here we develop an algorithm that uses the level set method to move the cell boundary and uses information stored in the distance map to construct a finite volume representation of the cell our method preserves cartesian connectivity of nodes in the finite volume representation while resolving the distorted cell geometry derivatives approximated using a taylor series expansion at finite volume interfaces lead to second order accuracy even on highly distorted quadrilateral elements a modified laplacian based interpolation scheme is developed that conserves mass while interpolating values onto nodes that join the cell interior as the boundary moves an implicit time stepping algorithm is used to maintain stability we use the algorithm to simulate two simple models for cellular crawling the first model uses depolymerization of the cytoskeleton to drive cell motility and suggests that the shape of a steady crawling cell is strongly dependent on the adhesion between the cell and the substrate in the second model we use a model for chemical signalling during chemotaxis to determine the shape of a crawling cell in a constant gradient and to show cellular response upon gradient reversal

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

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

a note on generalized nonlinear diagonal dominance in this paper an open problem proposed by a frommer about nonlinear generalized diagonal dominance is solved on some weak restriction a counterexample is presented if such a restriction is omitted and some new properties of nonlinear generalized diagonally dominant functions are investigated

adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound we consider 3 point numerical schemes that resolve scalar conservation laws that are oscillatory either to their dispersive or anti diffusive nature the spatial discretization is performed over non uniform adaptively redefined meshes we provide a model for studying the evolution of the extremes of the oscillations we prove that proper mesh reconstruction is able to control the oscillations we provide bounds for the total variation tv of the numerical solution we moreover prove under more strict assumptions that the increase of the tv due to the oscillatory behavior of the numerical schemes decreases with time hence proving that the overall scheme is tv increase decreasing tvi d we finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique 1 outline mesh adaptation techniques have been employed by several authors in the past it is worth noting the seminal works dd87 for88 hh83 luc85 luc86 tt03 where several properties of mesh adaptation were studied it was noticed in akm01 amt04 ams10 sfa09 that proper use of non uniform adaptively redefined meshes is capable of taming oscillations hence improving the stability properties of the numerical schemes to study the stabilization properties of mesh adaptation techniques we analyze the effect they have on the oscillations that oscillatory unstable numerical schemes produce the setting is the one dimensional scalar riemann problem ut f u x 0 x a b with the flux function f being smooth and convex for initial conditions we consider the single jump u0 x x 0 x0 x with x x0 0 1 we discretize spatially over a non uniform adaptively redefined mesh the mesh adaptation and the time evolution of the numerical solution are combined into the main adaptive scheme definition 1 1 main adaptive scheme mas given at time step n the mesh m x a x1 xn b and the approximations u u1 un the steps of the mas are 1 mesh reconstruction construct new mesh m x a x 1 x n b 2 solution update use the old mesh m x the approximations u n and the new mesh m x received by the editor september 19 2009 and in revised form september 2 2011 2010 mathematics subject classification primary 65 xx c 2012 american mathematical society reverts to public domain 28 years from publication

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

dimension splitting data points redistribution for meshless approximation to better approximate nearly singular functions with meshless methods we propose a data points redistribution method extended from the well known one dimensional equidistribution principle with properly distributed data points nearly singular functions can be well approximated by linear combinations of global radial basis functions the proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost in our numerical examples clear exponential convergence with respect to the numbers of data points can be observed