pollutant transport by shallow water equations on unstructured meshes hyperbolization of the model and numerical solution via a novel flux splitting scheme abstract the purpose of this paper is twofold first using the cattaneo s relaxation approach we reformulate the system of governing equations for the pollutant transport by shallow water flows over non flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms the proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection diffusion models this turns out to give a larger stability range for the choice of the time step second following a flux splitting approach we derive a novel numerical method to discretise the resulting problem in particular we propose a new flux splitting and study the associated two systems of differential equations called the hydrodynamic and the relaxed diffusive system respectively for the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the godunov type these schemes are simple to implement robust accurate and fast when compared with existing methods the resulting method is implemented on unstructured meshes and is systematically assessed for accuracy robustness and efficiency on a carefully selected suite of test problems including non flat topography and wetting and drying problems formal second order accuracy is assessed through convergence rates studies

numerical solution of a hyperbolic system of conservation laws with source term arising in a fluidized bed model abstract a model a gas fluidized bed is considered which leads to a hyperbolic system of conservation laws with a source term the system is solved numerically by a second order operator splitting technique based on a roe approximate riemann solver numerical experiments demonstrate the ability of the model to reproduce qualitatively the slugging phenomenon in the case when the bed is subject to a relatively large gas flux

convergence of sph method for scalar nonlinear conservation laws this paper is devoted to the study of the convergence of weighted particle approximation of nonlinear multidimensional conservation laws for euler equations the method is closely related to the smooth particle hydrodynamics sph method extension of the original algorithm is proposed we use approximate riemann solvers instead of artificial viscosity in order to stabilize the scheme the mathematical analysis is performed by connecting this new approach with the finite volume scheme convergence of the approximate solution in l loc p p infty towards the unique weak entropy solution of the cauchy problem is obtained in the scalar nonlinear case by using uniqueness of measure valued solutions

computation of fluid flows in non inertial contracting expanding and rotating reference frames we present the method for computation of fluid flows that are characterized by the large degree of expansion contraction and in which the fluid velocity is dominated by the bulk component associated with the expansion contraction and or rotation of the flow we consider the formulation of euler equations of fluid dynamics in a homologously expanding contracting and or rotating reference frame the frame motion is adjusted to minimize local fluid velocities such approach allows to accommodate very efficiently large degrees of change in the flow extent moreover by excluding the contribution of the bulk flow to the total energy the method eliminates the high mach number problem in the flows of interest an important practical advantage of the method is that it can be easily implemented with virtually any eulerian hydrodynamic scheme and adaptive mesh refinement amr strategy we also consider in detail equation invariance and existence of conservative formulation of equations for special classes of expanding contracting reference frames special emphasis is placed on extensive numerical testing of the method for a variety of reference frame motions which are representative of the realistic applications of the method we study accuracy conservativity and convergence properties of the method both in problems which are not its optimal applications as well as in systems in which the use of this method is maximally beneficial such detailed investigation of the numerical solution behavior is used to define the requirements that need to be considered in devising problem specific fluid motion feedback mechanisms

on metric generators of graphs we study generators of metric spaces sets of points with the property that every point of the space is uniquely determined by the distances from their elements such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces the two applications we present concern combinatorial search problems on false coins known from the borderline of extremal combinatorics and information theory and a problem known from combinatorial optimization connected joins in graphs we use results on the detection of false coins to approximate the metric dimension minimum size of a generator for the metric space defined by the distances of some particular graphs for which the problem was known and open in the opposite direction using metric generators we show that the existence of connected joins in graphs can be solved in polynomial time a problem asked in a survey paper of frank on the negative side we prove that the minimization of the number of components of a join is np hard we further explore the metric dimension with some problems the main problem we are led to is how to extend an isometry given on a metric generator of a metric space

metric locating dominating sets of graphs for constructing related subsets of vertices abstract a dominating set s of a graph is a metric locating dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of s and the minimum cardinality of such a set is called the metric location domination number in this paper we undertake a study that in general graphs and specific families relates metric locating dominating sets to other special sets resolving sets dominating sets locating dominating sets and doubly resolving sets we first characterize the extremal trees of the bounds that naturally involve metric location domination number metric dimension and domination number then we prove that there is no polynomial upper bound on the location domination number in terms of the metric location domination number thus extending a result of henning and oellermann finally we show different methods to transform metric locating dominating sets into locating dominating sets and doubly resolving sets our methods produce new bounds on the minimum cardinalities of all those sets some of them concerning parameters that have not been related so far

high order exactly well balanced numerical methods for shallow water systems abstract this paper is concerned with the development of high order methods for the shallow water system that preserve every stationary solution of the system and not only the water at rest ones the pursued strategy is based on the generalized hydrostatic reconstruction introduced in 10 and on the technique proposed in 8 to obtain high order well balanced reconstruction operators the more difficult step in the reconstruction procedure is the first one given a cell value a stationary solution of the system has to be found whose average at the cell equals the given value the implementation of this step is discussed in detail as well as the similarities and differences with some previous approaches to obtain well balanced methods for the shallow water system finally the well balanced properties of the developed numerical methods are checked in some test cases

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

node pair finite volume finite element schemes for the euler equation in cylindrical and spherical coordinates a numerical scheme is presented for the solution of the compressible euler equations in both cylindrical and spherical coordinates the unstructured grid solver is based on a mixed finite volume finite element approach equivalence conditions linking the node centered finite volume and the linear lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete euler equations in both the cylindrical and the spherical reference systems numerical simulations are presented for the explosion and implosion problems with spherical symmetry which are solved in both the axial radial cylindrical coordinates and the radial azimuthal spherical coordinates numerical results are found to be in good agreement with one dimensional simulations over a fine mesh

an evaluation of roe s scheme generalizations for equilibrium real gas flows the extension of roe s approximate riemann solver to equilibrium real gas is analyzed by means of a general formulation allowing us to clarify the inherent nonuniqueness of the average state and the influence of the functional form of the equation of state several generalizations of roe s scheme are then reviewed and their numerical performances are discussed by computing some 2d steady hypersonic flows the flow solvers are coupled with a newly developed efficient and robust procedure for thermochemical air properties evaluation all of the tested equilibrium solvers achieve very similar results they are found of comparable numerical efficiency the higher performances being associated with vinokur s and liou s solvers it is concluded that equilibrium simulations in 2d are by no means less robust than the perfect gas ones when coupled with the proposed procedure for properties evaluation

roe matrices for ideal mhd and systematic construction of roe matrices for systems of conservation laws in this paper the construction of a roe s scheme for the conservative system of ideal magnetohydrodynamics mhd is presented as this method relies on the computation of a roe matrix the problem is to find a matrixa ul ur which satisfies the following properties it is required to be consistent with the jacobian of the fluxf to have real eigenvalues a complete set of eigenvectors and to satisfy the relation f a ul ur u whereulandurare two admissible states and utheir difference for the ideal mhd system using eulerian coordinates a roe matrix is obtained without any hypothesis on the specific heat ratio especially its construction relies on an original expression of the magnetic pressure jump moreover a roe matrix is computed for lagrangian ideal mhd by extending the results of munz who obtained such a matrix for the system of lagrangian gas dynamics so this second matrix involves arithmetic averages unlike the eulerian one which contains classical roe averages like in eulerian gas dynamics in this paper a systematic construction of lagrangian roe matrices in terms of eulerian roe matrices for a general system of conservation laws is also presented this result applied to the above eulerian and lagrangian matrices for ideal mhd gives two new matrices for this system in the same way by applying this construction to the gas dynamics equations new roe matrices are also obtained all these matrices allow the construction of roe type schemes some numerical examples on the shock tube problem show the applicability of this method

solving shallow water systems in 2d domains using finite volume methods and multimedia sse instructions the goal of this paper is to construct efficient parallel solvers for 2d hyperbolic systems of conservation laws with source terms and nonconservative products the method of lines is applied at every intercell a projected riemann problem along the normal direction is considered which is discretized by means of well balanced roe methods the resulting 2d numerical scheme is explicit and first order accurate in m j castro j a garcia j m gonzalez c pares a parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products application to shallow flows comput methods appl mech engrg 196 2006 2788 2815 a domain decomposition method was used to parallelize the resulting numerical scheme which was implemented in a pc cluster by means of mpi techniques in this paper in order to optimize the computations a new parallelization of simd type is performed at each mpi thread by means of sse streaming simd extensions which are present in common processors more specifically as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations we use the intel c integrated performance primitives small matrix library to make easy the use of this library which is implemented using assembler and sse instructions we have developed a c wrapper of this library in an efficient way some numerical tests were carried out to validate the performance of the c small matrix wrapper the specific application of the scheme to one layer shallow water systems has been implemented on a pc s cluster the correct behavior of the one layer model is assessed using laboratory data

finite element method with optimal nodal velocity abstract the finite element method female 15 is outlined accuracy is much improved when the velocity of a finite element is equal to that of a local nonlinear wave in this paper we give a proof of the improvement of accuracy for the discretized equations the need for node annihilation is discussed for a hyperbolic system of equations and a restructuring algorithm of finite elements is proposed

renormalized meshfree schemes i consistency stability and hybrid methods for conservation laws this paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives the renormalized meshfree derivatives which improve the consistency of the original weighted particle methods the weak renormalized meshfree scheme built from the weak formulation of general conservation laws turns out to be l 2 stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field a time discretization is then performed by analogy with finite volume methods and the l 1 l infty and bv stabilities of the obtained time discretized scheme are studied from the same analogy with finite volume methods a hybrid particle scheme is built using the godunov method and is numerically compared to the weak renormalized scheme

adaptive triangular mesh generation a general adaptive grid algorithm is developed on triangular grids the adaptivity is provided by a combination of node addition dynamic node connectivity and a simple node movement strategy while the local restructuring process and the node addition mechanism take place in the physical plane the nodes are displaced on a monitor surface constructed from the salient features of the physical problem an approximation to mean curvature detects changes in the direction of the monitor surface and provides the pulling force on the nodes solutions to the axisymmetric grad shafranov equation demonstrate the capturing by triangles of the plasma vacuum interface in a free boundary equilibrium configuration

the riemann problem for the one dimensional free surface shallow water equations with a bed step theoretical analysis and numerical simulations in this paper the solution of the riemann problem for the one dimensional free surface shallow water equations over a bed step is analyzed both from a theoretical and a numerical point of view particular attention has been paid to the wave that is generated at the location of the bed discontinuity starting from the classical shallow water equations considering the bed level as an additional variable and adding to the system an equation imposing its time invariance we show that this wave is a contact wave across which one of the riemann invariants namely the energy is not constant this is due to the fact that the relevant problem is nonconservative we demonstrate that in this type of system riemann invariants do not generally hold in contact waves furthermore we show that in this case the equations that link the flow variables across the contact wave are the generalized rankine hugoniot relations and we obtain these for the specific problem from the numerical point of view we present an accurate and efficient solver for the step riemann problem to be used in a finite volume godunov type framework through a two step predictor corrector procedure the solver is able to provide solutions with any desired accuracy the predictor step uses a well balanced generalized roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution in order to show the effectiveness and the accuracy of the proposed approach we consider several step riemann problems and compare the exact solutions with the numerical results obtained by using a standard roe approach far from the step and the novel two step algorithm for the fluxes over the step achieving good results

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

coupling finite elements and finite volumes by lagrange multipliers for explicit dynamic fluid structure interaction this paper proposes a strategy for coupling a finite element description of structures with a finite volume fv description of fluids by means of the lagrange multipliers method for the numerical simulation of fluid structure interaction problems in fast transient dynamics the structural domain uses the classical explicit central difference cd time integration scheme while in the fluid domain a node centered finite volume scheme is adopted which achieves second order accuracy in space and in time by a predictor corrector algorithm combined with a muscl like technique the fv time marching scheme is suitably adapted to enable strong coupling with the cd scheme via lagrange multipliers numerical examples illustrate the effectiveness of the proposed procedure copyright 2010 john wiley sons ltd

shock stable roe scheme combining entropy fix and rotated riemann solver the roe scheme is an important shock capturing scheme that is known for its good performance however the classical roe scheme suffers from disastrous problems such as shock instability for hyper

on the metric dimension of corona product graphs given a set of vertices s v 1 v 2 v k of a connected graph g the metric representation of a vertex v of g with respect to s is the vector r v s d v v 1 d v v 2 d v v k where d v v i i 1 k denotes the distance between v and v i s is a resolving set for g if for every pair of distinct vertices u v of g r u s r v s the metric dimension of g dim g is the minimum cardinality of any resolving set for g let g and h be two graphs of order n 1 and n 2 respectively the corona product g h is defined as the graph obtained from g and h by taking one copy of g and n 1 copies of h and joining by an edge each vertex from the ith copy of h with the ith vertex of g for any integer k 2 we define the graph g kh recursively from g h as g kh g k 1h h we give several results on the metric dimension of g kh for instance we show that given two connected graphs g and h of order n 1 2 and n 2 2 respectively if the diameter of h is at most two then dim g kh n 1 n 2 1 k 1dim h moreover if n 2 7 and the diameter of h is greater than five or h is a cycle graph then dim g kh n 1 n 2 1 k 1dim k 1 h

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

optimal moving grids for time dependent partial differential equations abstract various adaptive moving grid techniques for the numerical solution of time dependent partial differential equations have been proposed the precise criterion for grid motion varies but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented we investigate moving grids on which the solutions of the linear heat conduction and viscous burgers equation in one space dimension are optimally approximated precisely we report the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of pde solutions in the least squares norm

a family of stable numerical solvers for the shallow water equations with source terms abstract in this work we introduce a multiparametric family of stable and accurate numerical schemes for 1d shallow water equations these schemes are based upon the splitting of the discretization of the source term into centered and decentered parts these schemes are specifically designed to fulfill the enhanced consistency condition of bermudez and vazquez necessary to obtain accurate solutions when source terms arise our general family of schemes contains as particular cases the extensions already known of roe and van leer schemes and as new contributions extensions of steger warming vijayasundaram lax friedrichs and lax wendroff schemes with and without flux limiters we include some meaningful numerical tests which show the good stability and consistency properties of several of the new methods proposed we also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods

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 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

hp multilevel monte carlo methods for uncertainty quantification of compressible navier stokes equations we propose a novel hp multilevel monte carlo method for the quantification of uncertainties in the compressible navier stokes equations using the discontinuous galerkin method as deterministic

the behavior of solutions of stochastic differential inequalities letx andz be d valued solutions of the stochastic differential inequalities dx t a t x t dt t x t dw t andb t z t dt t z t dw t dz t respectively with a fixed m valued wiener processw in this paper we give conditions ona b and under which the relationx0 z0 of the initial values leads to the same relation between the solutions with probability one further we discuss whether in general our conditions can be weakened or not then we deal with notions like maximal minimal solution of a stochastic differential inequality using the comparison result we derive a sufficient condition for the existence of such solutions as well as some gronwall type estimates

newton s method solver for the axisymmetric navier stokes equations a new two dimensional newton s method navier stokes solver is extended to the axisymmetric form of the equations roe s flux difference splitting is applied to the axisymmetric equations using glaister s approach a geometric conservation law correction term is added to insure satisfaction of the uniform stream reproduction criteria higher order accuracy is attained through the use of spekreijse s interpolation procedure with van albada s contiouous limiter harten and hyman s minimum eigenvalue correction is also included

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

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

on the stability of finite element methods for shock waves this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form 1 1 u t f u x u xx x 2 r i t 0 u 2 r i u delta 0 u 0 delta we treat systems which are strictly hyperbolic such systems possess a smooth travelling wave solution which is called a viscous p shock wave solution u x t oe x gamma oet x sigma1 oe x u sigma provided that the shock strength ffl j ju gamma u gamma j is small 19 the constant states u sigma and the wave speed oe are related by the rankine hugoniot condition 1 3a f u gamma gamma f u oe u gamma gamma u

on the k partition dimension of graphs abstract as a generalization of the concept of the partition dimension of a graph this article introduces the notion of the k partition dimension given a nontrivial connected graph g v e a partition of v is said to be a k partition generator of g if any pair of different vertices u v v is distinguished by at least k vertex sets of i e there exist at least k vertex sets s 1 s k such that d u s i d v s i for every i 1 k a k partition generator of g with minimum cardinality among all their k partition generators is called a k partition basis of g and its cardinality the k partition dimension of g a nontrivial connected graph g is k partition dimensional if k is the largest integer such that g has a k partition basis we give a necessary and sufficient condition for a graph to be r partition dimensional and we obtain several results on the k partition dimension for k 1 r

resolvability and the upper dimension of graphs abstract for an ordered set w w 1 w 2 w k of vertices and a vertex v in a connected graph g the metric representation of v with respect to w is the k vector r v w d v w 1 d v w 2 d v w k where d x y represents the distance between the vertices x and y the set w is a resolving set for g if distinct vertices of g have distinct representations a new sharp lower bound for the dimension of a graph g in terms of its maximum degree is presented a resolving set of minimum cardinality is a basis for g and the number of vertices in a basis is its metric dimension dim g a resolving set s of g is a minimal resolving set if no proper subset of s is a resolving set the maximum cardinality of a minimal resolving set is the upper dimension dim g the resolving number res g of a connected graph g is the minimum k such that every k set w of vertices of g is also a resolving set of g then 1 dim g dim g res g n 1 for every nontrivial connected graph g of order n it is shown that dim g res g n 1 if and only if g k n while dim g res g 2 if and only if g is a path of order at least 4 or an odd cycle the resolving numbers and upper dimensions of some well known graphs are determined it is shown that for every pair a b of integers with 2 a b there exists a connected graph g with dim g dim g a and res g b also for every positive integer n there exists a connected graph g with res g dim g n and dim g dim g n

a two dimensional mesh moving technique for time dependent partial differential equations abstract we discuss an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial boundary value problems for systems of partial differential equations in two space dimensions and time the mesh moving technique is based on an algebraic node movement function determined from the geometry and propagation of regions having significant discretization error indicators our procedure is designed to be flexible so that it can be used with many existing finite difference and finite element methods to test the mesh moving algorithm we implemented it in a system code with an initial mesh generator and a maccormack finite difference scheme on quadrilateral cells for hyperbolic vector systems of conservation laws results are presented for several computational examples the moving mesch scheme reduces dispersive errors near shocks and wave fronts and thereby reduces grid requirements necessary to compute accurate solutions while increasing computational efficiency

a parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products application to shallow flows the goal of this paper is to construct parallel solvers for 2d hyperbolic systems of conservation laws with source terms and nonconservative products more precisely finite volumes solvers on nonstructured grids are considered the method of lines is applied at every intercell a projected riemann problem along the normal direction is considered which is discretized by means of the numerical schemes presented in m j castro j macias c pares a q scheme for a class of systems of coupled conservation laws with source term application to a two layer 1 d shallow water system esaim m2an 35 1 2001 107 127 the resulting 2d numerical schemes are explicit and first order accurate the solver is next parallelized by a domain decomposition technique the specific application of the scheme to one and two layer shallow water systems has been implemented on a pcs cluster an efficient data structure based on oompi c object oriented extension of mpi has been developed to optimize the data exchange among the processors some numerical tests are next presented to validate the solver and the performance of its parallel implementation finally the two layer shallow water model is applied to the simulation of the steady exchange through the strait of gibraltar 2005 elsevier b v all rights reserved

modelling the transition between fixed and mobile bed conditions in two phase free surface flows in a two phase free surface flow the transition from a mobile bed condition to a fixed bed one and vice versa occurs at a sharp interface across which the relevant system of partial differential equations changes abruptly this leads to the possibility of conceiving a new type of riemann problem rp which we have called composite riemann problem crp where not only the initial constant values of the variables but also the system of equations change from left to right of a discontinuity in this paper we present a strategy for solving a crp by reducing it to a standard rp of a single composite system of equations this can be obtained by combining the two original systems by means of a suitable weighting function namely the erodibility variable and the introduction of an appropriate differential equation for this quantity in this way the crp problem can be analyzed theoretically with standard methods and the features of the solutions can be clearly identified in particular a stationary contact wave is able to correctly describe the sharp transition between mobile and fixed bed conditions a finite volume scheme based on the multiple averages generalized roe approach rosatti and begnudelli 2013 22 was used to numerically solve the fixed mobile crp several test cases demonstrate the effectiveness exact well balanceness and high accuracy of the scheme when applied to problems that fall within the physical range of applicability of the relevant mathematical model

a hybrid riemann solver for large hyperbolic systems of conservation laws this paper presents a riemann solver for nonlinear hyperbolic systems of conservation laws based on a krylov subspace approximation of the upwinding dissipation vector in the general case the solver does not require any detailed information of the eigensystem except an estimate of the global maximal propagation speed it uses successive flux function evaluations to obtain a numerical flux which is almost equivalent to that of a godunov scheme with complete upwinding the new krylov riemann solver is particularly efficient when used for large systems with many nonlinear equations such that typically no explicit expression for the eigensystem is available also no numerical procedures are necessary to compute the eigensystem numerical examples demonstrate the excellent performance of the solver with respect to other solvers

shock computations with adaptive mesh refinement and moving grids abstract highly accurate shock wave problems are presented using a combination of numerical techniques the new feature of this scheme is the treatment of discontinuities shocks or contact surfaces which are replaced by approximate step functions these approximations permit the solution of the euler equations within the discontinuity and permits a convenient way of treating source terms within a discontinuity the strong conservative law form is used throughout an explicit moving grid technique is used in which a moving frame is found in which the conservation law equations appear stationery in a least squares sense in this moving frame it is possible to drastically reduce or eliminate the temporal truncation errors thus the same time integration scheme can be used throughout the computational domain in the discontinuous as well as continuous regions for accurate physics of wave interactions a nonlinear riemann solver is applied in conjunction with the riemann solver as new waves from such interactions are created an adaptive mesh refinement scheme injects new mesh points automatically to define a post collision wave structure results are presented for the shock tube problem the von neumann spherical blast wave noh s spherical shock problem shock interactions with a set up and step down contact discontinuity

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

a new theory for one dimensional adaptive grid generation and its applications the theory of a new approach to adaptive grid generation in one dimension is developed the approach is based on approximating either the resolution or the grid spacing ratio on discrete lattice points by continuous variables the order of accuracy of these approximations in a suitable reference frame characterizes the various methods approximations that are first or second order accurate in a suitable reference coordinate are derived in this paper the free parameters associated with these methods provide flexibility in generating a large family of adaptive grids with smooth grid spacing ratio and high resolution a selected group of this family of adaptive grids may prove very useful in adaptive computations of partial differential equations the adaptive grids are numerically generated using these approximations numerical examples are given that exemplify the usefulness of these adaptive grids

coupled discontinuous and continuous galerkin finite element methods for the depth integrated shallow water equations abstract in this paper we investigate a new approach for the two dimensional fully non linear depth averaged shallow water equations by coupling continuous and discontinuous galerkin methods we apply a discontinuous galerkin method to the primitive continuity and momentum equations and couple this to a continuous method discretizing the so called wave continuity and momentum equation herein we present the formulation and derive an a priori error estimate and demonstrate some preliminary numerical results this work represents the culmination of a series of papers exploring coupled methods for the shallow water equations

automatic adaptive grid refinement for the euler equations a method of adaptive grid refinement for the solution of the steady euler equations for transonic flow is presented the algorithm automatically decides where the coarse grid accuracy is insufficient and creates locally uniform refined grids in these regions this typically occurs at the leading and trailing edges the solution is then integrated to steady state using the same integrator flo52 in the interior of each grid the boundary conditions needed on the fine grids are examined and the importance of treating the fine coarse grid interface conservatively is discussed numerical results indicate substantial computational savings for the same solution accuracy can be achieved

a singularities tracking conservation laws scheme for compressible duct flows a singularities tracking version of the grp scheme for the integration of the euler equations for compressible duct flow is presented flow singularities corresponding to contact material shock or gradient discontinuities are represented by special grid points that move through the fixed grid at the appropriate speed of propagation the primary modification to the eulerian grp scheme is the evaluation of fluxes at singular points in a unified grid containing the union of regular euler grid points and singular moving points interactions between singular points shock shock or shock contact interactions are treated accurately by solving the appropriate generalized riemann problem the new grp st scheme combines some merits of computation by characteristics with a robustness approaching that of a finite difference conservation laws scheme shock wave phenomena illustrating the capabilities of the tracking method are presented demonstrating improved resolution and accuracy at a given grid

pressure based compressible calculation method utilizing total variation diminishing schemes a pressure based implicit procedure to solve the euler and navier stokes equations on a nonorthogonal mesh with collocated finite volume formulation is described the boundedness criteria for this procedure are determined from total variation diminishing tvd schemes which are based on characteristic variables and are applied to the fluxes of the convected quantities including mass flow rate the procedure incorporates the e eddy viscosity turbulence model the algorithm is first tested for inviscid flows at different mach numbers ranging from subsonic to supersonic on a bump in a channel geometry where the results are compared with other existing numerical solutions the method is then validated against experiment and another numerical solution for the case of turbulent transonic flow through a gas turbine rotor blade cascade for which wind tunnel experimental data exist

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

a discontinuous galerkin finite element method for directly solving the hamilton jacobi equations in this paper we propose a new discontinuous galerkin finite element method to solve the hamilton jacobi equations unlike the discontinuous galerkin method of c hu c w shu a discontinuous galerkin finite element method for hamilton jacobi equations siam journal on scientific computing 21 1999 666 690 which applies the discontinuous galerkin framework on the conservation law system satisfied by the derivatives of the solution the method in this paper applies directly to the solution of the hamilton jacobi equations for the linear case this method is equivalent to the traditional discontinuous galerkin method for conservation laws with source terms thus stability and error estimates are straightforward for the nonlinear convex hamiltonians numerical experiments demonstrate that the method is stable and provides the optimal k 1 th order of accuracy for smooth solutions when using piecewise kth degree polynomials singularities in derivatives can also be resolved sharply if the entropy condition is not violated special treatment is needed for the entropy violating cases both one and two dimensional numerical results are provided to demonstrate the good qualities of the scheme

some aspects of high speed blunt body flow computations with roe scheme this paper discusses some aspects of three dimensional supersonic and hypersonic inviscid blunt body flow computations the method used is based on the solution of the euler equations employing an explicit upwind total variation diminishing formulation of the roe riemann solver within the cell centered finite volume approach a comparative study of various total variation diminishing limiters and entropy fixes is carried out to identify the most appropriate combination the effect of using cell spacing in the total variation diminishing extrapolations is highlighted furthermore a mach number dependence and grid sensitivity study is carried out on various total variation diminishing limiters scaled with mesh spacing local time stepping and code parallelization has been employed to accelerate convergence in terms of effective wall clock times

a linear time algorithm for metric dimension of cactus block graphs an undirected graph g v e has metric dimension at most k if there is a vertex set u v such that u k and u v v u v there is a vertex w u such that d g w u d g w v where d g u v is the distance the length of a shortest path in an unweighted graph between u and v the metric dimension of g is the smallest integer k such that g has metric dimension at most k a cactus block graph is an undirected graph whose biconnected components are either cycles or complete graphs we present a linear time algorithm for computing the metric dimension of cactus block graphs

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

riemann solver for relativistic hydrodynamics in this paper we construct an efficient accurate and rugged riemann solver for relativistic hydrodynamics the algorithm is an extension of the two shock approximation of colella to the relativistic regime the riemann solver constructed here is made to converge to the solution via iteration two different iterative techniques are presented one based on a secant method and the other on a newton method the method presented here provides an exact treatment of the transverse velocities across general oblique shocks this is a non trivial but very desirable property to have in a riemann solver for relativistic flow we also show the equivalence of our new formulation to the previous ones in the non relativistic limit

high resolution shock capturing schemes for inviscid and viscous hypersonic flows abstract a class of high resolution implicit total variation diminishing tvd type algorithms suitable for transonic multidimensional euler and navier stokes equations has been extended to hypersonic computations the improved conservative shock capturing schemes are spatially second and third order and are fully implicit they can be first or second order accurate in time and are suitable for either steady or unsteady calculations enhancement of stability and convergence rate for hypersonic flows is discussed with the proper choice of the temporal discretization and implicit linearization these schemes are fairly efficient and accurate for very complex two dimensional hypersonic inviscid and viscous shock interactions this study is complemented by a variety of steady and unsteady viscous and inviscid hypersonic blunt body flow computations due to the inherent stiffness of viscous flow problems numerical experiments indicated that the convergence rate is in general slower for viscous flows than for inviscid steady flows

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

mixed metric dimension of graphs abstract let g v e be a connected graph a vertex w v distinguishes two elements vertices or edges x y e v if dg w x dg w y a set s of vertices in a connected graph g is a mixed metric generator for g if every two distinct elements vertices or edges of g are distinguished by some vertex of s the smallest cardinality of a mixed metric generator for g is called the mixed metric dimension and is denoted by dim m g in this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds we also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph finally we prove that the problem of determining the mixed metric dimension of a graph is np hard in the general case

porous medium equation with absorption and a nonlinear boundary condition where is a bounded domain with smooth boundary is the outer normal derivative m 1 p and q are positive parameters and u0 is in l problems of this form arise in mathematical models in a number of areas of science for instance in models for gas or uid ow in porous media 3 and for the spread of certain biological populations 13 in the semilinear case that is for m 1 there is an extensive literature about global existence and blow up results for this type of problems see among others 5 9 16 and the literature therein for the degenerate case that is for m 1 with a nonlinear boundary condition local existence and uniqueness of weak solutions which are limit of solutions of nondegenerate problems has been established in 1 also in 2 existence and uniqueness of global weak solutions for a similar

godunov type method on non structured meshes for three dimensional moving boundary problems abstract this paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner the method uses the space time formulation by godunov while the discretization is conducted on non structured tetrahedral meshes using roe s approximate riemann solver an implicit time stepping and a muscl type interpolation the computation of the geometrical parameters required to take into account the movement of the boundaries is described examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method

solvability of boundary value problems for the general schrodinger equation via schrodinger type identity methods abstract in this article we consider a free boundary value problem for the general schrodinger equations with density dependent viscosity coefficients by applying a powerful technique based on the schrodinger type identity method under certain assumptions imposed on the initial data there exists a unique solution for this equation mentioned above finally we also give an application which reveals that the schrodinger type identity method is effective and simple

is it possible to determine a point lying in a simplex if we know the distances from the vertices abstract it is an elementary fact that if we fix an arbitrary set of d 1 affine independent points p 0 p d in r d then the euclidean distances x p j j 0 d determine the point x in r d uniquely in this paper we investigate a similar problem in general normed spaces which is motivated by this known fact namely we characterize those at least d dimensional real normed spaces x for which every set of d 1 affine independent points p 0 p d x the distances x p j j 0 d determine the point x lying in the simplex conv p 0 p d uniquely if d 2 then this condition is equivalent to strict convexity but if d 2 then surprisingly this holds only in inner product spaces the core of our proof is some previously known geometric properties of bisectors the most important of these theorem 1 is re proven using the fundamental theorem of projective geometry

variable neighborhood search for metric dimension and minimal doubly resolving set problems abstract in this paper two similar np hard optimization problems on graphs are considered the metric dimension problem and the problem of determining a doubly resolving set with the minimum cardinality both are present in many diverse areas including network discovery and verification robot navigation and chemistry for each problem a new mathematical programming formulation is proposed for solving more realistic large size instances a variable neighborhood search based heuristic is designed an extensive experimental comparison on five different types of instances indicates that the vns approach consistently outperforms a genetic algorithm the only existing heuristic in the literature designed for solving those problems

on the uniqueness of the solution of the evolution dam problem the purpose here is to investigate some aspects of the evolution dam problem partial chi alpha u partial t delta u rm div chi e in q chi in h u in q u geq 0 in q u phi in sigma 2 partial u partial v chi ev leq 0 on sigma 2 cap phi 0 e in bf r n e 0 cdots 0 1 partial u partial v chi ev 0 on sigma 1 here h is the heaviside maximal graph h s 1 if s 0 h s 0 1 if s 0 alpha is the storativity constant alpha 0 if the fluid is incompressible and alpha 0 if the fluid is compressible also q omega times 0 t where omega is a bounded domain in bf r n omega represents a porous medium separating a finite number of reservoirs with lipschitz boundary g where g is divided into two parts g 1 which is the impervious part and g 2 which is the pervious part with the following assumptions g 2 is an open subset of g g 2 not emptyset g 2 cap g 1 emptyset g 2 cup g 1 g and e nu leq 0 a e on g 1 sigma i denotes g i times 0 t i 1 2 the pressure at sigma 2 is known and is represented by a nonnegative function phi in c 0 1 overline q the above problem can be completed by prescribing an initial value to chi alpha u the existence of a weak solution of the evolution dam problem is well known however many properties of the solutions of the initial value dam problem in particular comparison and uniqueness theorems continue to be open for a general domain omega the results of this paper give precise information about the comparison theorem 4 7 and the uniqueness corollary 4 9 statements the key point to prove comparison and uniqueness results is that the function h mapsto chi x x n h t h x in bf r n 1 is a e nonincreasing

a fast adaptive quadtree scheme for a two layer shallow water model this paper presents a dynamically adaptive quadtree grid generation system for the solution of a two dimensional two layer shallow water model roe type two layer shallow water solvers require numerical approximation of the system eigenvalues as well as numerical balancing which increase computational cost considerably when a regular grid is used in order to improve computational efficiency we consider a dynamically adaptive quadtree grid generation system capable of increasing local resolution where high gradients occur in the physical flow variables test results show that satisfactory convergence can be obtained using the present scheme with the adaptive grid generator at a fraction of the cost incurred by a regular grid

computing the metric dimension of wheel related graphs abstract an ordered set w w 1 w k v g of vertices of g is called a resolving set or locating set for g if every vertex is uniquely determined by its vector of distances to the vertices in w a resolving set of minimum cardinality is called a basis for g and this cardinality is the metric dimension or location number of g denoted by g in this paper we study the metric dimension of certain wheel related graphs namely m level wheels an infinite class of convex polytopes and antiweb gear graphs denoted by w n m q n and awj 2 n respectively we prove that these infinite classes of wheel related graphs have unbounded metric dimension the study of an infinite class of convex polytopes generated by wheel denoted by q n also gives a negative answer to an open problem proposed by imran et al 2012 in 8 open problem is it the case that the graph of every convex polytope has constant metric dimension it is natural to ask for characterization of graphs with unbounded metric dimension

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

approximation for the minimum cost doubly resolving set problem locating source of diffusion in networks is crucial for controlling and preventing epidemic risks it has been studied under various probabilistic models in this paper we study source location from a deterministic point of view by modeling it as the minimum cost doubly resolving set drs problem which is a strengthening of the well known metric dimension problem let g be an undirected graph on n vertices where each vertex has a nonnegative cost a vertex subset s of g is a doubly resolving set drs of g if for every pair of vertices u v in g there exist x ys such that the difference of distances in terms of number of edges between u and x y is not equal to the difference of distances between v and x y the minimum cost drs problem consists of finding a drs in g with minimum total cost we establish lnn approximability of the minimum drs problem on general graphs for both weighted and unweighted versions this provides the first explicit lower and upper bounds on approximation for the minimum cost drs which are nearly tight moreover we design the first known strongly polynomial time exact algorithms for the minimum cost drs problem on general wheels and trees with additional constant k0 edges

on families of convex polytopes with constant metric dimension let g be a connected graph and d x y be the distance between the vertices x and y a subset of vertices w w 1 w 2 w k is called a resolving set for g if for every two distinct vertices x y v g there is a vertex w i w such that d x w i d y w i a resolving set containing a minimum number of vertices is called a metric basis for g and the number of vertices in a metric basis is its metric dimension dim g a family g of connected graphs is a family with constant metric dimension if dim g is finite and does not depend upon the choice of g in g in this paper we study the metric dimension of some classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes it is shown that these classes of convex polytopes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes

minimal doubly resolving sets and the strong metric dimension of some convex polytopes abstract in this paper we consider two similar optimization problems on graphs the strong metric dimension problem and the problem of determining minimal doubly resolving sets we prove some properties of strong resolving sets and give an integer linear programming formulation of the strong metric dimension problem these results are used to derive explicit expressions in terms of the dimension n for the strong metric dimension of two classes of convex polytopes d n and t n on the other hand we prove that minimal doubly resolving sets of d n and t n have constant cardinality for n 7

a new savage hutter type model for submarine avalanches and generated tsunami in this paper we present a new two layer model of savage hutter type to study submarine avalanches a layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid e g water the model is derived in a system of local coordinates following a non erodible bottom and takes into account its curvature we prove that the model verifies an entropy inequality preserves water at rest for a sediment layer and their solutions can be seen as particular solutions of incompressible euler equations under hydrostatic assumptions buoyancy effects and the centripetal acceleration of the grain movement due to the curvature of the bottom are considered in the definition of the coulomb term we propose a two step roe type solver to discretize the presented model it exactly preserves water at rest and no movement of the sediment layer when its angle is smaller than the angle of repose and up to second order all stationary solutions finally some numerical tests are performed by simulating submarine and sub aerial avalanches as well as the generated tsunami

tight integral duality gap in the chinese postman problem letg v e be a graph and letw be a weight functionw e z let t subseteq v be an even subset of the vertices ofg at cut is an edge cutset of the graph which dividest into two odd sets at join is a minimal subset of edges that meets everyt cut a generalization of solutions to the chinese postman problem the main theorem of this paper gives a tight upper bound on the difference between the minimum weightt join and the maximum weight integral packing oft cuts this difference is called the t join integral duality gap let w be the minimum weight of at join and letv w be the maximum weight of an integral packing oft cuts iff is a non empty minimum weightt join andn f is the number of components off then we prove that w v w n f 1

computing the k metric dimension of graphs abstract given a connected graph g v e a set s v is a k metric generator for g if for any two different vertices u v v there exist at least k vertices w 1 w k s such that dg u wi dg v wi for every i 1 k a metric generator of minimum cardinality is called a k metric basis and its cardinality the k metric dimension of g we make a study concerning the complexity of some k metric dimension problems for instance we show that the problem of computing the k metric dimension of graphs is np hard however the problem is solved in linear time for the particular case of trees

direct numerical simulation of interfacial instabilities a consistent conservative all speed sharp interface method we present a conservative and consistent numerical method for solving the navier stokes equations in flow domains that may be separated by any number of material interfaces at arbitrarily high density viscosity ratios and acoustic impedance mismatches subjected to strong shock waves and flow speeds that can range from highly supersonic to near zero mach numbers a principal aim is prediction of interfacial instabilities under superposition of multiple potentially active modes rayleigh taylor kelvin helmholtz richtmyer meshkov as found for example with shock driven immersed fluid bodies locally oblique shocks accordingly we emphasize fidelity supported by physics based validation including experiments consistency is achieved by satisfying the jump discontinuities at the interface within a conservative 2nd order scheme that is coupled in a conservative manner to the bulk fluid motions the jump conditions are embedded into a riemann problem solved exactly to provide the pressures and velocities along the interface which is tracked by a level set function to accuracy of o dx 5 dt 4 subgrid representation of the interface is achieved by allowing curvature of its constituent interfacial elements to obtain o dx 3 accuracy in cut cell volume with attendant benefits in calculating cell geometric features and interface curvature o dx 3 overall the computation converges at near theoretical o dx 2 spurious currents are down to machine error and there is no time step restriction due to surface tension our method is built upon a quadtree like adaptive mesh refinement infrastructure when necessary this is supplemented by body fitted grids to enhance resolution of the gas dynamics including flow separation shear layers slip lines and critical layers comprehensive comparisons with exact solutions for the linearized rayleigh taylor and kelvin helmholtz problems demonstrate excellent performance sample simulations of liquid drops subjected to shock waves demonstrate for the first time ab initio numerical prediction of the key interfacial features and phenomena found in recent experimental and theoretical studies of this class of problems t g theofanous aerobreakup of newtonian and viscoelastic liquids ann rev fluid mech 43 2011 661 690

grid adaptation and non iterative defect correction for improved accuracy of numerical solutions of pdes in this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations the methodology consists of analysis of leading order terms in the discretization error of any given finite difference method leading to a modified version of the original partial differential equation singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non iterative defect correction method is used to eliminate the leading order regular perturbation terms in the modified equation implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations the proposed approach is applied to four different canonical problems including the numerical solution of 1 liouville equation 2 inviscid burgers equation 3 nonlinear reaction advection equation and 4 a system of hyperbolic pdes when compared to exact solutions the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution

a hybrid adaptive eno scheme this paper describes a new hybrid adaptive eno scheme for partial differential equation in conservative form the scheme is hybrid because it combines finite difference approximations for points away from the shock and eno approximations near the shock the method is adaptive in two ways the first is that this method changes the computational grid in order to maintain a uniform approximation second the method determines where to use the conservative eno approximation to the derivative the combination of these produces a quick algorithm for the solution of conservation laws

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

an approximate riemann solver for shallow water equations and heat advection in horizontal centrifugal casting an approximate riemann solver was developed for solving modified shallow water equations swe and energy transport describing the average flow dynamics of a single layer spreading inside a horizontally rotating cylinder the numerical model was particularly developed for simulating the horizontal centrifugal casting hsc of the outer shell of a work roll the swe were derived in the rotating frame of reference therefore fictitious forces the centrifugal force and the coriolis force were considered in addition other forces such as the bed shear force the force of gravity the wind shear force and forces arising from the variable liquid solid interface were taken into account the jacobian matrix of the nonlinear hyperbolic system of pdes was decomposed into a set of eigenvalues and corresponding eigenvectors using standard and corrected roe averages a harten hyman entropy fix was used to prevent expansion shocks entropy violating solutions typically occurring during transonic rarefactions source terms were applied as a stationary discontinuity and were physically bounded and well balanced for steady states producing non oscillatory solutions each wave was upwinded using the explicit godunov s method the high resolution corrections with flux limiters were used to achieve second order of accuracy and dispersion free solutions at discontinuities in addition to the riemann solver a central scheme fv model was used to solve the heat diffusion inside the cylinder mold and partially solidified liquid layer coupled with the solidification model several simulations were performed results were analyzed and discussed

a note on the partition dimension of cartesian product graphs abstract let g v e be a connected graph the distance between two vertices u v v denoted by d u v is the length of a shortest u v path in g the distance between a vertex v v and a subset p v is defined as min d v x x p and it is denoted by d v p an ordered partition p 1 p 2 p t of vertices of a graph g is a resolving partition of g if all the distance vectors d v p 1 d v p 2 d v p t are different the partition dimension of g denoted by pd g is the minimum number of sets in any resolving partition of g in this article we study the partition dimension of cartesian product graphs more precisely we show that for all pairs of connected graphs g h pd g h pd g pd h and pd g h pd g dim h where dim h denotes the metric dimension of h consequently we show that pd g h dim g dim h 1

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche