high resolution schemes for conservation laws with locally varying time steps we develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward euler and second order time discretizations l infty stability is proven for both time stepping schemes for problems in one space dimension these methods are restricted by a local cfl condition rather than the traditional global cfl condition allowing local time refinement to be coupled with local spatial refinement numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions

on bounded lower λ variation abstract in the paper we introduce the new concept of variation which allows to work in the spaces of functions measurable in the lebesgue sense we define the banach space bv 0 1 and we provide some of its basic geometric and topological properties we also define the useful notion of good representatives of functions generating the suitable equivalence classes from the space bv 0 1 and raise the question of their existence as well as their properties moreover we investigate convolution and superposition operators acting in bv 0 1 and give some applications to linear differential and nonlinear integral equations

the time line interpolation method for large time step godunov type schemes abstract this paper describes the use of the time line interpolation procedure for the design of large time step godunov type schemes for systems of hyperbolic conservation laws in one dimension these schemes are based on a specific procedure to characterize the left and right states of the riemann problems at the cell interfaces when the courant number associated with the waves exceeds unity to do so the time line interpolation technique is used constant and linear reconstruction techniques are presented sonic or critical points are seen to be a source of difficulty in the algorithm and an appropriate treatment is proposed the algorithms are applied to the linear advection equation to the inviscid burgers equation and to the set of hyperbolic conservation laws that describe shallow water flow in one dimension these simulations show the superiority of the linear time reconstruction over the use of a constant time reconstruction when the linear reconstruction technique is used the modulus of the amplification factor of the scheme is equal to unity for all wave numbers inducing oscillations in the computed profile owing to the phase error the introduction of a slope limiter allows these oscillations to be eliminated but yields numerical diffusion thus restricting the range of applications of the scheme

error bounds for the large time step glimm scheme applied to scalar conservation laws in this paper we derive an l 1 error bound for the large time step i e large courant number version of the glimm scheme when used for the approximation of solutions to a genuinely nonlinear i e convex scalar conservation law for a generic class of piecewise constant data we show that the error is bounded by o delta x 1 2 vert log delta x vert for courant numbers up to 1 the order of the error is the same as that given by hoff and smoller 5 in 1985 for the glimm scheme under the restriction of courant numbers up to 1 2

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

on the characterization of compactness in the space of functions of bounded variation in the sense of jordan abstract the main goal of this paper is to prove a compactness criterion for subsets of the banach space of functions of bounded variation in the sense of jordan in comparison to the compactness criterion contained in dunford schwartz s monograph the conditions which appear in our criterion seems to be easier to verify moreover using this criterion we define a quasimeasure of noncompactness in that space some applications of quasimeasures of noncompactness to fixed point theorems are also presented

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

the discrete one sided lipschitz condition for convex scalar conservation laws physical solutions to convex scalar conservation laws satisfy a one sided lipschitz condition oslc that enforces both the entropy condition and their total variation boundedness consistency with this condition is therefore desirable for a numerical scheme and was proved for both the godunov and the lax friedrichs scheme also in a weakened version for the roe scheme all of them being only first order accurate a new fully second order scheme is introduced here which is consistent with the oslc the modified equation is considered and shows interesting features another second order scheme is then considered and numerical results are discussed

convergence of two stage and multisplitting methods using various types of splittings we first study the convergence of both two stage iterative method and multisplitting methods using various types of splittings for solving a linear system whose coefficient matrix is an m matrix or an h matrix we next study a parallel implementation of two stage multisplitting method associated with a special multisplitting and an application of the two stage multisplitting method to parallel preconditioner of krylov subspace methods lastly we provide parallel performance results of both two stage multisplitting method and krylov subspace method with a parallel preconditioner which is derived from two stage multisplitting method on the ibm p690 supercomputer

an inexact parallel splitting augmented lagrangian method for large system of linear equations parallel iterative methods are powerful in solving large systems of linear equations les the existing parallel computing research results focus mainly on sparse systems or others with particular structure most are based on parallel implementation of the classical relaxation methods such as gauss seidel sor and aor methods which can be efficiently carried out on multiprocessor system in this paper we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts then we convert the original problem les into a monotone linear variational inequality problem vi with separable structure finally an inexact parallel splitting augmented lagrangian method is proposed to solve the variational inequality problem vi to avoid dealing with the matrix inverse operator we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is o n 2 in addition the proposed method does not require any special structure of system of les under consideration convergence of the proposed methods in dealing with two and three separable operators respectively is proved numerical computations are provided to show the applicability and robustness of the proposed methods

on nonlocal monotone difference schemes for scalar conservation laws analyse d erreur d une classe de methodes aux differences finies monotones pour la solution approchee des lois de conservation hyperboliques

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

stability of interfaces with mesh refinement a study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite difference approximations first a root condition which implies stability for the initial boundary value problem for this type of interface is derived from the root condition the stability of several interface equations is proved using the maximum principle in some cases the final verification steps can be done analytically in other cases a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle using this method stability for lax wendroff with all the interface conditions considered and for leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved

a comparison theorem for the sor iterative method in 1997 kohno et al have reported numerically that the improving modified gauss seidel method which was referred to as the imgs method is superior to the sor iterative method in this paper we prove that the spectral radius of the imgs method is smaller than that of the sor method and gauss seidel method if the relaxation parameter w 0 1 as a result we prove theoretically that this method is succeeded in improving the convergence of some classical iterative methods some recent results are improved

dynamic choices of hyperbolic consumers laboratory and field studies of time preference find that discount rates are much greater in the short run than in the long run hyperbolic discount functions capture this property this paper solves the decision problem of a hyperbolic consumer who faces stochastic income and a borrowing constraint the paper uses the bounded variation calculus to derive the hyperbolic euler relation a natural generalization of the standard exponential euler relation the hyperbolic euler relation implies that consumers act as if they have endogenous rates of time preference that rise and fall with the future marginal propensity to consume e g discount rates that endogenously range from 5 to 41 for the example discussed in the paper

on entropy consistency of large time step schemes i the godunov and glimm schemes this paper shows for scalar conservation laws with a convex flux function that the large time step approximation of weak solutions gives entropy solutions in the limit if the courant number is between tfrac 1 2 and 1 for flux functions with constant curvature monotonicity or monotone initial data the results extend to larger courant numbers a continuity argument gives this result also for the case of fluxes with almost constant curvature

a class of high resolution difference schemes for nonlinear hamilton jacobi equations with varying time and space grids based on a simple projection of the solution increments of the underlying partial differential equations pdes at each local time level this paper presents a difference scheme for nonlinear hamilton jacobi h j equations with varying time and space grids the scheme is of good consistency and monotone under a local cfl type condition moreover one may deduce a conservative local time step scheme similar to osher and sanders scheme approximating hyperbolic conservation law cl from our scheme according to the close relation between cls and h j equations second order accurate schemes are constructed by combining the reconstruction technique with a second order accurate runge kutta time discretization scheme or a lax wendroff type method they keep some good properties of the global time step schemes including stability and convergence and can be applied to solve numerically the initial boundary value problems of viscous h j equations they are also suitable to parallel computing numerical errors and the experimental rate of convergence in the lp norm p 1 2 and infty are obtained for several one and two dimensional problems the results show that the present schemes are of higher order accuracy

on nonconservative algorithms for grid interfaces in computations of fluid flows by domain decomposition methods the necessity of conservation at grid interfaces is now widely claimed in this paper we consider nonconservative algorithms for the interfaces our investigation begins with discussions about typical nonconservative interface treatments for one dimensional calculations the analysis shows that the conservation error of a numerical solution caused by a nonconservative interface matching has an upper bound when the solution itself is bounded furthermore if the numerical solution converges as the mesh size goes to zero it converges to a weak solution of the problem under certain conditions that may be detected numerically also we acquire similar results for two dimensional calculations on grids intersecting with each other in an arbitrary way in order to illustrate the theoretical results we present numerical examples and demonstrate that under those conditions conservation error reduces and accuracy for jumps as well as locations of discontinuities improves as the mesh size decreases

a class of large time step godunov schemes for hyperbolic conservation laws and applications a large time step lts godunov scheme firstly proposed by leveque is further developed in the present work and applied to euler equations based on the analysis of the computational performances of leveque s linear approximation on wave interactions a multi wave approximation on rarefaction fan is proposed to avoid the occurrences of rarefaction shocks in computations the developed lts scheme is validated using 1 d test cases manifesting high resolution for discontinuities and the capability of maintaining computational stability when large cfl numbers are imposed the scheme is then extended to multidimensional problems using dimensional splitting technique the treatment of boundary condition for this multidimensional lts scheme is also proposed as for demonstration problems inviscid flows over naca0012 airfoil and onera m6 wing with given swept angle are simulated using the developed lts scheme the numerical results reveal the high resolution nature of the scheme where the shock can be captured within 1 2 grid points the resolution of the scheme would improve gradually along with the increasing of cfl number under an upper bound where the solution becomes severely oscillating across the shock computational efficiency comparisons show that the developed scheme is capable of reducing the computational time effectively with increasing the time step cfl number

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

bernstein and kantorovich polynomials diminish the λ variation abstract we prove the variation diminishing property of the bernstein and kantorovich polynomials next we apply this result to characterize the space c b v c as the closure of the space of polynomials in the norm a new proof of the separability of c b v c is given

new comparison results for parallel multisplitting iterative methods abstract the parallel multisplitting nonstationary iterative model a and models 1 2 were introduced by bru et al bru elsner neumann models of parallel chaotic iteration methods linear algebra appl 103 1988 175 192 and climent et al climent perea tortosa zamora convergence theorems for parallel alternating iterative methods appl math comput 148 2004 497 517 respectively using nonnegative splitting theory we present some new comparison theorems between two general parallel multisplittings of model a and models 1 2 when the coefficient matrix is a monotone matrix particularly we establish some sufficient conditions for strict inequalities in comparisons of spectral radii of different iteration matrices

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

letter to the editor a note on the upper bound of the spectral radius for sor iteration matrix recently wang and huang j comput appl math 135 2001 325 corollary 4 7 established the following estimation on the upper bound of the spectral radius for successive overrelaxation sor iteration matrix sor 1 gs under the condition that the coefficient matrix a is a nonsingular m matrix and 1 where sor and gs are the spectral radius of sor iteration matrix and gauss seidel iteration matrix respectively in this note we would like to point out that the above estimation is not valid in general

on integral operators and nonlinear integral equations in the spaces of functions of bounded variation abstract in this paper we introduce new conditions on a kernel of a linear fredholm integral operator which turn out to be sufficient and necessary for that operator to map the space of functions of bounded variation in the sense of jordan into itself furthermore we apply those conditions to deal with the problem of the existence of solutions to nonlinear hammerstein equations in that space to achieve our goals we will use a topological degree approach as well as a fixed point approach

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

a note on the improving modified gauss seidel imgs method abstract in this paper we first provide another proof for a theorem given by sun l sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 by showing that spectral radius of the gauss seidel method is not greater than that of the sor method when 0 1 next convergence results of the imgs method for both an h matrix and a symmetric positive definite matrix are provided

convergence of a large time step generalization of godunov s method for conservation laws a natural generalization of godunov s method for courant numbers larger than 1 is obtained by handling interactions between neighboring riemann problems linearly i e by allowing waves to pass through one another with no change in strength or speed this method is well defined for arbitrarily large courant numbers and can be written in conservation form it follows that if a sequence of approximations converges to a limit u x t as the mesh is refined then u is a weak solution to the system of conservation laws for scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence it is conjectured that in fact every sequence converges to the unique entropy solution provided the correct entropy solution is used for each riemann problem if the true riemann solutions are replaced by approximate riemann solutions which are consistent with the conservation law then the above convergence results for general systems continue to hold

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

large time step shock capturing techniques for scalar conservation laws for a scalar conservation law u t f u x with f of constant sign the first order upwind difference scheme is a special case of godonov s method the method is equivalent to solving a sequence of riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level the difference scheme is stable and the solutions to the associated sequence of riemann problems do not interact provided the courant number nu is less than 1 by allowing and explicitly handling such interactions it is possible to obtain a generalized method which is stable for nu much larger than 1 in many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods in particular shocks can be correctly computed with virtually no smearing the generalized method is rather unorthodox and still has some problems associated with it nonetheless preliminary results are quite encouraging

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

on entropy consistency of large time step schemes ii approximate riemann solvers it is shown that for scalar conservation laws with a convex flux function any sequence of approximate solutions constructed by large time step schemes using approximate riemann solvers as a building block gives the unique entropy solution in the limit of decreasing mesh size if the courant number is less than 1 when the curvature of the flux function is nearly constant the above results may be extended to courant numbers slightly larger than 1

letter to the editor a note on sor ω splitting of an m matrix recently wang and huang j comput appl math 135 2001 325 corollary 4 7 assumed that sor w splitting with w 1 is a weak regular splitting of an m matrix in this note we point out that sor w splitting with w 1 can never be a weak regular splitting of an m matrix this shows that chang s counterexample j comput appl math 167 2004 251 example 1 is not an appropriate one since the sor w splitting with w 1 given there is not a weak regular splitting

successive overrelaxation sor and related methods abstract covering the last half of the 20th century we present some of the basic and well known results for the sor theory and related methods as well as some that are not as well known most of the earlier results can be found in the excellent books by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 young iterative solution of large linear systems academic press new york 1971 and berman and plemmons nonnegative matrices in the mathematical sciences siam philadelphia pa 1994 while some of the most recent ones are given in the bibliography of this paper in this survey both the point and the block sor methods are considered for the solution of a linear system of the form ax b where a c n n and b c n 0 some general results concerning the sor and related methods are given and also some more specific ones in cases where a happens to possess some further property e g positive definiteness l m h matrix property p cyclic consistently ordered property etc

new convergence results for alternating methods abstract the alternating methods for solving the large system of linear equations ax b are investigated the convergence and the monotone convergence theories for the alternating method are formulated when the coefficient matrix is an h matrix or a monotone matrix sufficient conditions are established for the induced splitting by the alternating method to be a regular splitting furthermore new comparison theorems which improve previous comparison theorems are proved and several concrete applications are given

viscosity methods for piecewise smooth solutions to scalar conservation laws it is proved that for scalar conservation laws if the flux function is strictly convex and if the entropy solution is piecewise smooth with finitely many discontinuities which includes initial central rarefaction waves initial shocks possible spontaneous formation of shocks in a future time and interactions of all these patterns then the error of viscosity solution to the inviscid solution is bounded by o log in the l 1 norm which is an improvement of the o upper bound if neither central rarefaction waves nor spontaneous shocks occur the error bound is improved to o

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

convergence and comparison theorems for double splittings of matrices in this paper some convergence theorems for the double splitting of a monotone matrix or a hermitian positive definite matrix are presented two comparison theorems for two double splittings of a monotone matrix are obtained meanwhile we establish a new sufficient condition for convergence of the gauss seidel double sor method for an h matrix

h box methods for the approximation of hyperbolic conservation laws on irregular grids we study generalizations of the high resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization this godunov type scheme calculates fluxes at cell interfaces by solving riemann problems defined over boxes of a reference grid cell length h we discuss stability and accuracy of the resulting so called h box methods for one dimensional systems of conservation laws an extension of the method for the two dimensional case which is based on the multidimensional wave propagation algorithm is also described