obtuse cones and gram matrices with nonnegative inverse abstract we study some properties of gram matrices with nonnegative inverse which lead to the constructions of obtuse cones these constructions can find applications in optimization problems e g in methods of convex feasibility problems or of convex minimization problems

solvable separable nonlocal potential models we introduce a generalized yamaguchi rank one separable potential with three parameters which has the separable yukawa or central yamaguchi potentials as limiting cases the three parameters can be obtained when possible from the set of experimental data binding energy scattering length and effective range by solving a simple cubic equation we also discuss a rank two separable potential with one of the strengths infinite for both potentials the solutions can be expressed analytically

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

convergence analysis for operator splitting methods applied to conservation laws with stiff source terms we analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms we suppose that the source term q u is dissipative it is proved that the l 1 error introduced by the time splitting can be bounded by o d t vert q u 0 vert l 1 which is an improvement of the o q d t upper bound where d t is the splitting time step q is the lipschitz constant of q or q max u vert q u vert in case q is smooth a generic model with a special form of stiff source is also investigated we propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term our results are derived by using parabolic regularizations rather than using kuznetsov s approximation theory which has been employed as a standard approach for error analysis to the dimensional or time splitting methods numerical examples are presented to illustrate the theoretical results

on the accuracy of fractional step methods for conservation laws in two dimensions in this paper it is shown that on the assumption of the initial data belonging to bv bf r 2 cap l infty bf r 2 the convergence rates in l 1 bf r 2 for both godunov s spatial splitting method and strang s splitting method for scalar conservation laws in two dimensional spaces are of o delta t 1 2 where delta t is the splitting time step the author illustrates by numerical experiments that for some two dimensional riemann problems the rate is of the order o delta t it is also shown by counterexamples that if the initial data belongs to l infty bf r 2 no error estimate of the type o delta t beta for any beta 0 is valid

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

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

an error bound for the polygonal approximation of conservation laws with source terms abstract we give in this study an error bound for the polygonal approximation of hyperbolic conservation laws including a dissipative source term this is a generalization of results in the case of conservation laws without source terms to the inhomogeneous scalar conservation laws

scattering properties for a solvable model with a three dimensional separable potential of rank 2 the scattering by a three dimensional separable potential of rank 2 is described analytically the model potential is chosen so that the lippmann schwinger equation can be exactly solved and all calculations are carried out analytically the explicit expressions for a number of scattering properties such as phase shifts cross sections t and s matrices resonances and poles of the resolvent are calculated the second virial coefficient is also evaluated in terms of phase shifts and bound state energies for the particular potential model presented

on the convergence rate of operator splitting for hamilton jacobi equations with source terms we establish a rate of convergence for a semidiscrete operator splitting method applied to hamilton jacobi equations with source terms the method is based on sequentially solving a hamilton jacobi equation and an ordinary differential equation the hamilton jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit euler method we prove that the l infty error associated with the operator splitting method is bounded by mathcal o delta t where delta t is the splitting or time step this error bound is an improvement over the existing mathcal o sqrt delta t bound due to souganidis nonlinear anal 9 1985 pp 217 257 in the one dimensional case we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous hamilton jacobi equations it is proved that this fully discrete splitting method possesses a linear convergence rate moreover numerical results are presented to illustrate the theoretical convergence results

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

generalized inverses of partitioned matrices in banachiewicz schur form abstract the problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so called banachiewicz schur form is reconsidered theorem of marsaglia and styan sankhy ser a 36 1974 437 concerning the class of all generalized inverses the class of reflexive generalized inverses and the moore penrose inverse is strengthened and new results are established for the classes of outer inverses least squares generalized inverses and minimum norm generalized inverses

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

numerical wave propagation in an advection equation with a nonlinear source term the cauchy and initial boundary value problems are studied for a linear advection equation with a nonlinear source term the source term is chosen to have two equilibrium states one unstable and the other stable as solutions of the underlying characteristic equation the true solutions exhibit travelling waves which propagate from one equilibrium to another the speed of propagation is dependent on the rate of decay of the initial data at infinity a class of monotone explicit finite difference schemes are proposed and analysed the schemes are upwind in space for the advection term with some freedom of choice for the evaluation of the nonlinear source term convergence of the schemes is demonstrated and the existence of numerical waves mimicking the travelling waves in the underlying equation is proved the convergence of the numerical wave speeds to the true wave speeds is also established the behaviour of the scheme is studied when the monotonicity criteria are violated due to stiff source terms and oscillations and divergence are shown to occur the behaviour is contrasted with a split step scheme where the solution remains monotone and bounded but where incorrect speeds of propagation are observed as the stiffness of the problem increases

error bounds for fractional step methods for conservation laws with source terms fractional step methods have been used to approximate solutions of scalar conservation laws with source terms in this paper the stability and accuracy of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of nonhomogeneous scalar conservation laws the authors show that time splitting methods for conservation laws with source terms always converge to the unique weak solution satisfying the entropy condition in particular it is proved that the l 1 errors in the splitting methods are bounded by o sqrt delta t where delta t is the splitting time step the l 1 convergence rate of a class of fully discrete splitting methods is also investigated

more on generalized inverses of partitioned matrices with banachiewicz schur forms abstract necessary and sufficient conditions are derived for a 2 by 2 partitioned matrix to have 1 1 2 1 3 1 4 inverses and the moore penrose inverse with banachiewicz schur forms as applications the banachiewicz schur forms of 1 1 2 1 3 1 4 inverses and the moore penrose inverse of a 2 by 2 partitioned hermitian matrix are also given

error bounds for glimm difference approximations for scalar conservation laws on deduit des bornes d erreur pour l approximation aux differences de glimm pour la solution d une loi de conservation scalaire non lineaire

an error estimate for a finite difference scheme approximating a hyperbolic system of conservation laws a rigorous proof of an error estimate for a finite difference scheme applied to a 2 times 2 nonstrictly hyperbolic system of conservation laws is presented it is established that the error in the l 1 norm is of order delta x 1 2 where delta x denotes the mesh size this convergence estimate is optimal since for special data the system degenerates to a scalar equation for which a similar estimate is known to be optimal

the moore penrose inverse of a partitioned nonnegative definite matrix abstract consider an arbitrary symmetric nonnegative definite matrix a and its moore penrose inverse a partitioned respectively as a e f f h and a g 1 g 2 g 2 g 4 explicit expressions for g 1 g 2 and g 4 in terms of e f and h are given moreover it is proved that the generalized schur complement a g 4 g 1 g 2 g 4 g 2 is always below the moore penrose inverse a h of the generalized schur complement a h e fh f with respect to the lowner partial ordering

optimized high order splitting methods for some classes of parabolic equations we are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients it has been demonstrated that this se cond order barrier can be overcome by using splitting methods with complex valued coefficients with positive real parts in this way method s of orders 3 to 14 by using the suzuki yoshida triple and quadruple jump composition procedure have been explicitly built here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub optimal with respect to error constants as an alternative we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time even when low accuracies are desired we also show that in the general case 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8

bound preserving modified exponential runge kutta discontinuous galerkin methods for scalar hyperbolic equations with stiff source terms abstract in this paper we develop bound preserving modified exponential runge kutta rk discontinuous galerkin dg schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in zhang and shu 43 exponential strong stability preserving ssp high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions it is also straightforward to extend the method to two dimensions on rectangular and triangular meshes even though we only discuss the bound preserving limiter for dg schemes it can also be applied to high order finite volume schemes such as weighted essentially non oscillatory weno finite volume schemes as well

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

new representations of the moore penrose inverse of 2 2 block matrices abstract this paper presents a full rank factorization of a 2 2 block matrix without any restriction applying this factorization we obtain an explicit representation of the moore penrose inverse in terms of four individual blocks of the partitioned matrix which simplifies results of hung and markham 1975 12 and miao 1991 16 we also derive some important coincidence theorems including in the expressions of the moore penrose inverse of a sum of matrices all these results extend earlier works by various authors

finite volume schemes for nonhomogeneous scalar conservation laws error estimate in this paper we study finite volume schemes for the nonhomogeneous scalar conservation law u t rm div f x t u q x t u with initial condition u x 0 u0 x the source term may be either stiff or nonstiff in both cases we prove error estimates between the approximate solution given by a finite volume scheme the scheme is totally explicit in the nonstiff case semi implicit in the stiff case and the entropy solution the order of these estimates is h frac 1 4 in space time l 1 norm h denotes the size of the mesh furthermore the error estimate does not depend on the stiffness of the source term in the stiff case

adaptive multiresolution discontinuous galerkin schemes for conservation laws a multiresolution based adaptation concept is proposed that aims at accelerating discontinuous galerkin schemes applied to nonlinear hyperbolic conservation laws opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement instead of this a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients a central mathematical problem addressed in this work is then to show that the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh numerical comparisons demonstrate the efficiency of the concept and provide reliable estimates of the actual error in the numerical solution

general expressions for the moore penrose inverse of a 2 2 block matrix abstract the moore penrose inverse of a 2 2 block matrix m m a c b d is discussed general expressions for the moore penrose inverse for the block matrix m in terms of the individual blocks a b c d are delivered without any restrictions imposed under some conditions the moore penrose inverse can be simplified the results extend earlier work by various authors

on convergence of numerical schemes for hyperbolic conservation laws with stiff source terms we deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms we suppose that the source term is dissipative but it is not necessarily a lipschitzian function the convergence of the approximate solution towards the entropy solution is established for first and second order accurate muscl and for splitting semi implicit methods

scattering phase shift for relativistic exponential type separable potentials the j matrix method of scattering is used to obtain analytic expressions for the phase shift of two classes of relativistic exponential type separable potentials whose radial component is of the general form r 1e r 2 or r 2 e 2r2 2 where is a range parameter and 0 1 or 2 the rank of these separable potentials is 1 the nonrelativistic limit is obtained and shown to be identical to the nonrelativistic phase shift an exact numerical evaluation for higher order potentials 3 can also be obtained in a simple way as illustrated for the case 3