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

two a posteriori error estimates for one dimensional scalar conservation laws in this paper we propose a posteriori local error estimates for numerical schemes in the context of one dimensional scalar conservation laws we first consider the schemes for which a consistent in cell entropy inequality can be derived then we extend this result to second order schemes written in viscous form satisfying weak entropy inequalities as an illustration we show several numerical tests on the burgers equation and we propose an adaptive algorithm for the selection of the mesh

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

aliasing truncation errors in sampling approximations of sub gaussian signals this paper starts with new aliasing truncation error upper bounds in the sampling theorem for non bandlimited stochastic signals then it investigates l p 0 t approximations of the sub gaussian random signals explicit truncation error upper bounds are established the obtained rate of convergence provides a constructive algorithm for determining the sampling rate and the sample size in the truncated whittaker kotel nikov shannon expansions to ensure the approximation of the sub gaussian signals with given accuracy and reliability some numerical examples are presented

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

stability theorem and truncation error analysis for the glimm scheme and for a front tracking method for flows with strong discontinuities typical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations we consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws we obtain the stability consistency theorems and upper bounds of the truncation errors for the glimm scheme and for a front tracking method the major error in the glimm scheme is the error generated by the strong discontinuity this error is reduced when a front tracking method is applied to follow the location of the strong discontinuity this demonstrates an advantage of front tracking methods in one space dimension

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

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

the regularized chapman enskog expansion for scalar conservation laws rosenao r has recently proposed a regularized version of the chapmanenskog expansion of hydrodynamics this regularized expansion resembles the usual navier stokes viscosity terms at low wave numbers but unlike the latter it has the advantage of being a bounded macroscopic approximation to the linearized collision operator in this paper we study the behavior of the rosenau regularization of the chapman enskog expansion r c e in the context of scalar conservation laws we show that this r c e model retains the essential properties of the usual viscosity approximation e g existence of travelling waves monotonicity upper lipschitz continuity etc and at the same time it sharpens the standard viscous shock layers we prove that the regularized r c e approximation converges to the underlying inviscid entropy solution as its mean free path e 0 and we estimate the convergence rate

positive solutions for one dimensional p laplacian boundary value problems with dependence on the first order derivative abstract this paper presents sufficient conditions for the existence and multiplicity of positive solutions to the one dimensional p laplacian differential equation p u a t f u u 0 subject to some boundary conditions we show that it has at least one or two or three positive solutions under some assumptions by applying the fixed point theorem

a smoothness indicator for adaptive algorithms for hyperbolic systems the formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed in this paper a new smoothness indicator si is used to identify rough solution regions and is implemented in locally adaptive algorithms the si is based on the weak local truncation error of the approximate solution it was recently reported in s karni and a kurganov local error analysis for approximate solutions of hyperbolic conservation laws where error analysis and convergence properties were established the present paper is concerned with its implementation in scheme adaption and mesh adaption algorithms the si provides a general framework for adaption and is not restricted to a particular discretization scheme the implementation in this paper uses the central upwind scheme of a kurganov s noelle and g petrova siam j sci comput 23 707 2001 the extension of the si to two space dimensions is given numerical results in one and two space dimensions demonstrate the robustness of the proposed si and its potential in reducing computational costs and improving the resolution of the solution

existence of positive solutions for singular boundary value problems involving the one dimensional p laplacian this paper studies the existence of positive solutions for singular dirichlet boundary value problems these results are obtained by using the global continuation theorem fixed point index theory and approximate method

convergence of a splitting scheme applied to the ruijgrok wu model of the boltzmann equation this paper deals with upwind splitting schemes for the ruijgrok wu model physica a 113 1982 401 416 of the kinetic theory of rare7ed gases in the 8uid dynamic scaling we prove the stability and the convergence for these schemes the relaxation limit is also investigated and the limit equation is proved to be a 7rst order quasi linear conservation law the loss of quasi monotonicity of the present model makes it necessary to give a more careful analysis of its structure we also obtain global error estimates in the spaces w s p for 16s61 p 16p6 and pointwise error estimates for the approximate solution the proof naturally uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 1505 1519 due to the convexity of the 8ux function c

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

sharp conditions for the existence of sign changing solutions to equations involving the one dimensional p laplacian abstract we consider the boundary value problem involving the one dimensional p laplacian u p 2 u a x f u 0 0 x 1 u 0 u 1 0 where p 1 we establish sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio f s s p 1 at infinity and zero our argument is based on the shooting method together with the qualitative theory for half linear differential equations

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 iteration of monotone positive solutions for multipoint boundary value problem with p laplacian operator in the paper we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p laplacian operator f p u t q t f t u t 0 0

radial solutions for the p laplacian equation abstract we study the existence of positive solutions of the following nonlinear equation 1 a a p u u in 0 where p 1 0 p x x x p 2 and a satisfies some appropriate conditions our purpose is to give two existence results for the above equation subject to some boundary conditions where the nonlinear term is a nonnegative continuous function in 0 0 satisfying some adequate hypotheses

the existence of positive solutions for some nonlinear equation systems abstract in this paper we consider the existence of positive solutions of the following boundary value problem 1 x a t f x y 0 2 y b t g x y 0 t 0 1 1 x 0 1 x 0 0 2 y 0 2 y 0 0 1 x 1 1 x 1 0 2 y 1 2 y 1 0 where 1 2 r r are the increasing homeomorphism and positive homomorphism and 1 0 0 2 0 0 we show the sufficient conditions for the existence of positive solutions by using the nome type cone expansion expression fixed point theorem

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

triple positive solutions of a boundary value problem for nonlinear singular second order differential equations of mixed type with p laplacian in this paper we establish the existence of triple positive solutions of a two point boundary value problem for the nonlinear singular second order differential equations of mixed type with a p laplacian operator we also demonstrate that the results obtained can be applied to study certain higher order mixed boundary value problems finally an example is given to demonstrate the use of the main results of this paper

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

positive solutions for a fourth order p laplacian boundary value problem abstract in this paper we study the existence multiplicity and uniqueness of positive solutions for the fourth order p laplacian boundary value problem u p 1 u f t u u 2 i 0 u 2 i 1 0 i 0 1 here p 0 and f c 0 1 r r r 0 based on a priori estimates achieved by utilizing properties of concave functions we use fixed point index theory to establish our main results

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

convergence rate of approximate solutions to conservation laws with initial rarefactions the authors address the question of local convergence rate of conservative textit lip stable approximations u varepsilon x t to the entropy solution u x t of a genuinely nonlinear conservation law this question has been answered in the case of rarefaction free i e textit lip bounded initial data it has been shown that by post processing u varepsilon pointwise values of u and its derivatives may be recovered with an error as close to o varepsilon as desired where varepsilon measures in w 1 1 the truncation error of the approximate solution u varepsilon this paper extends the previous results by including textit lip unbounded initial data specifically it is shown that for arbitrary l infty cap bv initial data u and its derivatives may be recovered with an almost optimal modulo a spurious log factor error of o varepsilon ln varepsilon this analysis relies on obtaining new lip stability estimates for the speed a

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

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

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

existence and iteration of positive solutions for some p laplacian boundary value problems abstract this paper deals with the existence and iteration of positive solutions for the following p laplacian differential equation p u t q t f t u t u t 0 t 0 1 subject to one of the following boundary conditions p u 0 p u 0 0 p u 1 p u 1 0 or u 0 g 1 u 0 0 u 1 g 2 u 1 0 the main tool is the monotone iterative technique we obtain not only the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

convergence via oslc of the godunov scheme for a scalar conservation law with time and space flux discontinuities this paper deals with a godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time we extend the definition of vanishing viscosity solution of karlsen and towers j hyperbolic differ equ 14 671 702 2017 which applies to a flux with a spatial discontinuity in order to accommodate the addition of temporal flux discontinuities and prove that this extended definition implies uniqueness we prove convergence of the godunov approximations to the unique vanishing viscosity solution as the mesh size converges to zero thus establishing well posedness for the problem the novel aspect of this paper is the use of a discrete one sided lipschitz condition oslc in the discontinuous flux setting in the classical setting where flux discontinuities are not present the oslc is well known to produce an immediate regularizing effect with a local spatial variation bound resulting at any positive time we show that the oslc also produces a regularizing effect at any finite distance from the spatial flux discontinuity this regularizing effect is not materially affected by temporal flux discontinuities when combined with a cantor diagonal argument these regularizing effects imply convergence of the godunov approximations with this new method it is possible to forgo certain assumptions about the flux that seem to be required when using two commonly used convergence methods

convergence results for an inhomogeneous system arising in various high frequency approximations this paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of hamilton jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the schrodinger equation we first state an existence and uniqueness result in the framework of viscosity and duality solutions then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence as an illustration some practical computations are provided

the lp stability of relaxation rarefaction profiles we consider the large time behavior of solutions for a hyperbolic relaxation system for a certain class of initial data the solution is shown to converge to relaxation rarefaction profiles at a determined asymptotic rate the result is established without the smallness conditions of the wave strength and the initial disturbances 2001 academic press

convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 1 linearized approximations and linearized output functionals this paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one dimensional hyperbolic equation with a convex flux function a simple modified lax friedrichs discretization is used on a uniform grid and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined it is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data and also convergence in the discrete approximation of linearized output functionals in part 2 m giles and s ulbrich siam j numer anal 48 2010 pp 905 921 we extend the results to dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations in addition we present numerical results illustrating the asymptotic behavior which is analyzed

on convergence of minmod type schemes a class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered this class of methods contains the classical lax friedrichs and the second order nessyahu tadmor schemes in the case of linear flux new l2 stability results and error estimates for the methods are proved numerical experiments confirm that these methods are one sided l2 stable for convex flux instead of the usual lip stability

positive solutions for a system of second order quasilinear boundary value problems abstract in this paper we study the existence and multiplicity of positive solutions for the system of second order quasilinear boundary value problems u p 1 f t u v v q 1 g t u v u 0 u 1 0 v 0 v 1 0 where p q 1 and f g c 0 1 r 2 r r 0 based on a priori estimates achieved by utilizing jensen s inequality for nonnegative concave functions and homogeneous operators we use fixed point index theory to establish the main results

whittaker kotel nikov shannon approximation of φ sub gaussian random processes abstract the article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes then it investigates l p 0 t and uniform approximations of sub gaussian random processes by finite time sampling sums explicit truncation error upper bounds are established some specifications of the general results for which the assumptions can be easily verified are given direct analytical methods are employed to obtain the results

a priori error estimates for approximate solutions to convex conservation laws summary we introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite difference approximation calculated with the scheme of engquistosher lax friedrichs or godunov this technique is a discrete counterpart of the duality technique introduced by tadmor siam j numer anal 1991 the error is related to the consistency error of cell averages of the entropy weak solution this consistency error can be estimated by exploiting a regularity structure of the entropy weak solution one ends up with optimal error estimates

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

positive solution of three point boundary value problem for the one dimensional p laplacian with singularities abstract in the paper we deal with positive solutions of the following nonlinear three point singular boundary value problem with a p laplacian operator p u q t f t u 0 0 t 1 subject to u 0 g u 0 0 u 1 u 0 or u 0 u 0 u 1 g u 1 0 where f t u may be singular at u 0 and q t may be singular at t 0 1 new existence principles are established

total variation and error estimates for spectral viscosity approximations we study the behavior of spectral viscosity approximations to nonlinear scalar conservation laws we show how the spectral viscosity method compromises between the total variation bounded viscosity approximations which are restricted to first order accuracy and the spectrally accurate yet unstable fourier method in particular we prove that the spectral viscosity method is l sup 1 stable and hence total variation bounded moreover the spectral viscosity solutions are shown to be lip sup stable in agreement with oleinik s e entropy condition this essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution and we provide convergence rate estimates of both global and local types 16 refs

positive solutions of higher order four point boundary value problem with p laplacian operator in this paper we consider the existence of positive solutions of the higher order four point singular boundary value problem and show the sufficient conditions for the existence of positive solutions by using the fixed point theorem due to bai and ge some new existence results are obtained

time shifted aliasing error upper bounds for truncated sampling cardinal series abstract time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific l p p 1 classes analogous results are obtained in multidimensional regular sampling truncation error analysis is provided in all cases considered moreover sharpness of bounding inequalities convergence rates and various sufficient conditions are discussed

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

twin positive solutions for the one dimensional p laplacian boundary value problems abstract in this paper we study the existence of multiple positive solutions for the equation g u e t f u 0 where g v v p 2 v p 1 subject to nonlinear boundary conditions we show the existence of at least two positive solutions by using a new three functionals fixed point theorem in cones

effects of a saturating dissipation in burgers type equations we propose and study a new variant of the burgers equation with dissipation fluxes that saturate as the gradients become unbounded if the upstream downstream transition is above a critical threshold the corresponding riemann problem admits a weak solution wherein part of the transit is accomplished by a jump it is shown that the solution to a cauchy problem with sufficiently small compact or periodic initial data preserves its initial smoothness c 1997 john wiley sons inc

weakly nonoscillatory schemes for scalar conservation laws a new class of godunov type numerical methods called here weakly nonoscillatory or wno for solving nonlinear scalar conservation laws in one space dimension is introduced this new class generalizes the classical nonoscillatory schemes in particular it contains modified versions of min mod and uno under certain conditions convergence and error estimates for wno methods are proved

double positive solutions for a nonlinear four point boundary value problem with a p laplacian operator in this paper we study the existence of positive solutions for a nonlinear four point boundary value problem with a p laplacian operator by using a three functionals fixed point theorem in a cone the existence of double positive solutions for the nonlinear four point boundary value problem with ap laplacian operator is obtained this is different than previous results

bifurcation of sign changing solutions for one dimensional p laplacian with a strong singular weight p sublinear at abstract in this paper we consider one dimensional p laplacian problems with a singular weight which may not be in l 1 using the properties of eigenfunctions and the global bifurcation theory we prove existence uniqueness nonexistence and multiplicity results of positive solutions as well as sign changing solutions specially when the nonlinear term is p linear near 0 and p sublinear at

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

positive solutions for a system of p laplacian boundary value problems this paper is concerned with the existence and multiplicity of positive solutions for the system of p laplacian boundary value problems u i p i 1 f i t u 1 u n u i 0 u i 1 0 i 1 n where n 2 p i 1 f email protected c 0 1 xr n r i 1 n r 0 based on a priori estimates achieved by utilizing the jensen integral inequalities and r n monotone matrices we use fixed point index theory to establish the existence and multiplicity of positive solutions for the above problem

the large time behavior of the scalar genuinely nonlinear lax friedrichs scheme the lax friedrichs scheme approximating the scalar genuinely nonlinear conservation law u sub t f sub x u 0 where f u is say strictly convex double dot f dot a sub asterisk 0 is studied the divided differences of the numerical solution at time t do not exceed 2 t dot a sub asterisk to the 1 this one sided lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case in particular it is independent of the initial amplitude in sharp contrast to liner problems it guarantees the entropy compactness of the scheme in this case as well as providing a quantitive insight into the large time behavior of the numerical computation

a fundamentally simple approach to a singular boundary value problem abstract existence results by means of a simple approach for the 3rd order differential equation x t t f t x t x t x t 0 t 1 satisfying the three point boundary value conditions x 0 x x 1 0 are given where 0 1 is a fixed point in contrast to the usual case 1 2 1 f t x y z 0 for any t 0 1 x 0 y z r and the map t is positive on 0 1 and suitable singular at the end points furthermore the obtained solution x t is positive and concave on 0 1 in addition without any monotonicity assumption on the nonlinearity we prove the existence of a sequence of such solutions with lim n x n 0 our principal tools are very simple applications on the plane of the well known from combinatorial topology sperner s lemma in comparison with other classical approaches like fixed point or degree theorems

generalized monotone schemes discrete paths of extrema and discrete entropy conditions solutions of conservation laws satisfy the monotonicity property the number of local extrema is a non increasing function of time and local maximum minimum values decrease increase monotonically in time this paper investigates this property from a numerical standpoint we introduce a class of fully discrete in space and time high order accurate difference schemes called generalized monotone schemes convergence toward the entropy solution is proven via a new technique of proof assuming that the initial data has a finite number of extremum values only and the flux function is strictly convex we define discrete paths of extrema by tracking local extremum values in the approximate solution in the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum as a corollary we obtain a proof of convergence for a muscl type scheme that is second order accurate away from sonic points and extrema

convergence rate of approximate solutions to weakly coupled nonlinear systems we study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms such weakly coupled 2 x 2 systems appear for example in the context of resonant waves in gas dynamics equations this work is an extension of our previous scalar analysis this analysis asserts that a one sided lipschitz condition oslc or lip stability together with w 1 1 consistency imply convergence to the unique entropy solution moreover it provides sharp convergence rate estimates both global quantified in terms of the w s p norms and local we focus our attention on the lip stability of the viscosity regularization associated with such weakly coupled systems we derive sufficient conditions interesting for their own sake under which the viscosity and hence the entropy solutions are lip stable in an appropriate sense equipped with this we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of lip stability

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows

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