odd order brauer manin obstruction on diagonal quartic surfaces abstract we determine the odd order torsion subgroup of the brauer group of diagonal quartic surfaces over the field of rational numbers we show that a non constant brauer element of odd order always obstructs weak approximation but never the hasse principle

a level set based eulerian method for paraxial multivalued traveltimes we apply the level set methodology to compute multivalued solutions of the paraxial eikonal equation in both isotropic and anisotropic metrics this paraxial equation is obtained from 2d stationary eikonal equations by using one of the spatial directions as the artificial evolution direction the advection velocity field used to move level sets is obtained by the method of characteristics therefore the motion of level sets is defined in a phase space and the zero level set yields the location of bicharacteristic strips in the reduced phase space the multivalued traveltime is obtained from solving another advection equation with a source term the complexity of the algorithm is o n3 logn in the worst case and o n3 in the average case where n is the number of the sampling points along one of the spatial directions numerical experiments including the well known marmousi synthetic model illustrate the accuracy and the efficiency of the eulerian method

a field space based level set method for computing multi valued solutions to 1d euler poisson equations we present a field space based level set method for computing multi valued solutions to one dimensional euler poisson equations the system of these equations has many applications and in particular arises in semiclassical approximations of the schrodinger poisson equation the proposed approach involves an implicit eulerian formulation in an augmented space called field space which incorporates both velocity and electric fields into the configuration both velocity and electric fields are captured through common zeros of two level set functions which are governed by a field transport equation simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data the averaged density is then resolved by the integration of the obtained f against the dirac delta function of two level set functions in the field space moreover we prove that such obtained averaged density is simply a linear superposition of all multi valued densities and the averaged field quantities are weighted superposition of corresponding multi valued ones computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method

3 d traveltimes and amplitudes by gridded rays seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium they can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite difference methods unfortunately this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used we propose a new method for computing traveltimes and amplitudes in 3 d media that is simple fast unconditionally stable and robust defining the slowness vector as p px py pz and assuming an isotropic medium the ray velocity v is related to the slowness vector by the relation v p p p rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane the components of the slowness vector are then interpolated to gridpoints on this next horizontal plane this is termed regriddi

multiple reflection and transmission phases in complex layered media using a multistage fast marching method traditional grid based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only however later arrivals can be numerous and of greater amplitude making them a potentially valuable resource for practical applications such as seismic imaging the aim of this paper is to introduce a grid based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media a finite difference eikonal solver known as the fast marching method fmm is used to propagate wavefronts from one interface to the next by treating each layer that the wavefront enters as a separate computational domain one obtains a refracted branch by reinitializing fmm in the adjacent layer and a reflected branch by reinitializing fmm in the incident layer to improve accuracy a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high several examples are presented which demonstrate the viability of the new method in highly complex layered media even in the presence of velocity variations as large as 8 1 and interfaces of high curvature wavefronts composed of many reflection and transmission events are tracked rapidly and accurately this is because the scheme retains the two desirable properties of a single stage fmm computational speed and stability local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost

a new eulerian method for the computation of propagating short acoustic and electromagnetic pulses a new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics the pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections the method has many of the advantages of lagrangian ray tracing but is completely eulerian typically using a uniform cartesian grid accordingly it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse also information describing the pulses which are treated as continuous surfaces can be available throughout the computational grid rather than only at isolated individual markers the method uses a new type of representation which we call dynamic surface extension the basic idea is to propagate or broadcast defining fields from each pulse surface through a computational grid these fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes such as amplitude thus the emphasis is on the dynamics of these propagating defining fields which obey only local eulerian equations at each node the dynamic surface extension representation can be thought of as dual to level set representation the defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces in this way the new method overcomes the inability of level set or eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other while still using only single valued variables propagating thin pulse surfaces in 1 d 2 d and 3 d that can reflect from boundaries and pass through each other are computed using the new method the method was first presented as a new general representation of surfaces filaments and particles by j steinhoff and m fan 1998 eulerian computation of evolving surfaces curves and discontinuous fields utsi preprint

index of fibrations and brauer classes that never obstruct the hasse principle abstract let x be a smooth projective variety over a number field with a fibration into varieties that either satisfy a certain condition we study the classes in the brauer group of x that never obstruct the hasse principle for x we prove that if the generic fiber has a zero cycle of degree d over the generic point then the brauer classes whose orders are prime to d do not play a role in the brauer manin obstruction as a result we show that the odd torsion brauer classes never obstruct the hasse principle for del pezzo surfaces of degree 2 certain k3 surfaces and kummer varieties

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

the brauer group of kummer surfaces and torsion of elliptic curves in this paper we are interested in computing the brauer group of k3 surfaces to an element of the brauer grothendieck group br x of a smooth projective variety x over a number field k class field theory associates the corresponding brauer manin obstruction which is a closed condition satisfied by k points inside the topological space of adelic points of x see 20 ch 5 2 if such a condition is non trivial x is a counterexample to weak approximation and if no adelic point satisfies this condition x is a counterexample to the hasse principle the computation of br x is thus a first step in the computation of the brauer manin obstruction on x let k be an arbitrary field with a separable closure k gal k k recall that for a variety x over k the subgroup br0 x br x denotes the image of br k in br x and br1 x br x denotes the kernel of the natural map br x br x where x x k k in 22 we showed that if x is a k3 surface over a field k finitely generated over q then br x br0 x is finite no general approach to the computation of br x br0 x seems to be known in fact until recently there was not a single k3 surface over a number field for which br x br0 x was known one of the aims of this paper is to give examples of k3 surfaces x over q such that br x br q we study a particular kind of k3 surfaces namely kummer surfaces x kum a constructed from abelian surfaces a let br x n denote the n torsion subgroup of br x section 1 is devoted to the geometry of kummer surfaces we show that there is a natural isomorphism of modules br x br a proposition 1 3 when a is a product of two elliptic curves the algebraic brauer group br1 x often coincides with br k see proposition 1 4 section 2 starts with a general remark on the etale cohomology of abelian varieties which may be of independent interest proposition 2 2 it implies that if n is an odd integer then for any abelian variety a the group br a n br1 a n is canonically isomorphic to the quotient of het a n by ns a n where ns a is the neron severi group corollary 2 3 for any n 1 we prove that br x n br1 x n is a subgroup of br a n br1 a n and this inclusion is an equality for odd n see

ordered upwind methods for static hamilton jacobi equations abstract we introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static hamilton jacobi equations with dirichlet boundary conditions standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation these techniques are of complexity o m log m where m is the total number of points in the domain we consider anisotropic test problems in optimal control seismology and paths on surfaces

fast phase space computation of multiple arrivals we present a fast general computational technique for computing the phase space solution of static hamilton jacobi equations starting with the liouville formulation of the characteristic equations we derive escape equations which are static time independent eulerian pdes they represent all arrivals to the given boundary from all possible starting configurations the solution is numerically constructed through a one pass formulation building on ideas from semi lagrangian methods dijkstra like methods for the eikonal equation and ordered upwind methods to compute all possible trajectories corresponding to all possible boundary conditions the technique is of computational order o n log n where n is the total number of points in the computational phase space domain any particular set of boundary conditions then is extracted through rapid post processing suggestions are made for speeding up the algorithm in the case when the particular distribution of sources is provided in advance as an application we apply the technique to the problem of computing first multiple and most energetic arrivals to the eikonal equation

an improved mesh generation scheme for the wavefront construction method wavefront construction is an effective tool for the rapid calculation of ray fields in anisotropic media the method explicitly tracks the propagation of a wavefront through a model mapping it to a computational mesh that is interpolated when accuracy criteria based on paraxial ray methods are violated takeoff angles are used often to define the initial ray directions but uniform sampling in the two angles leads to oversampling of the ray field in the direction of the axis such sampling can lead also to numerical instability associated with vanishing derivatives with respect to the azimuthal angle we suggest a new wavefront mesh definition using the cubed sphere mesh which is a coordinate system used to solve partial differential equations in spherical geometries when using this mesh ray directions are assigned by mapping points on a regular discretization of the faces of a cube surrounding the source to corresponding rays this scheme produces a nearly uniform distribution of rays with minimal effort and using the cubed sphere coordinates as ray parameters to calculate partial derivatives completely eliminates the singularities that arise when takeoff angles are used as ray parameters numerical results for quantities related to seismic amplitudes confirm that this new mesh does provide more stable and reliable results

big ray tracing different traveltimes are identified as viscosity solutions of the eikonal equation on locally selected domains called big rays big rays are constructed by an ad hoc algorithm based on a coarse ray tracing viscosity solutions in the big rays are computed using an upwind finite difference scheme this hybrid method allows an easy and fast computation of multivalued traveltime fields

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

computational high frequency waves through curved interfaces via the liouville equation and geometric theory of diffraction we construct a class of numerical schemes for the liouville equation of geometric optics coupled with the geometric theory of diffractions to simulate the high frequency linear waves with a discontinuous index of refraction in this work s jin x wen a hamiltonian preserving scheme for the liouville equation of geometric optics with partial transmissions and reflections siam j numer anal 44 2006 1801 1828 a hamiltonian preserving scheme for the liouville equation was constructed to capture partial transmissions and reflections at the interfaces this scheme is extended by incorporating diffraction terms derived from geometric theory of diffraction into the numerical flux in order to capture diffraction at the interface we give such a scheme for curved interfaces this scheme is proved to be positive under a suitable time step constraint numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

3 d traveltime computation using huygens wavefront tracing traveltime computation is widely used in seismic modeling imaging and velocity analysis the two most commonly used methods are ray tracing and numerical solutions to the eikonal equation eikonal solvers are fast and robust but are limited to computing only the first arrival traveltimes ray tracing can compute multiple arrivals but lacks the robustness of eikonal solvers we propose a robust and complete method of traveltime computation it is based on a system of partial differential equations which is equivalent to the eikonal equation but formulated in the ray coordinates system we use a first order discretization scheme that is interpreted very simply in terms of the huygens s principle our explicit finite difference solution to the eikonal equation solved in the ray coordinates system delivers both computational speed and stability since we use more than one point on the current wavefront at every time step the finite difference method has proven to be a robust alternative to conventional ray tracing while being faster and having a better ability to handle rough velocity media and penetrate shadow zones

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials we propose a level set method for the semiclassical limit of the schrodinger equation with discontinuous potentials the discontinuities in the potential corresponds to potential barriers at which incoming waves can be partially transmitted and reflected previously such a problem was handled by jin and wen using the liouville equation which arises as the semiclassical limit of the schrodinger equation with an interface condition to account for partial transmissions and reflections s jin x wen siam j num anal 44 2006 1801 1828 however the initial data are dirac delta functions which are difficult to approximate numerically with a high accuracy in this paper we extend the level set method introduced in s jin h liu s osher r tsai j comp phys 210 2005 497 518 for this problem instead of directly discretizing the delta functions our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow and hence can be solved much more easily using conventional high order discretization schemes two ideas are introduced here 1 the solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections for problems involving only complete transmission or complete reflection the method of jlot applies and is used in our simulations 2 a reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems this is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight which can be easily implemented numerically we carry out numerical experiments in both one and two space dimensions to verify this new algorithm

prestack wavefield approximations abstractthe double square root dsr relation offers a platform to perform prestack imaging using an extended single wavefield that honors the geometrical configuration between sources receivers and the image point or in other words prestack wavefields extrapolating such wavefields nevertheless suffers from limitations chief among them is the singularity associated with horizontally propagating waves i have devised highly accurate approximations free of such singularities which are highly accurate specifically i use pade expansions with denominators given by a power series that is an order lower than that of the numerator and thus introduce a free variable to balance the series order and normalize the singularity for the higher order pade approximation the errors are negligible additional simplifications like recasting the dsr formula as a function of scattering angle allow for a singularity free form that is useful for constant angle gather imaging a dynamic form of this dsr formula ca

good reduction of the brauer manin obstruction for a smooth and projective variety over a number field with torsion free geometric picard group and finite transcendental brauer group we show that only the archimedean places the primes of bad reduction and the primes dividing the order of the transcendental brauer group can turn up in the description of the brauer manin set ams subject classification 14f22 14g05 11g35 11g25

an eikonal based formulation for traveltime perturbation with respect to the source location traveltime calculations amount to solving the nonlinear eikonal equation for a given source location the relationship between the eikonal solution and its perturbations is analyzed with respect to the source location and a partial differential equation is developed that relates the traveltime field for one source location to that for a nearby source this linear first order equation in one form depends on lateral changes in velocity and in another form is independent of the velocity field and relies on second order derivatives of the original traveltime field for stable finite difference calculations this requires the velocity field to be smooth in a sense similar to ray tracing requirements our formulation for traveltime perturbation has several potential applications such that as traveltime calculation by source location perturbation velocity independent interpolation including datuming and velocity estimation additionally higher order expansions provide parameters necessary for gaussian beam computations

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

the phase flow method this paper introduces the phase flow method a novel accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations the method operates by initially constructing the phase map for small times using a standard ode integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow the computational complexity of building up the complete phase map is usually that of tracing a few rays in addition the phase flow method is provably and empirically very accurate once the phase map is available integrating the ode for initial conditions on the invariant manifold only makes use of local interpolation thus having constant complexity the paper develops applications in the field of high frequency wave propagation and shows how to use the phase flow method to 1 rapidly propagate wave fronts 2 rapidly calculate wave amplitudes along these wave fronts and 3 rapidly evaluate multiple wave arrival times at arbitrary locations

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

on the dimensions of certain ldpc codes based on q regular bipartite graphs an explicit construction of a family of binary low density parity check ldpc codes called lu 3 q where q is a power of a prime was recently given a conjecture was made for the dimensions of these codes when q is odd the conjecture is proved in this note the proof involves the geometry of a four dimensional 4 d symplectic vector space and the action of the symplectic group and its subgroups

kirchhoff migration using eikonal based computation of traveltime source derivatives abstractthe computational efficiency of kirchhoff type migration can be enhanced by using accurate traveltime interpolation algorithms we addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location we adopted a first order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently unlike methods that rely on finite difference estimations the accuracy of the eikonal based derivative did not depend on input source sampling for smooth velocity models the first order traveltime source derivatives enabled a cubic hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into kirchhoff antialiasing schemes we provided an implementation of the proposed method to first arrival traveltimes by modifying the fast marching eikonal solver severa

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy