stochastic quasilinear viscoelastic wave equation with degenerate damping and source terms abstract in this paper we consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and source term we prove the blow up of solution for stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense

adaptive hybrid grid methods an adaptive refinement and redistribution method is presented for use on hybrid grids composed of prismatic pyramidal and tetrahedral elements the combined refinement and redistribution scheme minimizes the computational time and memory required for the solver and maximizes the numerical accuracy grid refinement is achieved via a priori source placement and also solution based splitting of the edges of the grid elements the prismatic and pyramidal elements are divided directionally and the tetrahedral elements may be refined isotropically or directionally the prismatic grid redistribution scheme attracts or repels points that are close to the wall surface so as to better capture boundary layers a method is developed to place newly created boundary nodes on the actual splined surface instead of the previous discretized surface this ensures that the grid refinement always represents the geometric model accurately several examples are presented to demonstrate the a priori source placement the solution based grid refinement and the prismatic grid redistribution

general decay of solutions of a viscoelastic equation abstract in this paper we consider the following viscoelastic equation u t t u 0 t g t u d 0 in a bounded domain and establish a general decay result which is not necessarily of exponential or polynomial type this work generalizes and improves earlier results in the literature

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

prismatic grid generation for three dimensional complex geometries the paper describes generation of semiunstructu red prismatic grids for three dimensional complex geometries a new method of generation is presented which couples both algebraic and elliptic steps this combination yields low computation cost as well as smoothness of the grid main advantages of the developed grid generator are its direct control of grid orthogonality and spacing as well as its generality for treatment of three dimensional geometries the method marches a surface consisting of triangular faces away from the body surface employment of prismatic elements is a relatively new approach toward three dimensional complex geometry numerical simulations of viscous flows a body surface is covered with triangles which gives the geometric flexibility whereas the structure of the mesh in the direction normal to the surface provides better resolution of the viscous regions an f 16a aircraft geometry was included in the applications to investigate efficiency and to demonstrate robustness of the method in handling relatively complex topologies grid generation time for the entire aircraft domain was of the order of minutes on a workstation

hybrid prismatic tetrahedral grid generation for viscous flow applications a method for the automatic generation of unstructured grids composed of tetrahedra and prisms is proposed the prismatic semistructured grid is generated around viscous boundary surfaces and covers viscous regions whereas the tetrahedral grid covers the rest of the computational domain the delaunay approach for tetrahedral grid generation is used the proposed prismatic grid is structured in directions normal to the boundary faces but the number of prisms generated from one boundary face is variable from face to face unlike conventional prismatic grid generators this technique works well even in regions of cavities and gaps the delaunay background grid generated for surface nodes serves as an efficient data structure to check possible intersections of prisms particular attention is given to the boundary constraining problem a robust algorithm for the boundary recovery by edge swapping followed by a direct subdivision of tetrahedra is used grid examples for internal and external flow problems of complex shapes demonstrate the efficiency of the method

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

surface boundary conditions for the numerical solution of the euler equations we consider the implementation of boundary conditions at solid walls in inviscid euler solutions by upwind finite volume methods we review some current methods for the implementation of surface boundary conditions and examine their behavior for the problem of an oblique shock reflecting off a planar surface we show the importance of characteristic boundary conditions for this problem and introduce a method of applying the classical flux difference splitting of roe as a characteristic boundary condition consideration of the equivalent problem of the intersection of two equal and opposite oblique shocks was very illuminating on the role of surface boundary conditions for an inviscid flow and led to the introduction of two new boundary condition procedures denoted as the symmetry technique and the curvature corrected symmetry technique examples of the effects of the various surface boundary conditions considered are presented for the supersonic blunt body problem and the subcritical compressible flow over a circular cylinder dramatic advantages of the curvature corrected symmetry technique over the other methods are shown with regard to numerical entropy generation total pressure loss drag and grid convergence

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

an algebro geometric realization of equivariant cohomology of some springer fibers we give an explicit affine algebraic variety whose coordinate ring is isomorphic as a w algebra with the equivariant cohomology of some springer fibers

hybrid prismatic tetrahedral grid generation for viscous flows around complex geometries a method for the generation of hybrid prismatic tetrahedral grids is described for complex three dimensional geometries including multibody domains and employed for viscous flows around an aircraft the prisms cover the region close to each body s surface while tetrahedra are created elsewhere a special method is presented that allows the generation of single block nonoverlapping prismatic meshes even in complex geometries that contain narrow gaps and cavities examples of cases treated by this method include wing engine configurations as well as multi element wings the second development is a combined octree advancing front method for the generation of the tetrahedra of the hybrid mesh the main feature of this octree based tetrahedra generator is that it does not require the creation of a background mesh by the user for the determination of the grid spacing and stretching parameters these are determined via an automatically generated octree the hybrid grid generator is robust geometry independent and requires no user intervention the suitability of hybrid meshes for capturing viscous flow phenomena is demonstrated by simulation of viscous flows around a high speed civil transport type of aircraft configuration reduction in computer resources has been substantial allowing flow simulations to be performed on workstations rather than supercomputers

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star shaped domain we study the asymptotic stability and give the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type

optimal g closure bounds via stability under lamination optimal bounds for a two dimensional g closure problem are proved they are obtained by combining together bounds derived using fine properties of quasiconformal mappings and a stability property of the g closure which is known as stability under lamination

exponential energy decay of solutions of viscoelastic wave equations abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u p 1 u in 0 with initial conditions and dirichlet boundary condition under some appropriate assumption on g by introducing a potential well we establish the global existence result for the equation above furthermore we investigate explicit exponential energy decay of the viscoelastic wave equation under the potential well

overset adaptive cartesian prism grid method for stationary and moving boundary flow problems t he use of unstructured grids in computational fluid dynamics cfd has become widespread during the last two decades due to their ability to discretize arbitrarily complex geometries and the flexibility in supporting solution based grid adaptations to enhance the solution accuracy and efficiency 1 7 in the early days of unstructured grid development triangular tetrahedral grids were employed the most in dealing with complex geometries recently mixed or hybrid grids including many different cell types have gained popularity because of the improved efficiency and accuracy over pure tetrahedral grids for example hybrid prism tetrahedral grids 8 mixed grids including tetrahedral prism pyramid hexahedral cells 9 and adaptive cartesian grid methods 10 15 have been used in many applications with complex configurations the success demonstrated by unstructured grids for steady flow problems has prompted their applications to unsteady movingboundary flow problems a powerful approach for moving boundary flow problems is the overset chimera grid method 16 originally the chimera grid method was used to simplify domain decomposition for complex geometries using structured grids the method is particularly useful for moving boundary flow simulations because grid remeshing can be avoided 17 however frequent hole cutting and donor cell searching may be necessary to facilitate communications between the moving chimera grids with continuous improvement over the last decade and a half the chimera grid method has achieved tremendous success in handling very complex moving boundary flow problems more recently to further simplify the grid generation process unstructured grids are also used in achimera grid system for moving boundary flow computations making the approach even more flexible in handling complex geometries 18 in this technical note we advocate the use of an overset adaptive cartesian prism grid method for moving boundary flow computations themethod combines the advantage of the adaptive cartesian prism grid in geometry flexibility with that of the chimera approach in tacklingmoving boundary flowwithout grid remeshing there are several reasons why an adaptive cartesian grid is used for movingboundary problems 1 cartesian cells are more efficient at filling space given a certain length scale than triangular tetrahedral cells 2 searching operations can be performed very efficiently with the octree data structure 3 solution based and geometry based grid adaptations are straightforward to carry out the grid generation process is as follows body fitted prism grids are generated first near solid bodies to resolve viscous boundary layers using the advancing layer approach by marching the prism layer grid in the approximate surface normal directions the prism grid generation approaches and their limitations have been well researched in the literature in the past two decades for example in 6 8 18 19 and the current implementation follows similar ideas therefore interested readers should consult these references for prism grid generation methods and their limitations an adaptive cartesian grid is then generated to cover the outer domain and serve as the background grid for bridging the gaps between the prism grids the outer boundaries of the prism grids are used to generate holes in the adaptive cartesian grid to facilitate data communication if the bodies move the prism grids move with the bodies whereas the cartesian grid remains stationary after a few tens of time steps new holes are cut out of the cartesian grids and new donor cells are also identified solution fields are interpolated from the old cartesian grid to the new grid using a cellwise linear reconstruction technique this technical note is organized as follows in the next section the overset adaptive cartesian prism grid generation and holecutting approaches will be presented together with illustrative examples in sec iii several steady and unsteady moving boundary flow problems are computed grid refinement studies are performed to ensure the computational solutions are grid independent computational results are compared with experimental data and other simulations whenever possible finally conclusions from this study are summarized in sec iv

second order accurate boundary conditions for compressible flows the accuracy and the consistency of the methods commonly used for the numerical application of the boundary conditions on the surface of solid bodies immerged in an inviscid compressible flow is examined a new method is proposed for the application of second order accurate boundary conditions in the case of isentropic irrotational compressible flows the method is applied to a circular cylinder in uniform subsonic and low transonic flows the conservation law equations are solved using the second order accurate difference scheme of maccormack the results indicate a definite improvement over lower accuracy methods the method can be easily generalized to apply on three dimensional solid surfaces

existence and decay of solutions of a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u tt u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove a local existence theorem and show for certain initial data and suitable conditions on g and that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g

two block springer fibers of types c and d a diagrammatic approach to springer theory we explain an elementary topological construction of the springer representation on the homology of topological springer fibers of types c and d in the case of nilpotent endomorphisms with two jordan blocks the weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types b and d we determine the decomposition of the representations into irreducibles and relate our construction to classical springer theory as an application we obtain presentations of the cohomology rings of all two block springer fibers of types c and d moreover we deduce explicit isomorphisms between the kazhdan lusztig cell modules attached to the induced trivial module and the irreducible specht modules in types c and d

global existence and global nonexistence of solutions of the cauchy problem for a nonlinearly damped wave equation abstract in this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form formula

general decay rate estimates for viscoelastic dissipative systems abstract the linear viscoelastic equation is considered we prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity the proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates

blow up of positive initial energy solutions of a nonlinear viscoelastic hyperbolic equation abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u t m 2 u u p 2 with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g and for p m we prove that there are solutions with positive initial energy that blow up in finite time

an implicit form for the osher upwind scheme conservative upwind schemes for the euler equations such as the osher scheme accurately resolve flow discontinuities and correctly model the physics of the problem however these schemes require many more arithmetic operations per integration step than simple central difference schemes and hence result in large computing times an implicit version of the first order and second order accurate osher schemes in two spatial dimensions and generalized coordinates is developed in this study because implicit schemes permit the use of large integration steps in many cases they require fewer integration steps to reach steady state especially in calculations on grids with widely varying mesh cell sizes the implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered test cases include quasi one dimensional nozzle flow and supersonic flow past a cylinder

global nonexistence of positive initial energy solutions for a viscoelastic wave equation abstract in this paper we consider the nonlinear viscoelastic equation u t u t t u 0 t g t u d u t m 2 u t u p 2 u in 0 t with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g we prove the nonexistence of global solutions with positive initial energy

algebraic turbulence modeling for adaptive unstructured grids algebraic turbulence models are quite common in flow numerical simulations their simplicity and low computation cost as well as their applicability to specific classes of flows have made them attractive compared to the more expensive multiple equation models adaptive unstructured grid algorithms have become quite popular over the last years due to their flexibility in treating complex geometries as well their high resolution capability for local flow phenomena however application of algebraic turbulence models with unstructured grids proves to be nontrivial the difficulties associated with implementation of those models with such grids have generally impeded use of unstructured grids for turbulent flow simulations this paper presents a novel method of implementing an algebraic turbulence model with embedded unstructured grids the baldwinlomax model was chosen in order to test the implementation modifications to the application of the original model on structured grids are suggested as well comparisons with both experiments and other numerical simulations are employed in order to evaluate accuracy of the implementation method onsiderable progress has been made in the development of numerical methods for the solution of the navier stokes equations however most of those methods are impractical for complicated flows in a design environment the primary reason is poor efficiency which makes it difficult to obtain accurate results very fine resolution is needed which results in long computation times even with the use of available supercomputers in general the selection of the equations to be solved the scheme and the grid all are determined a priori by the user before starting the solution procedure and quite often some and even all of these factors must be modified by the user in order to improve the results the robustness of current numerical schemes as well as present computer capabilities has recently allowed a dramatic change in this philosophy general algorithms have been developed that are flexible enough to adaptively adjust those parameters during the solution procedure without intervention by the user one of the most efficient adaptive methods consists of local refinement by dividing grid cells an initially coarse grid is embedded in regions with large flow gradients e g boundary layers shocks wakes etc the algorithm senses high gradient regions and automatically divides grid cells in such regions this approach has been used for the resolution of shocks in flowfields described by euler equations 1 2 the embedded scheme was used as part of an adaptive algorithm for turbulent flows 3 6 algebraic turbulence models are quite common in aerodynamics numerical simulations their simplicity and low computation cost as well as their accuracy for specific classes of flows have made them attractive compared to the more expensive multiple equation models however such models require nonlocal information that characterizes an entire shearlayer profile the adaptive algorithm results in an unstructured mesh which is a major issue in the implementation of an algebraic turbulence model grid interfaces frequently interrupt normal to wall meshlines and therefore models that employ information from an entire shear layer profile cannot be applied similarly in the case of triangular meshes such normal grid lines generally do not exist a new method of

general decay of the solution energy in a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u t t u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove that for certain class of relaxation functions and certain initial data the rate of decay of energy is similar to that of g this result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered

boundary approximations for implicit schemes for one dimensional inviscid equations of gasdynamics the applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary value problems of the hyperbolic type for the numerical experiments the one dimensional inviscid gasdynamic equations in conservation law form are selected a class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space time extrapolations as implicit or explicit boundary schemes is emphasized some numerical examples with various inflow outflow conditions highlight the commonly discussed issues explicit vs implicit boundary schemes and unconditionally stable schemes hen finite difference schemes are used to solve initial boundary value problems for the equations of fluid dynamics it is well known that most methods require more boundary conditions than those required by the governing partial differential equations these additional boundary conditions for the finite difference equations are often called numerical boundary conditions the numerical boundary conditions cannot be imposed arbitrarily but are determined in general using interior information for example by extrapolation or uncentered approximations in this paper any numerical procedure used to provide a numerical boundary condition will be called a boundary scheme whatever schemes are used for the numerical boundary conditions it is a common practice to assume that the boundary scheme has a local effect and will not affect the solution globally during the early 1970s kreiss 1 2 osher 3 gustafsson et al 4 varah 5 and gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with boundary schemes included since then further progress has been made in the theory of linear difference approximations for initial boundary value problems of the hyperbolic and parabolic type 7 12 because improper treatment of the boundary conditions can lead to instability and inaccuracy even though we start with a stable interior scheme i e scheme for the interior points it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and boundary schemes surveys of recent developments and extensive bibliographies are included in papers by coughran13 and yee 14 the purpose of this paper is to examine the applicability to practical calculations for nonlinear gasdynamic problems of recent theoretical stability analyses of implicit difference approximations for initial boundary value problems of the hyperbolic type as numerical computations have progressed the use of the conservative form of the gasdynamic equations has gained popularity for physical reasons it is sometimes desirable to specify boundary conditions in the nonconservative variables and to compute with conservative variables in the interior we will consider the additional complications introduced by this procedure

octree advancing front method for generation of unstructured surface and volume meshes amethodisdescribedforthegeneration ofunstructuredgridsforcomplexthree dimensionalgeometriesincluding multibodydomains acombinedoctree advancing frontmethod ispresentedforgenerationoftheunstructured surface mesh as well as the tetrahedral volume mesh the unique feature of this generator is that a special octree provides the local length scales of the mesh which can vary signi cantly this octree is automatically created and eliminates the need for a user constructed background mesh to determine the grid spacing and stretching parameters the grid generator isrobust isgeometry independent and requires minimal userinteraction a techniqueis also presented for generation of anisotropic surface meshes that result in a signi cant reduction in the number of triangular faces automatic local remeshing is also employed to improve the quality of the tetrahedral mesh the geometry independence of the octree advancing front method is demonstrated through both surface and volume

exponential and polynomial decay for a quasilinear viscoelastic equation abstract we consider the following nonlinear viscoelastic equation u t u t t u u t t 0 t g t s u s d s 0 in a bounded domain for 0 we show that the dissipation induced by the integral term is strong enough to stabilize the solution this result improves an earlier one given by cavalcanti et al in m m cavalcanti v n d cavalcanti j ferreira existence and uniform decay for nonlinear viscoelastic equation with strong damping math methods appl sci 24 2001 1043 1053

blow up in a nonlinear viscoelastic wave equation with strong damping abstract in this paper we consider the nonlinear viscoelastic equation u t t u 0 t g t u d u t u p 2 u in 0 t with initial conditions and dirichlet boundary conditions for nonincreasing positive functions g we show the finite time blow up of some solutions whose initial data have arbitrarily high initial energy

curvature based wall boundary condition for the euler equations on unstructured grids a curvature corrected symmetry technique developed by dadone and grossman aiaa j vol 32 1994 for a structured grid euler solver has been extended to unstructured grids and to arbitrary curved boundaries in this paper the local curvature is estimated numerically with a robust procedure and is used successfully in the boundary treatment numerical results for 2d inviscid flows around circular and elliptic cylinders and the naca0012 airfoil with the new boundary condition showed dramatic improvements over those with a conventional non curvature corrected approach in all cases spurious entropy productions with the new boundary treatment are significantly reduced and sometimes by several orders of magnitude

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

euler equations implicit schemes and boundary conditions implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady euler equations this new boundary point treatment is based on the mathematical theory of characteristics for hyperbolic systems of equations along with the theoretical background the practical application of the method to several types of boundaries is also explained using several examples the specific boundary conditions covered include subsonic inflow and outflow surface tangency and shock waves the example problems include one dimensional laval nozzle flow dual throat rocket engine nozzle flow and supersonic flow past a sphere the implicit boundary treatment permits the use of large time steps allowing the finite difference algorithm to converge to the asymptotic steady state much faster than schemes that use explicitly applied boundary conditions at least an order of magnitude increase in computational speed is demonstrated in the examples shown background t he growing popularity of solutions to the euler equations in transonics and their continued application in supersonics have increased the need for quicker solutions the potential of implicit schemes in this direction has not been fully exploited for want of correct implicit application of boundary conditions the predominant use of implicit algorithms for the navier stokes equations has partly been responsible for the neglect of implicit boundary point treatment for the euler equations thus there is a need for correct and stable procedures for the easy implicit application of boundary conditions such methods will serve the two purposes of 1 reaching time asymptotic steady state faster and 2 permitting a time step for truly unsteady flow that is not necessarily restricted by the cfl stability criterion but is based upon the magnitude of the transients for clues and information on how to construct such boundary condition procedures one must turn to the mathematical theory of characteristi cs for hyperbolic systems of equations the unsteady euler equations belong to this category the theory for hyperbolic systems is rich with information on signal propagation directions the characteristics theory clearly points to the number of boundary conditions that may and need be prescribed without overdetermining the solution boundary condition procedures based on this theory have been known and applied for several years by kentzer 1 porter and coakley 2 de neef 3 and others in earlier work by this author 4 5 easily understood and implementable methods for boundary point treatment were presented however all of the above techniques were developed for explicit finite difference schemes it seems that it must be easy to extend such methodologies based on mathematical theory for hyperbolic systems to implicit finite difference schemes and indeed it is simple enough the rest of this paper describes such implicit boundary condition procedures the given examples illustrate in detail the application of the proposed methodology to specific types of boundaries and demonstrate the merits of the new scheme

general decay and blow up of solution for a quasilinear viscoelastic problem with nonlinear source abstract in this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous dirichlet boundary condition we prove that for certain class of relaxation functions and certain initial data in the stable set the decay rate of the solution energy is similar to that of the relaxation function this result improves earlier ones obtained by messaoudi and tatar s a messaoudi n e tatar global existence and uniform stability of solutions for a quasilinear viscoelastic problem math methods appl sci 30 2007 665 680 in which only the exponential and polynomial decay rates are considered conversely for certain initial data in the unstable set there are solutions that blow up in finite time the last result is new since it allows a larger class of initial energy which may take positive values