the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

multidomain spectral solution of the euler gas dynamics equations abstract we present interface treatments for computing the euler gas dynamics equations by a multidomain chebyshev spectral collocation method these treatments are suitable for use at the corners of subdomains and for patched or overlapping subdomains they can be applied to interfaces placed in subsonic supersonic or transonic regions of a flow computed results indicate that the solutions are spectrally accurate can be more accurate and more efficient than a single domain calculation and that reflections of waves at interfaces are not significant

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

arma models prewhitening and minimum cross entropy the problem of spectral estimation on the basis of observations from a finite stretch of a stationary time series is considered in connection with knowledge of a prior estimate of the spectral density a reasonable posterior spectral density estimate would be the density that is closest to the prior according to some measure of divergence while at the same time being compatible with the data the cross entropy has often been proposed to serve as such a measure of divergence a correction of the original minimum cross entropy spectral analysis mcesa method of j e shore see ieee trans acoust speech signal process vol 29 p 230 7 1981 to traditional prewhitening techniques and to autoregressive moving average arma models is pointed out and a fast approximate solution of the minimum cross entropy problem is proposed the solution is in a standard multiplicative form that is the posterior is equal to the prior multiplied by a correction factor

optimizing the geometrical accuracy of curvilinear meshes this paper presents a method to generate valid high order meshes with optimized geometrical accuracy the high order meshing procedure starts with a linear mesh that is subsequently curved without taking care of the validity of the high order elements an optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two dimensional meshes but they prove computationally too costly for optimization purposes a fast estimate of the geometrical accuracy based on taylor expansions of the curves is introduced an unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes as measured by the standard hausdorff distance between the geometrical model and the mesh several examples illustrate the beneficial impact of this method on cfd solutions with a particular role of the enhanced mesh boundary smoothness

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

spectral stability of noncharacteristic isentropic navier stokes boundary layers building on the work of barker humpherys lafitte rudd and zumbrun in the shock wave case we study stability of compressive or shock like boundary layers of the isentropic compressible navier stokes equations with law pressure by a combination of asymptotic ode estimates and numerical evans function computations our analytical results include convergence of the evans function in the shock and large amplitude limits and stability in the large amplitude limit the first rigorous stability result for other than the nearly constant case for all gamma geqq 1 together with these analytical results our numerical investigations indicate stability for 1 3 for all compressive boundary layers independent of amplitude save for inflow layers in the characteristic limit not treated expansive inflow boundary layers have been shown to be stable for all amplitudes by matsumura and nishihara using energy estimates besides the parameter of amplitude appearing in the shock case the boundary layer case features an additional parameter measuring displacement of the background profile which greatly complicates the resulting case structure inflow boundary layers turn out to have quite delicate stability in both large displacement shock and large amplitude limits necessitating the additional use of a mod two stability index studied earlier by serre and zumbrun in order to decide stability

a computational methodology for the acoustic characterization of orifices under grazing flow conditions this paper deals with the numerical prediction of the aeroacoustic behavior of orifices under grazing flow conditions a hybrid computational aeroacoustics approach is adopted where the steady incompressible mean flow over the orifice is obtained from a reynolds averaged navier stokes rans simulation in a next step the mean flow variables are used to solve the linearized navier stokes equations using a runge kutta discontinuous galerkin method in this way the linear interaction mechanisms between the aerodynamic and acoustic fluctuations are studied which enables an aeroacoustic characterization of the orifice a methodology is presented involving a virtual impedance tube and two computations for each geometrical configuration one with the presence of a mean flow and one for a quiescent medium this allows to isolate the contribution of the mean flow to the orifice impedance the method is verified against theoretical models and experimental data from literature and is used to study the influence of orifice geometry variations such as the orifice length the plate thickness and the edge rounding on the mean flow contribution to the impedance

curved boundary treatments for the discontinuous galerkin method applied to aeroacoustic propagation a discontinuous galerkin method is applied to unstructured grids to simulate aeroacoustic propagation modeled by the linearized euler equations on triangular and tetrahedral elements with straight edges the quadrature free form of the discontinuous galerkin method is used in addition to the classical linear treatment of wall boundaries two treatments involving a second order representation of the geometry are presented the simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved elements overcomes this restriction the benefits of higher order treatments are also assessed for more realistic geometries namely a high lift airfoil and an elliptical muffler

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

stable spectral methods for conservation laws on triangles with unstructured grids this paper presents an asymptotically stable scheme for the spectral approximation of linear conservation laws defined on a triangle lagrange interpolation on a general two dimensional nodal set is employed and by imposing the boundary conditions weakly through a penalty term the scheme is proven stable in l 2 this result is established for a general unstructured grid in the triangle a special case for which the nodes along the edges of the triangle are chosen as the legendre gauss lobatto quadrature points is discussed in detail the eigenvalue spectrum of the approximation to the advective operator is computed and is shown to result in an o n 2 restriction on the time step when considering explicit time stepping c 1999 elsevier science s a all rights reserved

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations

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

a local refinement finite element method for one dimensional parabolic systems vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement galerkin finite element procedure piecewise linear polynomials are used for the spatial representation of the solution and the backward euler method is used for temporal integration a local error indicator based on an estimate of the local discretization error is used to control an adaptive feedback mesh refinement strategy where finer space time meshes are recursively added to coarser ones in regions where greater solution resolution is needed a posteriori estimates of the local discretization error are obtained by a p refinement procedure that uses piecewise quadratic hierarchic finite element approximations in space and trapezoidal rule integration in time superconvergence properties of the finite element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure further improvements in computational e

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

stability of detonation profiles in the znd limit confirming a conjecture of lyng raoofi texier zumbrun we show that stability of strong detonation waves in the znd or small viscosity limit is equivalent to stability of the limiting znd detonation together with stability of the viscous profile associated with the component neumann shock moreover on bounded frequencies the nonstable eigenvalues of the viscous detonation wave converge to those of the limiting znd detonation while on frequencies of order one over viscosity they converge to one over viscosity times those of the associated viscous neumann shock this yields immediately a number of examples of instability and hopf bifurcation of reacting navier stokes detonations through the extensive numerical studies of znd stability in the detonation literature

a stability index for detonation waves in majda s model for reacting flow abstract using evans function techniques we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in siam j math anal 32 2001 929 commun pure appl math 51 7 1998 797 yielding useful necessary conditions for stability here we carry out the analysis in the context of the majda model a simplified model for reacting flow the method is extended to the full navier stokes equations of reacting flow in g lyng one dimensional stability of detonation waves doctoral thesis indiana university 2002 g lyng k zumbrun stability of detonation waves preprint 2003 the resulting stability condition is satisfied for all nondegenerate i e spatially exponentially decaying weak and strong detonations of the majda model in agreement with numerical experiments of siam j sci statist comput 7u 1986 1059 and analytical results of commun math phys 204 3 1999 551 commun math phys 202 3 1999 547 for a related model of majda and rosales we discuss also the role in the znd limit of degenerate subalgebraically decaying weak detonation and for a modified bump type ignition function deflagration profiles as discussed in siam j math anal 24 1993 968 siam j appl math 55 1995 175 for the full equations

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

finite difference method for a combustion model we study a projection and upwind finite difference scheme for a combustion model problem convergence to weak solutions is proved under the courant friedrichs lewy condition more assumptions are given on the ignition temperature then convergence to strong detonation wave solutions or to weak detonation wave solutions is proved

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

stability of undercompressive viscous shock profiles of hyperbolic parabolic systems abstract extending to systems of hyperbolic parabolic conservation laws results of howard and zumbrun for strictly parabolic systems we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral evans function conditions for linearized stability established by mascia and zumbrun are also sufficient for linearized and nonlinear phase asymptotic stability yielding detailed pointwise estimates and sharp rates of convergence in l p 1 p

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

hopf bifurcation of viscous shock waves in compressible gas dynamics and mhd extending our previous results for artificial viscosity systems we show under suitable spectral hypotheses that shock wave solutions of compressible navier stokes and magnetohydrodynamics equations undergo hopf bifurcation to nearby time periodic solutions the main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in h s for data in h s we accomplish this by a novel energy estimate carried out in lagrangian coordinates interestingly this estimate is false in eulerian coordinates at the same time we greatly sharpen and simplify the analysis of the previous work

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

weak and strong detonation profiles for a qualitative model abstract in this article the unidirectional travelling wave fronts for a qualitative model in combustion are studied the model is considered for a simple r p kind of chemical reaction by proving a general theorem of ode s the weak and strong detonations for exothermic reactions are shown to exist the uniqueness of these waves are also considered

high frequency asymptotics and one dimensional stability of zel dovich von neumann doring detonations in the small heat release and high overdrive limits we establish one dimensional spectral or normal modes stability of zel dovich von neumann doring detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed verifying numerical observations made by erpenbeck in the 1960s the key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite regular perturbation problem and a careful high frequency analysis depending uniformly on model parameters the latter recovers and extends to arbitrary amplitudes the important result of high frequency stability established by erpenbeck by somewhat different techniques notably the techniques used here yield quantitative estimates well suited for numerical stability investigation

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

on the detonation of a combustible gas this paper is concerned with the existence of detonation waves for a combustible gas the equations are those of a viscous heat conducting polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas see 1 the reaction is assumed to be of the simplest form a b i e there is a single product and a single reactant the main result see theorem 2 1 is a rigorous existence theorem for strong and under certain conditions weak detonation waves for explicit ranges of the viscosity heat conduction and species diffusion coefficients in other words a class of admissible viscosity matrices is determined the problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points the proof employs topological methods including conley s index of isolated invariant sets

pointwise green function bounds for shock profiles of systems with real viscosity following the pointwise semigroup approach of zh mz 1 we establish sharp pointwise green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic parabolic systems of conservation laws of dissipative type under the necessary assumptions z 1 z 3 z 4 of spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the connecting profile and hyperbolic stability of the corresponding ideal shock of the associated inviscid system with no additional assumptions on the structure or strength of the shock these bounds are used in a companion paper mz 2 to establish nonlinear stability of small amplitude lax shocks of symmetrizable hyperbolic parabolic systems

nonlinear stability of strong detonation waves for a dissipative model we study the asymptotic behavior of solutions to a combustion model which contains the effect of convection viscosity chemical reaction and diffusion it is proved that if a perturbation of a travelling strong detonation wave is small then the solution to this system converges to a strong detonation wave which is the original one with a shift as t infinity the rate of convergence is also estimated c 1999 academic press

dynamics and stability of a weak detonation wave one dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable i e initially perturbed waves tend asymptotically to translated weak detonation waves this model system was derived as the low mach number limit of the one component reactive navier stokes equations by majda and roytburd siam j sci stat comput 43 1086 1118 1983 and its weak detonation waves have been numerically observed as stable the analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave the shock translation solves a nonlinear integral equation obtained by green s function techniques and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder the inverse operator of the convolution and detailed properties of the traveling wave reduce by monotonicity the remainder to a small l1 perturbation

an information theoretic spectral density an information theoretic spectrum which minimizes the kullback leibler 1951 information number subject to the first p 1 autocovariance terms is proposed the kl spectrum includes the maximum entropy spectrum and the arma autoregressive moving average spectrum as special cases a method is proposed for modifying a spectral estimate on the basis of the results for the kl spectrum so that the revised spectral estimate is more loyal to the given observations than the primary estimate

a time domain interpretation of the kl spectrum an information theoretic spectrum minimizing the frequency domain kullback leibler kl information number has been proposed by b s choi ieee trans acoust speech and signal processing vol 38 p 717 21 1990 based on the kl spectrum a method of modifying a spectrum estimate was suggested some numerical examples illustrated that the kl spectrum estimate is superior to the initial estimate i e the autocovariances obtained by the inverse fourier transformation of the kl spectrum estimate are closer to the sample autocovariances of the given observations than those of the initial spectrum estimate it is shown that a gaussian autoregressive process of order p is the closest in the time domain kullback leibler sense to independently and identically distributed normal random variables subject to the first p 1 autocovariance constraints a rationale for the existence of the kl spectrum is given

stability of noncharacteristic boundary layers in the standing shock limit we investigate one and multi dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave u x1 x 1 0 obtaining necessary conditions of i weak stability of the limiting shock ii weak stability of the constant layer u u lim z z and iii nonnegativity of a modified lopatinski determinant similar to that of the inviscid shock case for lax 1 shocks we obtain equally simple sufficient conditions for p shocks p 1 the situation appears to be more complicated using these results we determine the stability of certain gas dynamical boundary layers generalizing earlier work of serre zumbrun and costanzino humphreys nguyen zumbrun

global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data we prove the global existence of solutions of the navier stokes equations describing the dynamic combustion of a compressible exothermically reacting fluid and we study the large time behavior of solutions giving necessary and sufficient conditions for complete combustion in certain cases the adiabatic constants and specific heats of the burned product and unburned reactant fluids may differ and the initial data may be large and discontinuous

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

optimal runge kutta schemes for discontinuous galerkin space discretizations applied to wave propagation problems we study the performance of methods of lines combining discontinuous galerkin spatial discretizations and explicit runge kutta time integrators with the aim of deriving optimal runge kutta schemes for wave propagation applications we review relevant runge kutta methods from literature and consider schemes of order q from 3 to 4 and number of stages up to q 4 for optimization from a user point of view the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method two scenarios are defined in the first one the element size is totally free and a 8 stage fourth order runge kutta scheme is found to minimize a cost measure depending on both accuracy and stability in the second one the elements are assumed to be constrained to such a small size by geometrical features of the computational domain that accuracy is disregarded we then derive one 7 stage third order scheme and one 8 stage fourth order scheme that maximize the stability limit the performance of the three new schemes is thoroughly analyzed and the benefits are illustrated with two examples for each of these runge kutta methods we provide the coefficients for a 2n storage implementation along with the information needed by the user to employ them optimally

stability of large amplitude viscous shock profiles of hyperbolic parabolic systems we establish nonlinear l1 h3 lp orbital stability 2 p with sharp rates of decay of large amplitude lax type shock profiles for a class of symmetric hyperbolic parabolic systems including compressible gas dynamics and magnetohydrodynamics mhd under the necessary conditions of strong spectral stability i e a stable point spectrum of the linearized operator about the wave transversality of the profile and hyperbolic stability of the associated ideal shock this yields in particular together with the spectral stability results of 50 the nonlinear stability of arbitrarily large amplitude shock profiles of isentropic navier stokes equations for a gamma law gas as 1 the first complete large amplitude stability result for a shock profile of a system with real i e partial viscosity a corresponding small amplitude result was established in 53 54 for general systems of kawashima class by a combination of kawashima type energy estimates and pointwise green function bounds where the small amplitude assumption was used only to close the energy estimates here under the mild additional assumption that hyperbolic characteristic speeds relative to the shock are not only nonzero but of a common sign we close the estimates instead by use of a goodman type weighted norm 25 26 designed to control estimates in the crucial hyperbolic modes

hybrid fourier pseudospectral discontinuous galerkin time domain method for wave propagation the fourier pseudospectral time domain fourier pstd method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized euler equations lee but is limited to real valued frequency independent boundary conditions and predominantly staircase like boundary shapes this paper presents a hybrid approach to solve the lee coupling fourier pstd with a nodal discontinuous galerkin dg method dg exhibits almost no restrictions with respect to geometrical complexity or boundary conditions the aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of dg at the boundaries while keeping the efficient fourier pstd in the bulk of the domain the hybridization approach is based on conformal meshes to avoid spatial interpolation of the dg solutions when transferring values from dg to fourier pstd while the data transfer from fourier pstd to dg is done utilizing spectral interpolation of the fourier pstd solutions the accuracy of the hybrid approach is presented for one and two dimensional acoustic problems and the main sources of error are investigated it is concluded that the hybrid methodology does not introduce significant errors compared to the fourier pstd stand alone solver an example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach finally no instabilities were found during long time calculation using the current hybrid methodology on a two dimensional domain a method for wave propagation in fluid domains with complex geometries is presented the method combines fourier pstd in the bulk of the domain and dg at boundaries the main application of interest is atmospheric sound propagation in cities the method shows no significant additional error compare with a fourier pstd solver no indication of instability in long time calculations has been detected

spectrum estimation by iterative minimization of the i divergence measure a method for spectrum estimation either narrowband or broadband based on minimization of csiszar s 1975 i divergence measure is introduced the blurring effect of the observation window or antenna array is minimized by applying a nonlinear deconvolution procedure this algorithm has subsequently been shown to minimize the i divergence between the spectrum and its estimate subject to a non negativity constraint here the method is applied to the spectrum estimation problem for stationary processes a reblurring procedure is used to regularize the method simulations show that the method offers error performance comparable to that of music although the resolution performance is inferior the method is iterative allowing a tradeoff between resolution and error performance and can be implemented using fast fft hardware

transition to longitudinal instability of detonation waves is generically associated with hopf bifurcation to time periodic galloping solutions we show that transition to longitudinal instability of strong detonation solutions of reactive compressible navier stokes equations is generically associated with hopf bifurcation to nearby time periodic galloping or pulsating solutions in agreement with physical and numerical observation in the process we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated evans function and obtain the first complete nonlinear stability result for strong detonations of the reacting navier stokes equations in the limit as amplitude hence also heat release goes to zero the analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works

on the relation between the maximum entropy probability density function and the autoregressive model the problem of maximizing the entropy of an n variate random vector subject to constraints on the first p 1 autocovariance terms is examined it is shown that the maximum is achieved by the gaussian autoregressive process of order p satisfying the autocovariance constraints this solution provides burg s theorem about the maximum entropy spectral density as a special case

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

stable spectral methods on tetrahedral elements a framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented although most of the developments are of a general nature an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron in the first part the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the jacobi polynomials to problems beyond the line this allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and subsequently for the formulation of stable spectral schemes on such unstructured nodal sets the second part of this work is devoted to a discussion of weakly imposed boundary conditions and energy stable schemes are formulated for a wide class of problems exemplified by advection problems advection diffusion problems and linear symmetric hyperbolic systems finally in the third part issues related to computational efficiency and implementation of the schemes are discussed the spectral accuracy of the approximation is confirmed through an example and factorization methods for the efficient computation of derivatives on the general nodal sets within the d simplex are developed ensuring that the proposed schemes are competitive with tensor product based methods in this last part we also show that the advective operator results in an cal o n 2 restriction on the time step similar to that of spectral collocation methods employing a tensor product based approximation the performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain confirming the expected accuracy and stability

acoustic behaviour of elliptical chamber mufflers the present work studies in detail the acoustic behaviour of mufflers with elliptical cross section the effect of the chamber length and the location of the inlet and outlet ports is considered in addition the eccentricity of the ellipse is taken into account since it generates important discrepancies with respect to those results obtained for the case of circular chambers mainly when practical values of eccentricity are considered the analytical description of the acoustic behaviour of chambers with elliptical cross section has been obtained via the truncated modal superposition method first the problem of natural frequencies and mode shapes has been approached by considering the helmholtz equation in an elliptical domain the solution of which can be expressed by means of mathieu functions then the frequency response functions of the muffler have been evaluated through truncated modal superposition the results obtained are shown to compare well with those obtained from finite element calculations and experimental measurements in addition simple polynomial expressions have been developed in order to evaluate the cut off frequency of concentric and non concentric elliptical chambers which is an important feature since higher order modes start to propagate without attenuation above this frequency a simple polynomial expression is also developed to find the optimum location of the outlet port in order to extend the range of acoustic attenuation of the muffler the influence of the geometry on the acoustic performance and the propagation of elliptic modes is studied for two configurations of interest such as the expansion chamber and the reversing chamber muffler

spectral stability of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation in the korteweg de vries limit we study the spectral stability of a family of periodic wave trains of the korteweg de vries kuramoto sivashinsky equation tv v xv 3v 2 xv 4 xv 0 0 in the korteweg de vries limit 0 a canonical limit describing small amplitude weakly unstable thin lm ow more precisely we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non empty set of parameter space the main technical diculty

nonlinear stability of time periodic viscous shocks in order to understand the nonlinear stability of many types of time periodic travelling waves on unbounded domains one must overcome two main difficulties the presence of embedded neutral eigenvalues and the time dependence of the associated linear operator this problem is studied in the context of time periodic lax shocks in systems of viscous conservation laws using spatial dynamics and a decomposition into separate floquet eigenmodes it is shown that the linear evolution for the time dependent operator can be represented using a contour integral similar to that of the standard time independent case by decomposing the resulting green s distribution the leading order behavior associated with the embedded eigenvalues is extracted sharp pointwise bounds are then obtained which are used to prove that time periodic lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues the latter conditions hold for example for small oscillation time periodic waves that emerge through a supercritical hopf bifurcation from a family of time independent lax shocks of possibly large amplitude

conditional stability of unstable viscous shock waves in compressible gas dynamics and mhd extending our previous work in the strictly parabolic case we show that a linearly unstable lax type viscous shock solution of a general quasilinear hyperbolic parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 3 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by mascia and zumbrun the main new difficulty in the hyperbolic parabolic case is to construct an invariant manifold in the absence of parabolic smoothing

existence and stability of noncharacteristic boundary layers for the compressible navier stokes and viscous mhd equations for a general class of hyperbolic parabolic systems including the compressible navier stokes and compressible mhd equations we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical navier stokes boundary conditions our first main result using the abstract framework established by the authors in the companion work gues et al in j differ equ 244 309 387 2008 is to show that existence and stability of arbitrary amplitude exact boundary layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an evans function uniform evans stability whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with residual boundary conditions our second is to show that uniform evans stability for small amplitude layers is equivalent to evans stability of the limiting constant layer which in turn can be checked by a linear algebraic computation finally for a class of symmetric dissipative systems including the physical examples mentioned above we carry out energy estimates showing that constant and thus small amplitude layers always satisfy uniform evans stability this yields existence of small amplitude multi dimensional boundary layers for the compressible navier stokes and mhd equations for both equations these appear to be the first such results in the compressible case

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

existence of chapman jouguet detonation for a viscous combustion model in siam j appl math 41 1981 70 93 majda proposed a model for the interaction between chemical reactions and compressible fluid dynamics this model is a low mach number limit of the one component reactive navier stokes equations siam j appl math 43 1983 1086 1118 and was extended to the case where the diffusion coefficient is positive by larrouturou nonlinear anal 77 2001 405 418 in this paper the existence of a one dimensional chapman jouguet detonation wave or equivalently a heteroclinic orbit for the extended model is proven the proof is based on an application of topological arguments to a system of ordinary differential equations which is obtained from the partial differential equations describing the interaction

natural mode analysis of hollow and annular elliptical cylindrical cavities analytical solutions for natural frequencies and mode shapes of elliptical cylindrical acoustic cavities were obtained by solving the homogeneous wave equation in elliptical cylindrical co ordinates both hollow cylinder and annular cylinder cases were considered the characteristic equation was obtained in terms of mathieu functions natural frequencies and mode shapes were calculated for cavities of the size used as a typical hermetic compressor shell or a small automotive muffler the results were compared with those of circular cylindrical cavities which have the same cross sectional area and volume the influence of the eccentricity on the general characteristics of cylindrically shaped acoustic devices is discussed

a high order immersed boundary method for acoustic wave scattering and low mach number flow induced sound in complex geometries a new sharp interface immersed boundary method based approach for the computation of low mach number flow induced sound around complex geometries is described the underlying approach is based on a hydrodynamic acoustic splitting technique where the incompressible flow is first computed using a second order accurate immersed boundary solver this is followed by the computation of sound using the linearized perturbed compressible equations lpce the primary contribution of the current work is the development of a versatile high order accurate immersed boundary method for solving the lpce in complex domains this new method applies the boundary condition on the immersed boundary to a high order by combining the ghost cell approach with a weighted least squares error method based on a high order approximating polynomial the method is validated for canonical acoustic wave scattering and flow induced noise problems applications of this technique to relatively complex cases of practical interest are also presented

remarks on a model for combustion waves discussion d un modele unidimensionnel simplifie propose par majda pour l etude qualitative des ondes de combustion

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

stability of isentropic navier stokes shocks in the high mach number limit by a combination of asymptotic ode estimates and numerical evans function calculations we establish stability of viscous shock solutions of the isentropic compressible navier stokes equations with law pressure i in the limit as mach number m goes to infinity for any 1 proved analytically and ii for m 2 500 gamma in 1 2 5 or m 13 000 gamma in 2 5 3 demonstrated numerically this builds on and completes earlier studies by matsumura nishihara and barker humpherys rudd zumbrun establishing stability for low and intermediate mach numbers respectively indicating unconditional stability independent of shock amplitude of viscous shock waves for law gas dynamics in the range gamma in 1 3 other values may be treated similarly but have not been checked numerically the main idea is to establish convergence of the evans function in the high mach number limit to that of a pressureless or infinitely compressible gas with additional upstream boundary condition determined by a boundary layer analysis recall that low mach number behavior is formally incompressible

tetrahedralhpfinite elements we present a new discretisation for the incompressible navier stokes equations that extends spectral methods to three dimensional complex domains consisting of tetrahedral subdomains the algorithm is based on standard concepts ofhpfinite elements as well as tensorial spectral elements this new formulation employs a hierarchical modal basis constructed from a new apex co ordinate system which retains a generalised tensor product these properties enable the development of computationally efficienct algorithms for use on standard finite volume unstructured meshes a detailed analysis is presented that documents the stability and exponential convergence of the method and several flow cases are simulated and compared with analytical and experimental results

one dimensional stability of parallel shock layers in isentropic magnetohydrodynamics abstract extending investigations of barker humpherys lafitte rudd and zumbrun for compressible gas dynamics and freistuhler and trakhinin for compressible magnetohydrodynamics we study by a combination of asymptotic ode estimates and numerical evans function computations the one dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters shock amplitude strength of imposed magnetic field viscosity magnetic permeability and electrical resistivity for a law gas with 1 3 other values may be treated similarly but were not checked numerically depending on magnetic field strength these shocks may be of fast lax intermediate overcompressive or slow lax type however the shock layer is independent of magnetic field consisting of a purely gas dynamical profile in each case our results indicate stability interesting features of the analysis are the need to renormalize the evans function in order to pass continuously across parameter values where the shock changes type or toward the large amplitude limit at frequency 0 and the systematic use of winding number computations on riemann surfaces

theoretical and numerical structure for unstable one dimensional detonations the spatio temporal structure of unstable detonations in a single space dimension is studied through a combination of numerical and asymptotic methods a new high resolution numerical method for computing unstable detonations is developed this method combines the piecewise parabolic method ppm with conservative shock tracking and adaptive mesh refinement a new nonlinear asymptotic theory for the spatio temporal growth of instabilities is also developed this asymptotic theory involves a nonclassical hopf bifurcation because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free boundary problem for a nonlinear hyperbolic equation an interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio temporal mechanisms of nonlinear stability near the transition boundary in particular a quantitative qualitative explanation is developed for the experimentally observed instabilities

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

pointwise green function bounds and stability of combustion waves abstract generalizing similar results for viscous shock and relaxation waves we establish sharp pointwise green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both majda s model and the full reacting compressible navier stokes equations with artificial viscosity with general multi species reaction and reaction dependent equation of state under the necessary conditions of strong spectral stability i e stable point spectrum of the linearized operator about the wave transversality of the profile as a connection in the traveling wave ode and hyperbolic stability of the associated chapman jouguet square wave approximation notably our results apply to combustion waves of any type weak or strong detonations or deflagrations reducing the study of stability to verification of a readily numerically checkable evans function condition together with spectral results of lyng and zumbrun this gives immediately stability of small amplitude strong detonations in the small heat release i e fluid dynamical limit simplifying and greatly extending previous results obtained by energy methods by liu ying and tesei tan for majda s model and the reactive navier stokes equations respectively