a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

an optimized spectral difference scheme for caa problems in the implementation of spectral difference sd method the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes the errors incurred in using extrapolation and interpolation would result in instability on the other hand the difference between the left and right conserved variables at the edge interface will introduce dissipation to the sd method when applying a riemann solver to compute the flux at the element interface in this paper an optimization of the extrapolation and interpolation schemes for the fourth order sd method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers the optimized coefficients of the extrapolation and interpolation are presented and the dispersion error of the original and optimized schemes is plotted and compared an improvement of the dispersion error over the resolvable wavenumber range of sd method is obtained the stability of the optimized fourth order sd scheme is analyzed it is found that the stability of the 4th order scheme with chebyshev gauss lobatto flux points which is originally weakly unstable has been improved through the optimization the weak instability is eliminated completely if an additional second order filter is applied on selected flux points one and two dimensional linear wave propagation analyses are carried out for the optimized scheme it is found that in the resolvable wavenumber range the new sd scheme is less dispersive and less dissipative than the original scheme and the new scheme is less anisotropic for 2d wave propagation the optimized sd solver is validated with four computational aeroacoustics caa workshop benchmark problems the numerical results with optimized schemes agree much better with the analytical data than those with the original schemes

practical aspects of higher order numerical schemes for wave propagation phenomena this paper examines practical issues related to the use of compact difference based fourth and sixth order schemes for wave propagation phenomena with focus on maxwell s equations of electromagnetics an outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches transformed plane method and physical space differencing in the first technique the truncation error expansion for the sixth order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available physical space differencing formulas are derived for the five point stencil by expressing the coefficients in terms of local spacing ratios the order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes to ensure stability for a broad range of problems fourier analysis is employed to develop a single parameter family of up to tenth order tridiagonal based spatial filters the implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1 d cavity where they are employed to suppress a late time instability the paper concludes after demonstrating the application of the scheme to several 3 d canonical problems utilizing cartesian as well as curvilinear meshes published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

a solution to the stability issues with block norm summation by parts operators finite difference operators approximating first and second derivatives and satisfying a summation by parts sbp rule have been derived for the 4 t h 6 t h 8 t h and 10 t h order case by using the symbolic mathematics software maple the operators are based on block norms to avoid the curse of losing accuracy at the boundaries that are present for corresponding operators based on diagonal norms the reason why block norm operators have not been used in realistic applications before is related to the well known stability issues on curvilinear grids to avoid this problem an additional boundary stabilization is introduced that removes unstable eigenvalues without interfering with accuracy and stiffness the superior accuracy properties of the newly derived block norm sbp operators will be demonstrated for the second order wave equation in 1 d and for the compressible euler equations in complex 2 d geometries

interface conditions of finite difference compact schemes for computational aeroacoustics the interface condition provides flexibility to handle complex geometries on a set of structured grids however when used with a compact scheme to apply in computational aeroacoustics its accuracy and stability may be degraded seriously by the boundary treatment that arises at a grid interface to minimize this effect a modified characteristic condition is incorporated in the finite difference compact scheme by using an explicit difference form at the boundary the extension of this approach is naturally attained through the coordinate transformation of convective terms across an interface in multidimensions the validity of current development is demonstrated through a simulation of trailing edge noise generation from an airfoil at a low mach number

on the outflow conditions for spectral solution of the viscous blunt body problem the purpose of this paper is to study and identify suitable outflow boundary conditions for the numerical simulation of viscous supersonic hypersonic flow over blunt bodies governed by the compressible navier stokes equations with an emphasis motivated primarily by the use of spectral methods without any filtering the subsonic supersonic composition of the outflow boundary requires a dual boundary treatment for well posedness all compatibility relations modified to undertake the hyperbolic parabolic behaviour of the governing equations are used for the supersonic part of the outflow regarding the unknown downstream information in the subsonic region different subsonic outflow conditions in the sense of the viscous blunt body problem are examined a verification procedure is conducted to make out the distinctive effect of each outflow condition on the solution detailed comparisons are performed to examine the accuracy and performance of the outflow conditions considered for two model geometries of different surface curvature variations numerical simulations indicate a noticeable influence of pressure from subsonic portion to supersonic portion of the boundary layer it is demonstrated that two approaches for imposing subsonic outflow conditions namely 1 extrapolating all flow variables and 2 extrapolation of pressure along with using proper compatibility relations are more suitable than the others for accurate numerical simulation of viscous high speed flows over blunt bodies using spectral collocation methods

parallel implementation of block boundary value methods for odes the parallel solution of initial value problems for odes has been the subject of much research in the last thirty years and different approaches to the problem have been devised in this paper we examine the parallel methods derived by block boundary value methods bvms recently introduced for approximating hamiltonian problems here we restrict the analysis of the methods when applied to linear problems since their nonlinear parallel implementation deserves further study however for linear problems the methods can reach a high parallel efficiency some of these solvers can also be adapted for approximating continuous two point boundary value problems numerical tests carried out on a distributed memory parallel computer are reported

a parameter uniform implicit difference scheme for solving time dependent burgers equations abstract a numerical study is made for solving one dimensional time dependent burgers equation with small coefficient of viscosity burgers equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows for high coefficient of viscosity a number of solution methodology exist in the literature 6 7 8 9 and 14 but for the sufficiently low coefficient of viscosity the exist solution methodology fail and a discrepancy occurs in the literature in this paper we present a numerical method based on finite difference which works nicely for both the cases i e low as well as high viscosity coefficient the method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh the quasilinearzation process is used to tackle the non linearity an extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter to illustrate the method numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity

implicit high order accurate in space algorithms for the navier stokes equations high order accurate alternating direction implicit schemes have been developed these schemes can be used to obtain efe ciently high resolution numerical solutions of the compressible euler and navier stokes equations implicit solvers compatible with both centered and upwind schemes for the discretization of the convective terms arediscussed forcenteredspacediscretizationoftheconvectivee uxes fourth orderspatialaccuracyoftheimplicit operators is obtained at no additional computing cost by performing compact space differentiation this implicit compact scheme which requires block tridiagonal matrix inversions is used to obtain viscous and inviscid e ow solutions in curvilinear coordinates in addition high order accurate in space implicit operators are obtained for diagonalized schemes which require scalar matrix inversions both high order algorithms improve convergence characteristics for subsonic and transonic e ow compared to second order accurate in space implicit schemes high order accurate in space implicit algorithms compatible with upwind high resolution shock capturing total variation diminishing schemes are also developed third order accurate upwind biased diagonalized algorithms were found to be as accurate and more efe cient than the standard block tridiagonal implicit solvers

an asymptotically stable compact upwind biased finite difference scheme for hyperbolic systems asymptotic stability of high order finite difference schemes for linear hyperbolic systems is investigated using the nyquist criterion of linear system theory this criterion leads to a sufficient stability condition which is evaluated numerically a fifth order compact upwind biased finite difference scheme is developed which is asymptotically stable according to the nyquist criterion for linear 2x2 systems moreover this scheme is optimised with respect to its dispersion properties the suitability of the scheme for discretisation of the compressible navier stokes equations is demonstrated by computing inviscid and viscous eigensolutions of compressible couette flow

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains a finite difference time dependent numerical method for the wave equation supported by recently derived novel elliptic grids is analyzed the method is successfully applied to single and multiple two dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries the new grids have nearly uniform cell area j grids and nearly uniform grid line spacing email protected numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state the restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing as a consequence moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost also numerical solutions for wave problems inside annular regions of complex shapes are obtained the use of the new grids results in late time stability in contrast with other classical finite difference time dependent methods

toward the construction of a fourth order difference scheme for transient em wave simulation staggered grid approach a compact central difference approximation in conjunction with the yee 1966 grid is used to compute the spatial derivatives in maxwell s equations to advance the semi discrete equations the four stage runge kutta rk integrator is invoked this combination of spatial and temporal differencing leads to a scheme that is fourth order accurate conditionally stable and highly efficient moreover the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor an attribute not found in other higher order methods results are provided that quantify the spectral properties of the method simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type results from these simulations are compared with exact closed form solutions the agreement between these results is consistent with numerical analysis

fourth order difference methods for hyperbolic ibvps abstract in this paper we consider fourth order difference approximations of initial boundary value problems for hyperbolic partial differential equations we use the method of lines approach with both explicit and compact implicit difference operators in space the explicit operator satisfies an energy estimate leading to strict stability for the implicit operator we develop boundary conditions and give a complete proof of strong stability using the laplace transform technique we also present numerical experiments for the linear advection equation and burgers equation with discontinuities in the solution or in its derivative the first equation is used for modeling contact discontinuities in fluid dynamics the second one is used for modeling shocks and rarefaction waves the time discretization is done with a third order runge kutta tvd method for solutions with discontinuities in the solution itself we add a filter based on second order viscosity in case of the non linear burgers equation we use a flux splitting technique that results in an energy estimate for certain difference approximations in which case also an entropy condition is fulfilled in particular we shall demonstrate that the unsplit conservative form produces a non physical shock instead of the physically correct rarefaction wave in the numerical experiments we compare our fourth order methods with a standard second order one and with a third order tvd method the results show that the fourth order methods are the only ones that give good results for all the considered test problems

nonlinearly stable compact schemes for shock calculations in this paper the authors discuss the applications of high order compact finite difference methods for shock calculations the main idea is the definition of a local mean that serves as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while keeping the formal accuracy of the scheme for scalar conservation laws the resulting schemes can be proven total variation stable in one space dimension and maximum norm stable in multispace dimensions numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks the idea in this paper can also be applied to other implicit schemes such as the continuous galerkin finite element methods

convergence of the compact finite difference method for second order elliptic equations abstract in this paper we give a fourth order compact finite difference scheme for the general forms of two point boundary value problems and two dimensional elliptic partial differential equations pde s by decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix thus we prove the compact finite difference scheme converges with the fourth order of accuracy to solve the discretized block tri diagonal matrix equations of the two dimensional elliptic pde s we develop an efficient iterative method full multigrid method in our numerical experiments we compare the compact finite difference method with the finite element method and crank nicolson finite difference method the results show that the compact finite difference scheme is a highly efficient and accurate method

a high order compact difference algorithm for half staggered grids for laminar and turbulent incompressible flows abstract the paper presents a novel efficient and accurate algorithm for laminar and turbulent flow simulations the spatial discretisation is performed with help of the compact difference schemes up to 10th order for collocated and half staggered grid arrangements the time integration is performed by a predictor corrector approach combined with the projection method for pressure velocity coupling at this stage a low order discretisation is introduced which considerably decreases the computational costs it is demonstrated that such approach does not deteriorate the solution accuracy significantly following boersma b j 13 the interpolation formulas developed for staggered uniform meshes are used also in the computations with a non uniform strongly varying nodes distribution in the proposed formulation of the projection method such interpolation is performed twice it is shown that it acts implicitly as a high order low pass filter and therefore the resulting algorithm is very robust its accuracy is first demonstrated based on simple 2d and 3d problems an inviscid vortex advection a decay of taylor green vortices a modified lid driven cavity flow and a dipole wall interaction in periodic flow problems the first two cases the solution accuracy exhibits the 10th order behaviour in the latter cases the 3rd and the 4th order is obtained robustness of the proposed method in the computations of turbulent flows is demonstrated for two classical cases a periodic channel with re 395 and re 590 and a round jet with re 21 000 the solutions are obtained without any turbulence model and also without any explicit techniques aiming to stabilise the solution the results are in a very good agreement with literature dns and les data both the mean and r m s values are predicted correctly

quasi disjoint pentadiagonal matrix systems for the parallelization of compact finite difference schemes and filters this paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite difference schemes and filters based on domain decomposition the proposed approach allows a pentadiagonal banded matrix system to be split into quasi disjoint subsystems by using a linear algebraic transformation technique as a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo exchange process the proposed matrix transformation leads to new subdomain boundary sb compact schemes and filters that require three halo terms to exchange with neighboring subdomains the internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven point discretization stencils the new sb compact schemes and filters demand additional arithmetic operations compared to the original serial ones however it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils compared to earlier published results the proposed sb compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency the overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests

structure preserving algorithms for the two dimensional sine gordon equation with neumann boundary conditions abstract this paper presents two kinds of strategies to construct structure preserving algorithms with homogeneous neumann boundary conditions for the sine gordon equation while most existing structure preserving algorithms are only valid for zero or periodic boundary conditions the first strategy is based on the conventional second order central difference quotient but with a cell centered grid while the other is established on the regular grid but incorporated with summation by parts sbp operators both the methodologies can provide conservative semi discretizations with different forms of hamiltonian structures and the discrete energy however utilizing the existing sbp formulas schemes obtained by the second strategy can directly achieve higher order accuracy while it is not obvious for schemes based on the cell centered grid to make accuracy improved easily further combining the implicit midpoint method and the scalar auxiliary variable sav approach we construct symplectic integrators and linearly implicit energy preserving schemes for the two dimensional sine gordon equation respectively extensive numerical experiments demonstrate their effectiveness with the homogeneous neumann boundary conditions

direct numerical simulation of spatial transition to turbulence using fourth order vertical velocity second order vertical vorticity formulation a highly accurate algorithm has been developed to study the process of spatial transition to turbulence the algorithmic details of the direct numerical simulation dns of transition to turbulence in a boundary layer based on a formulation in terms of vertical velocity and vertical vorticity are presented issues concerning the boundary conditions are discussed the linear viscous terms are discretized using an implicit crank nicholson scheme and a low storage runge kutta method is used for the nonlinear terms for the spatial discretization fourth order compact finite differences have been used as these have been found to have better resolution compared to explicit differencing schemes of comparable order the number of grid points that are needed per wavelength is close to the theoretical optimum for any numerical scheme the resulting time discretized fourth order equations are split up into two second order equations resulting in helmholtz and poisson type equations the boundary conditions for the laplacian of the vertical velocity are determined using an influence matrix method a robust multigrid algorithm has been developed to solve the resulting anisotropic elliptical equations for the outflow boundary a buffer domain method which smoothly reduces the disturbances to zero in conjunction with parabolization of the navier stokes equations has been used the validation of the results for the dns solver is made both for linear and weakly nonlinear cases

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

convergence and stability of boundary value methods for ordinary differential equations a usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula if the formula has k steps k values are needed to obtain the discrete solution the continuous problem provides only the initial value it is customary to impose the additional k 1 conditions at the successive k 1 initial points however the class of methods obtained in this way suffers from heavy limitations summarized by the two dahlquist barriers it is also possible to impose the additional conditions at different grid points for example some conditions can be imposed at the initial points and the remaining ones at the final points the obtained methods called boundary value methods bvms do not have barriers whatsoever in this paper the question of convergence of bvms is discussed along with the linear stability theory some numerical examples on stiff test problems are also presented

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

strict stability of high order compact implicit finite difference schemes temporal or strict stability of approximation to pdes is much more difficult to achieve than the classical lax stability in this paper we present a class of finite difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability the approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part boundary conditions are imposed by using penalty like terms fourth and sixth order compact implicit finite difference schemes are constructed and analyzed computational efficacy of the approach is corroborated by a series of numerical tests in 1 d and 2 d scalar problems

constrained open mapping theorem with applications abstract we prove a generalization of graves open mapping theorem for a class of mappings which can be approximated at a reference point by a set valued one having particular properties the nonlinear mapping is restricted to a closed convex subset of a banach space we apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping a slight generalization of a sufficient condition by j klamka on so called constrained exact local controllability of nonlinear and semi linear dynamic systems is also proved

convergence and stability of multistep methods solving nonlinear initial value problems convergence and stability of initial and boundary value multistep methods are analyzed for a class of nonlinear problems satisfying a one sided lipschitz condition the linear multistep methods are recast to handle the numerical solution globally on the time interval this allows us to use the theory of toeplitz matrices to show that under suitable assumptions global properties of the exact solution are preserved by its numerical approximation in particular a new concept of stability which avoids the unpleasant passage to one leg methods is introduced

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

an accurate compact treatment of pressure for colocated variables we propose a new scheme for the pressure treatment in computations of incompressible flow using a colocated grid arrangement to avoid oscillations associated with the sparse non compact stencil we introduce a compact fourth order equivalent of this stencil and study its advantages over the classical second order averaging procedure

high order cyclo difference techniques the summation by parts energy norm is used to establish a new class of high order finite difference techniques referred to here as cyclo difference techniques these techniques are constructed cyclically from stable subelements and require no special numerical procedures near the boundaries when coupled with the simultaneous approximation term sat boundary treatment they are time asymptotically stable for an arbitrary hyperbolic system these techniques are similar to spectral multi domain techniques and are ideally suited for parallel implementation but they do not require special collocation points the principal focus of this work is on methods of sixth order formal accuracy or less however these methods could be extended in principle to any arbitrary order of accuracy

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

implementation of a high order compact finite difference lattice boltzmann method in generalized curvilinear coordinates abstract in this work the implementation of a high order compact finite difference lattice boltzmann method cfdlbm is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non uniform grids the incompressible form of the discrete boltzmann equation with the bhatnagar gross krook bgk approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates herein the spatial derivatives in the resulting lattice boltzmann lb equation in the computational plane are discretized by using the fourth order compact finite difference scheme and the temporal term is discretized with the fourth order runge kutta scheme to provide an accurate and efficient incompressible flow solver a high order spectral type low pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions flow non linearities and grid non uniformity all boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates the accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems a sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are unsteady couette flow and steady flow in a 2 d cavity with non uniform grid and steady and unsteady flows over a circular cylinder and the naca0012 hydrofoil at different flow conditions results obtained for the above test cases are in good agreement with the existing numerical and experimental results the study shows the present solution methodology based on the implementation of the high order compact finite difference lattice boltzmann method cfdlbm in the generalized curvilinear coordinates is robust efficient and accurate for solving steady and unsteady incompressible flows over practical geometries

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

an extension of a theorem of r datko to the case of non uniform exponential stability of linear skew product semiflows abstract in 1970 r datko proved that the trajectories of a c 0 semigroup t t t 0 on a hilbert space x exhibit an exponential decay if and only if they stay in l 2 r x datko s theorem was crucial in extending the lyapunov operator equation to the case of autonomous systems x a x with unbounded a extensions have been done to the case of uniform exponential stability of evolution families and more recently of lsps linear skew product semiflow the aim of the present paper is to extend datko s result to the general case of non uniform exponential stability of the lsps that does not necessarily possess a uniform exponential growth

compact fourth order finite volume method for numerical solutions of navier stokes equations on staggered grids the development of a compact fourth order finite volume method for solutions of the navier stokes equations on staggered grids is presented a special attention is given to the conservation laws on momentum control volumes a higher order divergence free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes three forms of the nonlinear correction for staggered grids are proposed and studied the accuracy of each approximation is assessed comparatively in fourier space the importance of higher order approximations of pressure is discussed and numerically demonstrated fourth order accuracy of the complete scheme is illustrated by the doubly periodic shear layer and the instability of plane channel flow the efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations the proposed scheme is highly accurate and efficient at the same level of accuracy the fourth order scheme can be ten times faster than the second order counterpart this gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost

high order finite volume schemes for the advection diffusion equation a high order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one dimensional and two dimensional advection diffusion equation evolution equations for the mean values of each control volume are integrated in time by a classical fourth order runge kutta since our work focuses on the behaviour of the spatial discretization the time step is chosen small enough to neglect the time integration error two dimensional interpolants are built by means of one dimensional interpolants it is shown that when the degree of the one dimensional interpolant q is odd the proper selection of a fixed stencil gives rise to centred schemes of order q 1 in order not to lose precision due to the change of stencil near boundaries the degree of the interpolants close to boundaries is raised to q 1 four test cases with small values of diffusion are integrated with high order methods it is shown that the spatial discretization of the advection diffusion equation with periodic boundary conditions leads to normal discretization matrices and asymptotic stability must be assured to bound the spatial discretization error once the asymptotic stability is assured by means of the spectra of the discretization matrix the spatial error is of the order of the truncation error however it is shown that the discretization of the advection diffusion equation with arbitrary boundary conditions gives rise to non normal matrices if asymptotic stability is assured the spatial order of steady solutions is of the order of the truncation error but for transient processes the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization the use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution copyright 2001 john wiley sons ltd

nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces abstract in this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew product flows in infinite dimensional spaces we treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families we show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary our results generalize the previous studies on this topic and provide as particular cases many interesting situations among which we mention the detection of the exponential trichotomy of general non autonomous systems

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

high order non uniform grid schemes for numerical simulation of hypersonic boundary layer stability and transition the direct numerical simulation of receptivity instability and transition of hypersonic boundary layers requires high order accurate schemes because lower order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields the main limiting factor in the application of high order schemes to practical boundary layer flow problems is the numerical instability of high order boundary closure schemes on the wall this paper presents a family of high order non uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary layer transition by using an appropriate grid stretching and clustering grid points near the boundary high order schemes with stable boundary closures can be obtained the order of the schemes ranges from first order at the lowest to the global spectral collocation method at the highest the accuracy and stability of the new high order numerical schemes is tested by numerical simulations of the linear wave equation and two dimensional incompressible flat plate boundary layer flows the high order non uniform grid schemes up to the 11th order are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge the steady and unsteady results show that the new high order schemes are stable and are able to produce high accuracy for computations of the nonlinear two dimensional navier stokes equations for the wall bounded supersonic flow

the asymptotic spectra of banded toeplitz and quasi toeplitz matrices toeplitz matrices occur in many mathematical as well as scientific and engineering investigations this paper considers the spectra of banded toeplitz and quasi toeplitz matrices with emphasis on n

explicit and implicit multidimensional compact high resolution shock capturing methods two families of explicit and implicit compact high resolution shock capturing methods for the multidimensional compressible euler equations for fluid dynamics are constructed some of these schemes can be fourth and sixth order accurate away from discontinuities for the semi discrete case their shock capturing properties are of the total variation diminishing tvd total variation bounded tvb total variation diminishing in the mean tvdm essentially nonoscillatory eno or positive type of scheme for 1d scalar hyperbolic conservation laws and are positive schemes in more than one dimension these higher order compact schemes require the same grid stencil per spatial direction as their second order noncompact cousins the added terms over the second order noncompact cousins involve extra vector additions but no added flux evaluations due to the construction these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1d extensions the extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the roe approximate riemann solver the generalized steger and warming flux vector splitting or the van leer type flux vector splitting modification to existing high resolution second or third order noncompact shock capturing computer codes is minimal high resolution shock capturing properties can also be achieved via a variant of the second order lax friedrichs numerical flux without the use of riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock capturing counterparts an efficient and compatible high resolution shock capturing filter for spatially fourth and sixth order classical compact and noncompact schemes is discussed the simplest extension to viscous flows can be achieved by using the standard fourth order compact or noncompact formula for the viscous terms

a fourth order accurate finite volume compact method for the incompressible navier stokes solutions abstract this paper presents a finite volume fourth order accurate compact scheme for discretization of the incompressible navier stokes equations in primitive variable formulation the numerical method of integrating the navier stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes the pressure velocity coupling is achieved via the coupled solution of the resulting system of equations the solution of the coupled set of equations is performed with an implicit newton krylov matrix free method for stationary problems for simulation of unsteady flows a standard fourth order runge kutta method was used for temporal discretization and the velocity pressure coupling was ensured at each stage also using the matrix free method several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method

optimised boundary compact finite difference schemes for computational aeroacoustics a set of optimised boundary closure schemes is presented for use with compact central finite difference schemes in computational aeroacoustics caa involving non trivial boundaries the boundary schemes are given in a form of non central compact finite differences they maintain fourth order accuracy a pentadiagonal matrix system and seven point stencil which the main interior scheme employs this paper introduces a new strategy to optimise the boundary schemes in the spectral domain and achieve the best resolution characteristics given a strict tolerance for the dispersion and dissipation errors the boundary schemes are derived from sophisticated extrapolation of solutions outside the domain the extrapolation functions are devised by combining polynomials and trigonometric series which contain extra control variables used to optimise the resolution characteristics the differencing coefficients of the boundary schemes are determined in association with the existing coefficients of the interior scheme which is also optimised through an improved procedure in this paper the accuracy of the proposed schemes is demonstrated by their application to caa benchmark problems

strong nonuniform behaviour a datko type characterization abstract we obtain a datko type characterization of several classes of a strong nonuniform exponential behaviour we consider separately the case of exponential contractions expansions and dichotomies in addition we consider both discrete and continuous time as a nontrivial application of our work we study the robustness property of strong nonuniform exponential contractions expansions and dichotomies

averaging theorems for the large time behavior of the solutions of nonautonomous systems abstract we obtain some averaging theorems for the large time behavior of an evolution family u t s t s 0 acting on a banach space it is known that if a trajectory u t 0 x 0 is asymptotically stable then its p mean tends to zero we will show here that if the uniformly weighted p means of all the trajectories starting on the unit sphere are bounded then u t s t s 0 is uniformly exponentially stable while the converse statement is a simple verification discrete time versions of this result are given also variants for the uniform exponential blow up are obtained thus we generalize some known results obtained by r datko a pazy and v pata

on the properties of matrices defining some classes of bvms it is known that the matrices defining the discrete problem generated by a k step boundary value method bvm have a quasi toeplitz band structure 7 in particular when the boundary conditions are skipped they become toeplitz matrices in this paper by introducing a characterization of positive definiteness for such matrices we shall prove that the toeplitz matrices which arise when using the methods in the classes of bvms known as generalized bdf and top order methods have such property

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

parallel solution of almost block diagonal systems on a hypercube a new tearing type approach toward the solution of almost block diagonal systems on distributed memory parallel computers is presented its arithmetical complexity is examined and compared with other existing approaches

high order transverse schemes for the numerical solution of pdes many existing numerical schemes for the solution of initial boundary value problems for partial differential equations can be derived by the method of lines the pdes are converted into a system of ordinary differential equations either with initial conditions longitudinal scheme or with boundary conditions transverse scheme in particular this paper studies the performance of the transverse scheme in combination with boundary value methods moreover we do not restrict the semi discretization by the usual first or second order finite difference approximations to replace the derivative with respect to time but we use high order formulae

meagre functions and asymptotic behaviour of dynamical systems a measurable function x j r x x a metric space is said to be c meagre if c x is non empty and for every closed set k x with k c x 1 k has 5nite lebesgue measure this concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems by way of motivation consider the initial value problem f x x 0 rn with f rn rn locally lipschitz and denote its unique solution by x t t de5ned on its maximal interval of existence i 0 let g rn r be continuous assume that x is a bounded solution of the initial value problem in which case i 0 then g x is uniformly continuous if in addition g x l1 then we may conclude by an elementary observation frequently referred to as barb alat s lemma 5 see also 18 section 21 lemma 1 that g x t 0 as t 0 since by boundedness x t approaches as t its non empty compact invariant limit set b we also conclude that b g 1 0 it follows that x t tends as t to the largest invariant subset of g 1 0 this is a short proof of theorem 1 2 of 6 the proof in 6 is based on arguments involving properties of the dow generated by the dieerential

on weak exponential stability of evolution operators in banach spaces abstract the aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in banach spaces variants for weak exponential stability of some well known results in uniform stability theory buse and niculescu 2009 1 daleckij and krein 1974 2 datko 1973 3 rolewicz 1986 7 stoica and megan 2009 8 are obtained

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

wavenumber extended high order upwind biased finite difference schemes for convective scalar transport this paper proposes some new wavenumber extended high order upwind biased schemes the dispersion and dissipation errors of upwind biased finite difference schemes are assessed and compared by means of a fourier analysis of the difference schemes up to 11th order upwind biased schemes are analyzed it is shown that both the upwind biased scheme of order 2n 1 and the corresponding centered differencing scheme of order 2nhave the same dispersion characteristics thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation the new second order wavenumber extended scheme is tested and compared with some well known schemes the range of wavenumbers that are accurately treated by the upwind biased schemes is improved by using additional constraints from the fourier analysis to construct the new schemes the anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber extended upwind biased finite difference schemes

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

symmetrized compact scheme for receptivity study of 2d transitional channel flow in many physical problems one must maintain symmetry of numerical schemes the symmetry arising due to geometry and or physical conditions defining the flow the direct simulation of transitional plane poiseuille flow presents one such case that motivated the present study it has been shown here using the method of analyzing non periodic problems in sengupta et al 2003 that compact and other high accuracy schemes display strong directionality of the algorithm and such methods cannot be used for direct simulation of the physical flow this prevalent but unacceptable situation for dns is rectified in the present work and a new scheme is introduced that prevents this asymmetry from contaminating results the simulated transitional flow past 2d channel here using the new scheme are in agreement with experimental results and other recent views of sub critical instabilities

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

a version of a theorem of r datko for nonuniform exponential contractions abstract we point out versions of a well known theorem of r datko for nonuniform exponential contractions more precisely we obtain theorems that characterize the nonuniform exponential stability and the nonuniform exponential dichotomy for nonuniform exponential contractions thus we generalize the results in datko 1972 8 and preda and megan 1985 14

the spatial resolution properties of composite compact finite differencing demand for spectral like spatial routines to resolve fine scale physics is easily satisfied by compact finite differencing commonly the lower order multi parameter families at and near non periodic boundaries are independently tuned to meet or exceed the high order resolution character of the field stencil unfortunately that approach quantifies a false influence of the boundary scheme on the resultant interior dispersive and dissipative consequences of the compound template knowing that each composite template owns three ingredients that define their numerical character only their formal accuracy and global stability have been properly treated in a coupled fashion the present work presents a companion means for quantifying the resultant spatial resolution properties the procedure particularly focuses on the multi parameter families used to diminish the dispersive and dissipative errors at the non periodic boundaries the process introduces a least squares technique of the target field stencil to optimize the free parameters of the boundary scheme application of the optimized templates to both the linear convection and burgers equations at a fictitious non periodic boundary showed major reductions of the predictive error

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

high order accurate modeling of electromagnetic wave propagation across media grid conforming bodies the maxwell equations contain a dielectric permittivity e that describes the particular media for homogeneous materials at low temperatures this coefficient is constant within a material however it jumps at the interface between different media this discontinuity can significantly reduce the order of accuracy of the numerical scheme we solve the maxwell equations with an interface between two media using a fourth order accurate algorithm we regularize the discontinuous dielectric permittivity by a continuous function either locally near the interface or globally in the entire domain we study the effect of this regularization on the order of accuracy for a one dimensional time dependent problem we then implement this for the three dimensional maxwell equations in spherical coordinates with appropriate physical and artificial absorbing boundary conditions we use fourier filtering of the high frequency modes near the poles to increase the time step

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

method of lines transpose an implicit solution to the wave equation we present a new method for solving the wave equation implicitly our approach is to discretize the wave equation in time following the method of lines transpose sometimes referred to as the transverse method of lines or rothe s method we differ from conventional methods that follow this approach in that we solve the resulting system of partial differential equations using boundary integral methods our algorithm extends to higher spatial dimensions using an alternating direction implicit adi framework thus we develop a boundary integral solution that is competitive with explicit finite difference methods both in terms of accuracy and speed however it provides more flexibility in the treatment of source functions and complex boundaries we provide the analytical details of our one dimensional method herein along with a proof of the convergence of our schemes in free space and on a bounded domain we find that the method is unconditionally stable and achieves second order accuracy a caveat of the analysis is the derivation of a unique and novel optimal quadrature method which can be viewed as a lax type correction

on exponential h expansiveness of semigroups of operators in banach spaces this paper introduces the concept of exponential h expansiveness for semigroups of nonlinear operators which is an extension of classical concept of exponential expansiveness following the idea of obtaining an unitary treatment for stability and expansiveness necessary and sufficient conditions for exponential h expansiveness are given as particular cases the variants for exponential expansiveness of some well known stability results due to datko pazy ichikawa rolewicz and neerven are obtained

higher order solution procedure for three dimensional nonideal magnetogasdynamics a high resolution scheme developed for use in multidisciplinary applications is applied to solve the equations governingthethree dimensionale owofanonidealcompressibleconductinge uidinamagnetice eld both4th order and 6th order spatially implicit schemes are considered in conjunction with up to 10th order e lters the fourthorderclassicalrunge kuttaschemeisemployedfortimeintegration theaccuracyandversatility ofthealgorithm and the accompanying code are explored by solving a wide variety of problems discretized with uniform as well as curvilinear grids with particular emphasis on the unique physical phenomena arising in conducting e uids these

on uniform exponential stability for skew evolution semiflows on banach spaces the aim of this paper is to give several characterizations for the property of stability for skew evolution semiflows on banach spaces thus we obtain generalizations of some well known results due to barbashin datko and rolewicz in the case of evolution equations in banach spaces a unified treatment in the uniform setting is provided

strict stability of high order compact implicit finite difference schemes the role of boundary conditions for hyperbolic pdes ii abstract this paper deals with the problem of systems of hyperbolic pdes in one and two space dimensions using the theory of part i 7