the method of space time conservation element and solution element a new approach for solving the navier stokes and euler equations a new numerical framework for solving conservation laws is being developed this new framework differs substantially in both concept and methodology from the well established methods i e finite difference finite volume finite element and spectral methods it is conceptually simple and designed to overcome several key limitations of the above traditional methods a two level scheme for solving the convection diffusion equation u t a u x 2u x2 0 0 is constructed and used to illuminate major differences between the present method and those mentioned above this explicit scheme referred to as the a scheme has two independent marching variables unj and ux nj which are the numerical analogues of u and u x at j n respectively the a scheme has the unusual property that its stability is limited only by the cfl condition i e it is independent of also it can be shown that the amplification factors of the a scheme are identical to those of the leapfrog scheme if o and to those of the dufort frankel scheme if a o these coincidences are unexpected because the a scheme and the above classical schemes are derived from completely different perspectives and the a scheme does not reduce to the above classical schemes in the limiting cases the a scheme is extended to solve the 1d time dependent navier stokes equations of a perfect gas stability of this explicit solver also is limited only by the cfl condition in spite of the fact that it does not use i any techniques related to the high resolution upwind methods and ii any ad hoc parameter the current navier stokes solver is capable of generating highly accurate shock tube solutions particularly for high reynolds number flows shock discontinuities can be resolved within one mesh interval the inviscid 0 a scheme is reversible in time it also is neutrally stable i e free from numerical dissipation such a scheme generally cannot be extended to solve the euler equations thus the inviscid version is modified stability of this modified scheme referred to as the a scheme is limited by the cfl condition and 0 1 where is a special parameter that controls numerical dissipation moreover if 0 the amplification factors of the a scheme are identical to those of the leapfrog scheme which has no numerical dissipation on the other hand if 1 the two amplification factors of the a scheme become the same function of the courant number and the phase angle unexpectedly this function also is the amplification factor of the highly diffusive lax scheme note that because the lax scheme is very diffusive and it uses a mesh that is staggered in time a two level scheme using such a mesh is often associated with a highly diffusive scheme the a scheme which also uses a mesh staggering in time demonstrates that it can also be a scheme with no numerical dissipation the euler extension of the a scheme has stability conditions similar to those of the a epsiv scheme itself it has the unusual property that numerical dissipation at all mesh points can be controlled by a set of local parameters moreover it is capable of generating accurate shock tube solutions with the cfl number ranging from close to 1 to 0 022

on the hypersurface orbital varieties of sl n c abstract we study the structure of hypersurface orbital varieties of sl n c those that are hypersurfaces in the nilradical of some parabolic subalgebra and how information about this structure is encoded in the standard young tableau associated to it by the robinson schensted algorithm we present a conjecture for the exact form of the unique non linear defining equations of hypersurface orbital varieties and proofs of the conjecture in certain cases

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

improved block interface conditions for aeroacoustic problems with grid discontinuities aeroacoustic simulations are extremely sensitive to numerical errors to achieve a high order of accuracy optimized finite difference schemes such as the well known dispersion relation preservation schemes and multiblock structured meshes are often used challenges appear in communicating the flow information near grid discontinuities between neighboring blocks and sophisticated interface treatments are required the characteristic interface condition has been successful in applications in previous publications and yet limitations exist that are identified in the present study this paper proposes a new interface treatment based on the analogy between finite difference and finite volume discretizations one sided finite difference schemes are used to calculate the spatial derivatives in the normal direction and the residuals are reconstructed from a finite volume perspective applications to several benchmark cases show an encouraging improvement as compared to the previous characteristic interface

cartesian cut cell method for axisymmetric separating body flows a method for the calculation of unsteady axisymmetric compressible flows involving both fixed and separating bodies is presented the method uses a cartesian cut cell mesh approach and high resolution upwind finite volume scheme a stationary background cartesian mesh is generated on the computational domain with complex solid geometries represented by different types of cut cell solid bodies are allowed to move across the mesh using a finite volume scheme modified to deal with moving boundary problems the flow solver employed is a muscl hancock godunov type scheme in conjunction with an approximate riemann solver of the harten lax and van leer type for flow interfaces and an exact riemann solver for a moving piston for fixed or moving solid faces a cell merging technique is used to maintain numerical stability in the presence of arbitrarily small cut cells and to retain strict conservation at moving boundaries the method is applied to a muzzle blast flow and muzzle break problems involving both fixed and separating bodies

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

a cartesian grid finite volume method for the advection diffusion equation in irregular geometries we present a fully conservative high resolution finite volume algorithm for advection diffusion equations in irregular geometries the algorithm uses a cartesian grid in which some cells are cut by the embedded boundary a novel feature is the use of a capacity function to model the fact that some cells are only partially available to the fluid the advection portion then uses the explicit wave propagation methods implemented in clawpack and is stable for courant numbers up to 1 diffusion is modelled with an implicit finite volume algorithm results are shown for several geometries convergence is verified and the 1 norm order of accuracy is found to between 1 2 and 2 depending on the geometry and peclet number software is available on the web

orbital varieties in slnand the smith conjecture abstract let o be a nilpotent orbit in the lie algebra s l n c that is a class of nilpotent elements for conjugation bysln c let v be an orbital variety contained in o andpbe the largest parabolic subgroup ofsln c stabilizing v the smith conjecture asserts that v contains a denseporbit this is shown to fail in general and further those nilpotent orbits for which such a dense orbit exists are determined

regular article the space time conservation element and solution element method a new high resolution and genuinely multidimensional paradigm for solving conservation laws a new high resolution and genuinely multidimensional numerical method for solving conservation laws is being developed it was designed to avoid the limitations of the traditional methods and was built from ground zero with extensive physics considerations nevertheless its foundation is mathematically simple enough that one can build from it a coherent robust efficient and accurate numerical framework two basic beliefs that set the new method apart from the established methods are at the core of its development the first belief is that in order to capture physics more efficiently and realistically the modeling focus should be placed on the original integral form of the physical conservation laws rather than the differential form the latter form follows from the integral form under the additional assumption that the physical solution is smooth an assumption that is difficult to realize numerically in a region of rapid change such as a boundary layer or a shock the second belief is that with proper modeling of the integral and differential forms themselves the resulting numerical solution should automatically be consistent with the properties derived from the integral and differential forms e g the jump conditions across a shock and the properties of characteristics therefore a much simpler and more robust method can be developed by avoiding the explicit use of the above derived properties specifically to capture physics as fully as possible the method requires that i space and time be unified and treated as a single entity ii both local and global flux conservation in space and time be enforced and iii a multidimensional scheme be constructed without using the dimensional splitting approach such that multidimensional effects and source terms which are scalars can be modeled more realistically to simplify mathematics and broaden its applicability as much as possible the method attempts to use the simplest logical structures and approximation techniques specifically i it uses a staggered space time mesh such that flux at any interface separating two conservation elements can be evaluated internally in a simpler and more consistent manner without using a separate flux model ii it avoids the use of many well established techniques such as riemann solvers flux splittings and monotonicity constraints such that the complications and possibly even the limitations associated with them may be avoided and iii it avoids the use of special techniques that are not applicable to more general problems furthermore triangles in 2d space and tetrahedrons in 3d space are used as the basic building blocks of the spatial meshes such that the method i can be used to construct 2d and 3d nondissipative schemes in a natural manner and ii is compatible with the simplest unstructured meshes note that while numerical dissipation is required for shock capturing it may also result in annihilation of small disturbances such as sound waves and in the case of flow with a large reynolds number may overwhelm physical dissipation to overcome this difficulty two different and mutually complementary types of adjustable numerical dissipation are introduced in the present development

conservative smoothing on an adaptive quadrilateral grid the lax wendroff scheme can be freed of spurious oscillations by introducing conservative smoothing in this paper the approach is first tested in 1 d modeling equations and then extended to multidimensional flows by the finite volume method the scheme is discretized by a space splitting method on an adaptive quadrilateral grid the artificial viscosity coefficients in the conservative smoothing step are specially designed to capture slipstreams and vortices algorithms are programmed using a vectorizable data structure under which not only the flow solver but also the adaptation procedure is well vectorized the good resolution and high efficiency of the approach are demonstrated in calculating both unsteady and steady compressible flows with either weak or strong shock waves

lowner inequality of indefinite type abstract a selfadjoint involutive matrix j provides c n with an indefinite inner product x y jx y for a pair of j selfadjoint matrices a b the j order relation a j b is defined as ax x bx x for all x we will show that if a b are j selfadjoint matrices such that all eigenvalues of a b are real and contained in an interval then for any operator monotone function f t on the matrices f a f b are well defined by the riesz dunford integral and a j b f a j f b when j i and f t t 1 2 on 0 this is the classical lowner inequality

uniformly accurate finite difference schemes for p refinement the accuracy deficiency of high order conventional finite difference schemes in the high frequency range is attributed to the approximation property of a power polynomial interpolation over equispaced nodes to improve the numerical representation of high frequency solutions a new family of high order finite difference schemes is constructed by trigonometric polynomial interpolation as expected these new difference schemes can resolve a wider frequency range than conventional finite difference schemes with the same stencil in particular they remove the possibility of runge s phenomenon for uniform mesh the recently proposed optimized difference schemes are indeed types of mixed difference schemes which make a compromise between power and trigonometric polynomial interpolation

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

a study of finite difference approximations to steady state convection dominated flow problems abstract five different finite difference schemes first order upwind skew upwind second order upwind second order central differencing and quick approximating the convection terms in the transport equation with fluid motion have been studied three simple test problems are used to compare the performances by the five schemes for high cell peclet number flows they are also used to demonstrate the restraints on the accuracy of the numerical approximations set by the types of the boundary conditions by the presence of the source term in the flow region and by the skewness of the numerical grid lines the basic reasons behind the spurious oscillations in a numerical solution are studied among all five schemes studied the second order upwind is found to be in general the most satisfactory

designing an efficient solution strategy for fluid flows 1 a stable high order finite difference scheme and sharp shock resolution for the euler equations abstract we derive high order finite difference schemes for the compressible euler and navier stokes equations that satisfy a semidiscrete energy estimate and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution the formulation of the semidiscrete energy estimate is based on symmetrization of the equations a canonical splitting of the flux derivative vector and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate for the euler equations the symmetrization is designed such as to preserve the homogeneity of the flux vectors around discontinuities or sharp gradients refined grids are created on which the discrete equations are solved after adding artificial viscosity the positioning of the subgrids and computation of the viscosity are aided by a detection algorithm which is based on a multiscale wavelet analysis of the pressure grid function the wavelet theory provides easy to implement mathematical criteria to detect discontinuities sharp gradients and spurious oscillations quickly and efficiently as the detection algorithm does not depend on the numerical method used it is of general interest the numerical method described and the detection algorithm are part of a general solution strategy for fluid flows which is currently being developed by the authors and collaborators

on analogy and dissimilarity of dependence of stability on several parameters in flow simulations in the present paper we investigate the asymptotic behavior of numerical solutions in direct fluid simulations the incompressible navier stokes equations and the continuity equation are solved numerically by using the marker and cell method the model adopted in the present study is a flow around a two dimensional circular cylinder dependence of the unsteady structure of numerical solutions on several parameters are discussed by analyzing the behavior of numerical drag coefficient cd concretely we concentrate the dependence of bifurcation processes on the amplitude of second and fourth order viscosity terms and time increment though the numerical fourth order artificial viscosity has the stabilizing effect like the physical second order one the bifurcation processes are different furthermore it is clarified that adopting small time increment values does not always produce the reasonable results

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

multiphase dynamics in arbitrary geometries on fixed cartesian grids in this work a mixed eulerian lagrangian algorithm calledelafint eulerian lagrangian algorithm for interface tracking is developed further and applied to compute flows with solid fluid and fluid fluid interfaces the method is capable of handling fluid flows in the presence of both irregularly shaped solid boundaries and moving boundaries on a fixed cartesian grid the field equations are solved on the underlying fixed grid using a collocated variable pressure based formulation the moving boundary is tracked explicitly by the lagrangian translation of marker particles the moving boundary passes through the grid and the immersed boundary technique is used to handle its interaction with the underlying grid the internal solid boundaries are dealt with by using a cut cell technique particular attention is directed toward conservation and consistency in the vicinity of both phase boundaries the complex geometry feature has been tested for a variety of flow problems the performance of the immersed boundary representation is demonstrated in the simulation of newtonian liquid drops the combination of the two features is then employed in the simulation of motion of drops through constricted tubes the capabilities developed here can be useful for solving flow problems involving moving and stationary complex boundaries

optimized prefactored compact schemes the numerical simulation of aeroacoustic phenomena requires high order accurate numerical schemes with low dispersion and dissipation errors in this paper we describe a strategy for developing high order accurate prefactored compact schemes requiring very small stencil support these schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes the prefactorization strategy splits the central implicit schemes into forward and backward biased operators using fourier analysis we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components this ensures that when the forward and backward stencils are added the original central compact scheme is recovered to extend the resolution characteristic of the schemes an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh the resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth and eighth order compact schemes making them more suitable for high resolution numerical simulations in gas dynamics and computational aeroacoustics the efficiency accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems

on a reverse heinz kato furuta inequality abstract in the set up of minkowski spaces the schwarz inequality holds with the reverse inequality sign as a consequence the same occurs with the triangle inequality in this note extensions of this indefinite version of the schwarz inequality are presented namely a reverse heinz kato furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated

grid optimized dispersion relation preserving schemes on general geometries for computational aeroacoustics abstract in this paper we investigate the dispersion relation preserving property of a finite difference scheme on general geometries for computational aeroacoustics where nondispersive and nondissipative properties are of critical importance the analysis pertains to the application of the optimization algorithm of the dispersion relation preserving drp scheme in wave number space to the general geometries nonuniform cartesian and curvilinear grids in many computational aeroacoustics applications the drp schemes have often been favored for their accuracy and efficiency drp schemes however are implemented only on uniform cartesian grids practical problems in aeroacoustics however are seldom confined to uniform cartesian grids with the associated computing grids often being nonuniform cartesian or curvilinear grid optimized dispersion relation preserving godrp finite difference schemes are proposed based on optimization that gives finite difference equations locally the same dispersion relation as the original partial differential equations on the grid points in the nonuniform cartesian or curvilinear mesh this local dispersion relation preserving property guarantees global accuracy of numerical schemes in the wave number space over the full domain the basic idea behind mathematical formulations of godrp schemes is that the optimization in the wave number space is carried out not on the computational domain but on the physical domain because the properties of cartesian and curvilinear grids differ whether the coupling of the coordinate variables between the physical and computational domains exists or not different mathematical formulations are developed for each grid type to investigate the effectiveness of godrp schemes a sequence of benchmark problems is executed through many numerical test problems it is shown that the use of godrp schemes can broaden the application area of conventional drp schemes to aeroacoustic phenomena enhancing both the speed and accuracy of the computation using nonuniform cartesian or curvilinear grids

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically