periodic schwarz christoffel mappings with multiple boundaries per period we present an extension to the theory of schwarzchristoffel sc mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal straigh

chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

conformal mapping based coordinate generation method for flows in periodic configurations abstract a conformal mapping based coordinate generation method for periodic configurations has been developed a channel of an arbitrary shape in the physical plane representing internal configurations is mapped into a straight channel in the computational plane and a wall of an arbitrary shape in the physical plane representing external configurations is mapped into a straight wall in the computational plane the parameters of the transformations have to be determined through a method of successive approximations a simple iteration scheme that converges quite rapidly even with a poor initial guess is presented solutions for different configurations are displayed to illustrate the capabilities of the method

a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

conformal maps to multiply slit domains and applications by exploiting conformal maps to vertically slit regions in the complex plane a recently developed rational spectral method t w tee and l n trefethen siam j sci comput 28 2006 pp 1798 1811 is able to solve pdes with interior layer like behavior using significantly fewer collocation points than traditional spectral methods the conformal maps are chosen to enlarge the region of analyticity in the solution an idea which can be extended to other numerical methods based upon global polynomial interpolation here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries and apply them to some challenging differential equations

a fast conformal mapping algorithm with no fft abstract an algorithm is presented for the computation of a conformal mapping discretized on a non uniformly spaced point set useful for the numerical solution of many problems of fluid dynamics most existing iterative techniques both those having a linear and those having a quadratic type of convergence rely on the fast fourier transform fft algorithm for calculating a convolution integral which represents the most time consuming phase of the computation the fft however definitely cannot be applied to a non uniform spacing the algorithm presented in this paper has been made possible by the construction of a calculation method for convolution integrals which despite not using an fft maintains a computation time of the same order as that of the fft the new technique is successfully applied to the problem of conformally mapping a closely spaced cascade of airfoils onto a circle which requires an exceedingly large number of points if it is solved with uniform spacing

sound scattering by cylinders of noncircular cross section a conformal mapping approach a new method for predicting the far field scattered pressure due to a plane wave incident upon an infinitely long cylinder of noncircular cross section is presented the method referred to herein as the fourier matching method fmm involves conformally mapping the exterior and interior of a closed surface to a semi infinite strip this method is new in that the boundary conditions are satisfied using constraints described in the new angular variable the resultant formulation is numerically efficient much more so than the t matrix method under certain conditions for example and works well for both small and large deviations from a circular cross section as well as penetrable and impenetrable materials furthermore the basis functions generated in the calculation can also improve the efficiency of other methods such as the t matrix method example calculations are presented for elliptic square and three leaf clover cross sections for several types of boundary conditions in all cases the results compare extremely well with exact or high frequency asymptotic results

numerical conformal mapping based on the generalised conjugation operator an iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity the formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function the conventional approach is to use only one and ignore the other equation however the discrete version of the operator using the boundary element method bem leads to two non equivalent sets of linear equations forming an over determined system the generalised conjugation operator is introduced so that both sets of equations can be utilised and their least square solution determined without any additional computational cost a strategy largely responsible for the stability and efficiency of the proposed method numerical tests on various samples including problems with cracked domains suggest global convergence although this cannot be proved theoretically the computational efficiency appears significantly higher than that reported earlier by other investigators

a stable richardson iteration method for complex linear systems the richardson iteration method is conceptually simple as well as easy to program and parallelize this makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues we change the ordering of the relaxation parameters of a richardson iteration method proposed by eiermann niethammer and varga for the solution of such problems the new method obtained is shown to be stable and to have better convergence properties

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

a study of wave propagation in varying cross section waveguides by modal decomposition part i theory and validation the propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition the approach adopted is to construct two infinite first order differential equations for the components of the pressure and the velocity projected over the normal modes from these an infinite matricial riccati equation is derived for the impedance matrix these equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes the stiffness of the pressure velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide the method is checked using different examples where the solutions of the plane wave approximation or the finite element method are known results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare

multimodal admittance method in waveguides and singularity behavior at high frequencies this work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi trapped modes the coupled mode equations are obtained by projecting the helmholtz equation on the local transverse modes to solve this problem we integrate the riccati equation governing the admittance matrix dirichlet to neumann operator for many propagating modes i e at high frequencies the numerical integration of the riccati equation shows that the rule is that this matrix has quasi singularities associated to quasi trapped modes

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

diffusive transport across irregular and fractal walls diffusive transport of heat mass and momentum across walls with irregular and fractal geometry is discussed a configuration is considered in detail in which the transported variable diffuses across a two dimensional periodic irregular wall toward an overlying plane wall while the value of the variable over each wall is maintained at a constant value several families of self similar wall geometries with increasingly finer structure eventually leading to fractal shapes are considered in detail using an efficient numerical method that is based on conformal mapping the numerical procedure involves the iterative solution of a large system of nonlinear algebraic equations computed patterns of iso scalar contours reveal the precise effect of the shape size and total length of boundary irregularities on the local and total transport rates and illustrate the enhancement in transport efficacy with wall refinement the total rate of transport across walls with self similar irregularities is shown to be remarkably close to that across walls with random irregularities of same roughness height

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

the newton method for solving the theodorsen integral equation the newton method for the solution of the theodorsen integral equation in conformal mapping is studied one step of this method consists of solving a linear integral equation the solution of which is given explicitly as the result of a riemann hilbert problem quadratic convergence of the newton method is established under certain assumptions whereas in other methods a so called condition with 1 is required to hold our method converges also for 1 we will also present a numerical implementation in which the result of one step of the newton method is approximately by a vector in r2n which can be computed with 2n log n o n multiplications in comparison one step of the newton method for the discrete theodorsen equation requires o n3 multiplications

elliptic integrals and the schwarz christoffel transformation abstract the real elliptic integrals of the first and second kind in jacobi s normal form are computed efficiently using the convolution number in conjunction with the method of frobenius for this purpose certain treatments of the laurent series are included different regions of convergence on the real axis are determined and for each one a different series is developed the real elliptic integral of the third kind is solved within a limited parameter plane by the same method the integral of the schwarz christoffel transformation is solved in the complex variable by complex convolution number algebra using the unit disk as mapping region different regions of convergence of frobenius laurent and taylor series are determined to cover the whole disk the complex evaluation of the elliptic integral of the third kind is included a schwarz christoffel formula for an infinite periodic mapping is given the solutions for exterior interior periodic and cyclic polygons are separately treated examples of several polygon mappings are presented graphically and compared with previous numerically integrated solutions the parameter problem is solved by the newton raphson method using a quotient matrix as approximation for the jacobian matrix the coordinate relations are simplified by using an overdetermined system an exact analytical jacobian matrix is computed solving leibniz derivative of the schwarz christoffel integral and results are compared with the approximate quotient matrix method

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

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

iterative methods for discrete nonlinear riemann hilbert problems abstract let d denote the complex unit disk t d and let f t r 2 r be a given function the paper presents iterative methods for the computation of functions w u iv which are holomorphic in d and satisfy the boundary relation f t u t v t 0 for all t t it is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle t the proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution error estimates for the iterative solutions are given in the scale of the sobolev spaces wk2

the numerical solution of theodorsen integral equation abstract in this paper a numerical solution of the theodorsen integral equation is studied using an adequate quadrature formula which eliminates the singularity of the integral part of the theodorsen equation we obtain a system of nonlinear algebraical equations this system may be solved using a jacobi type method and the procedure can be easily implemented using a programming language with parallel facilities examples are given using ada eval and parallaxis a convergence result is established 1999 elsevier science ltd all rights reserved

applications of numerical conformal mapping abstract the application of conformal mapping methods to the solution of free surface flow problems is considered methods of numerical conformal mapping based on fourier series are extended to handle efficiently problems with time dependent boundaries they are shown to be practicable only for moderately distorted geometries extensions of the menikoff zemach method to breaking geometries are presented these latter methods are robust at quite large distortions but degrade prematurely in time dependent problems at amplitudes smaller than achieved by our recent vortex methods

lamb wave propagation in elastic waveguides with variable thickness the problem of lamb wave propagation in waveguides with varying height is treated by a multimodal approach the technique is based on a rearrangement of the equations of elasticity that provides a new system of coupled mode equations preserving energy conservation these coupled mode equations avoid the usual problem at the cut offs with zero wavenumber thereafter we define an impedance matrix that is governed by a riccati equation yielding a stable numerical computation of the solution incidentally the versatility of the multimodal method is exemplified by treating analytically the case of slowly varying guide and by showing how to get easily the green tensor in any geometry the method is applied for a waveguide whose height is described by a gaussian function and the energy conservation in verified numerically we determine the green tensor in this geometry

on the computation of the conjugate trigonometric rational function and on a related splitting problem we prove several splitting theorems related to the following one every continuous real trigonometric rational function of t with period 2 pi can be written as the real part of a rational function of z e it that is analytic for z leqq 1 as an application we obtain immediately a method for computing the conjugate function of a real trigonometric rational function

uber die anzahl der losungen der diskreten theodorsen gleichung discretization of the theodorsen integral equation t yields the discrete theodorsen equation t d a system of 2n nonlinear equations a so called condition may be fulfilled it is known that t has exactly one continuous solution this solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domaing it is also known that t d has one and only one solution if 0 and any domaing that fulfills an condition we will construct a domaing 1 in the neighbourhood ofg that fulfills a max 1 k condition such that t d forg 1 has an infinite number of solutions the underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of t d even in the case 1

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

multimodal method and conformal mapping for the scattering by a rough surface the scattering of a wave from a periodic rough surface in two dimensions is considered the proposed method is based on the use of a conformal mapping and of the multimodal admittance method the use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface then the multimodal admittance method reduces this problem to a riccati equation for the modal admittance matrix which is solved numerically with a magnus mobius scheme the method is shown to converge exponentially with the number of fourier modes the method also allows to find geometries having trapped modes or quasi trapped modes at a given frequency by looking at singularities or quasi singularities of this riccati equation besides a simple perturbation expansion based on a small roughness approximation is developed

coupling scattering from the sea surface to a one way marching propagation model via conformal mapping validation propagation models in underwater acoustics usually incorporate sea surface scattering effects in an ad hoc manner which in most cases requires making severe approximations in particular to include in a coherent manner in a marching acoustic propagation model the scattering that occurs at a rough sea surface poses a serious problem dozier j acoust soc am 75 1415 1432 1984 introduced a rigorous approach in the framework of the split step parabolic equation model which used a sequence of conformal mappings to flatten segments of the sea surface locally each conformal mapping preserved the elliptic form of the wave equation in each transformed space the parabolic approximation is made and the solution advanced one range step the method has the attractive feature of handling surface roughness within a propagation model in a mathematically consistent manner including refraction and multiple surface interactions when and where they occur in this work the technique developed by dozier is impleme

superposition laws for solutions of differential matrix riccati equations arising in control theory representation formulas are given for the general solution of the n times n matrix riccati equation dot w a wb cw wdw using n known solutions with n 1 5 n representations the 5 representation is a superposition formula in that it expresses the general solution explicitly as a function of five particular solutions and n2arbitrary constants n geq 2 using no further information the representation formulas can be used in numerical calculations the 4 and 5 representations are specially useful when a solution w t has a singularity for some finite t t 0 they also clarify the properties of the solution space the matrix elements of w t are meromorphic functions of t having simple poles as the only possible singularities the relation between the representation formulas and previously known results is discussed

discretized versions of newton type iterative methods for conformal mapping abstract it is shown that for nearly circular regions discretized versions of the newton type methods of wegmann 1978 and hubner 1986 converge locally to fixed points convergence is linear the rates can be determined approximately for several standard regions the dominant operator acts only on a subspace of high order harmonics therefore under conditions of smoothness and or symmetry convergence can be much faster the fixed points satisfy a discrete version of lavrentev s variational principle therefore the resulting approximations for conformal mapping are essentially as accurate as wittich s approximation for the conjugation operation

global boundary flattening transforms for acoustic propagation under rough sea surfaces this paper introduces a conformal transform of an acoustic domain under a one dimensional rough sea surface onto a domain with a flat top this non perturbative transform can include many hundreds of wavelengths of the surface variation the resulting two dimensional flat topped domain allows direct application of any existing acoustic propagation model of the helmholtz or wave equation using transformed sound speeds such a transform model combination applies where the surface particle velocity is much slower than sound speed such that the boundary motion can be neglected once the acoustic field is computed the bijective one to one and onto mapping permits the field interpolation in terms of the original coordinates the bergstrom method for inverse riemann maps determines the transform by iterated solution of an integral equation for a surface matching term rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the helmholt

propagation modeling over terrain using the parabolic wave equation we address the numerical solution of the parabolic wave equation over terrain using the fourier split step approach the method referred to as a shift map generalizes that of beilis and tappert 1979 who introduced a coordinate transformation technique to flatten the boundary this technique is extended to a wide angle form allowing larger propagation angles with respect to the horizon a new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave pwe using the previously developed mixed fourier transform it is also shown by example that in many cases of interest the boundary may be approximated by discrete piecewise linear segments without affecting the field solution a more accurate shift map solution of the pwe for a piecewise linear boundary is therefore developed for modeling propagation over digitally sampled terrain data the shift map solution is applied to various surface types including ramps wedges curved obstacles and actual terrain where possible comparisons are made between the numerical solution and an exact analytical form the examples demonstrate that the shift map performs well for surface slopes as large as 10 15 spl deg and discontinuous slope changes on the order of 15 20 spl deg to accommodate a larger range of slopes it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well known terrain masking knife edge diffraction approximation

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

on fornberg s numerical method for conformal mapping it is shown that fornberg s method 1 is a discretized version of an iterative process which is essentially the same we introduced and studied in 7 8 the main difference consists in the determination of the shift u k in each iterative step while we give an explicit representation of u k fornberg determines u k from a linear equation which involves a certain integral operator r we prove some properties and estimates for this operator which explain several experimental results reported in 1 especially the fact that the conjugate gradient method is so efficient the results of 8 concerning the convergence can be applied also for fornberg s method finally we give estimates for the numerical error

stationary and almost stationary iterative k l step methods for linear and nonlinear systems of equations we first discuss the solution of a fixed point equationx x of a frechet differentiable self mapping by iterative methods of the general form x m left phi left sum limits i 0 m 1 x i gamma i m 1 right sum limits j 0 m 1 x j beta j m right frac 1 beta m m m 1 2 ldots defined by two infinite nonsingular upper triangular matricesb j m andc j m with column sums 1 we show that two such methods defined byb c and tilde b tilde c respectively are in a certain sense equivalent if and only if c 1 b tilde c 1 tilde b in particular hat b c 1 and the unit matrix replacingc define an equivalent so called semiiterative method we introduce k l step methods as those whereb andc have upper bandwidthk andl 1 respectively they require storing max k l previous iterates only for stationary methodsb andc have toeplitz structure except for their first row which is chosen such that the column sum condition holds an euler method which may require to store all iterates is equivalent to a stationary k l step method if and only if the underlying conformal mapg is a rational function of the formg w w 0 k k v 0 v l 1 w 1 by choosingg withg 1 1 such that for some onto the exterior of some continuums known to contain the eigen values of the frechet derivative of one obtains a feasible procedure for designing a locally converging stationary k l step method custom made for a set of problems in the case x tx d with a linear operatort where one wants to solve a linear system of equations we show that the residual polynomials of a stationary semiiterative method are generalized faber polynomials with respect to a particular weight function using another weight function leads to what we call almost stationary methods the classical chebyshev iteration is an example of such a method we define equivalent almost stationary k l step methods and give a corresponding convergence result

the schwarz christoffel mapping to bounded multiply connected polygonal domains a formula for the generalized schwarz christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived the theory of classical schottky groups is employed the formula for the derivative of the mapping function contains a product of powers of schottky klein prime functions associated with a schottky group relevant to the circular pre image domain the formula generalizes in a natural way the known mapping formulae for simply and doubly connected polygonal domains

on some relations among numerical conformal mapping methods abstract gaier and gutknecht have shown that many numerical methods for producing the conformal map from the unit disk to a simply connected region share a common theoretical basis as solutions of nonlinear integral equations arising from the hilbert transform of a function of the boundary correspondence we give a brief presentation of this classification and extend it somewhat to include some equations for the inverse correspondence such as those of menikoff and zemach noble and schwarz christoffel the use of explicit maps and the method of bisshopp are also brought into this framework an example illustrating the use of explicit maps is given

spectrally accurate algorithm for the analysis of flows in two dimensional vibrating channels a spectral algorithm based on the immersed boundary conditions ibc concept has been developed for the analysis of flows in channels bounded by vibrating walls the vibrations take the form of travelling waves of arbitrary profile the algorithm uses a fixed computational domain with the flow domain immersed in its interior boundary conditions enter the algorithm in the form of constraints the spatial discretization uses a fourier expansion in the stream wise direction and a chebyshev expansion in the wall normal direction use of the galileo transformation converts the unsteady problem into a steady one an efficient solver which takes advantage of the structure of the coefficient matrix has been used it is demonstrated that the method can be extended to more extreme geometries using the overdetermined formulation various tests confirm the spectral accuracy of the algorithm

numerical computation of schwarz christoffel transformation for simply connected unbounded domain abstract a direct numerical conformal mapping based on the modified kantorovich method is developed to transform the upper half plane onto a bounded polygon in the complex domain the accessary parameters in the schwarz christoffel formula are determined iteratively from solving a set of nonlinear equations which are in terms of the side length ratios of the polygon both pure and compound gauss jacobi quadratures are employed to evaluate the improper integrals and their first order derivatives with end point singularities in addition to the parameter determination the computational schemes for forward and inverse mapping which can be applied to the numerucal grid generation for computational fluid dynamics are also presented several examples are provided in order to demonstrate the advantage of this approach

a remark on rickard complexes abstract in this paper we characterize a rickard complex which induces a rickard equivalence between the block algebras of a block b and its brauer correspondent and whose vertices have the same order as defect groups of the block b the homology of such a rickard complex vanishes at all degree but degree q and the homology at degree q induces a basic morita equivalence between the block algebras in the sense of puig

the flattened surface parabolic equation a simple surface flattening transformation is applied to the problem of underwater acoustic scattering from an irregular sea surface the flattened surface elliptic wave equation is factored to obtain a parabolic equation that has the same high angle capability as recent parabolic equation models the flattened surface parabolic equation contains an additional surface slope term but includes forward scattering from the irregular sea surface the transformation creates additional volume irregularities and irregularities in the water sediment interface that the range and depth mesh need to resolve the surface flattening transformation is easy to implement in a parabolic equation model it compares very well with results obtained using boundary integral equations an example is given of scattering at 400 hz from a sea surface with wave heights of 3 m or nearly a wavelength

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

an iterative method for solving nonlinear riemann hilbert problems abstract a geometrically motivated linearization is used to construct an iterative method for solving nonlinear boundary value problems for holomorphic functions the nondiscrete method is quadratically convergent results of some test calculations are reported

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

a formula for optimal integration in h 2 abstract the weights j of the optimal integration formula q j jf zj in h2 for given integration points zj are the exact integrals of the cardinal functions in the corresponding formula for optimal evaluation by writing these cardinal functions as sums of their principal values we very easily obtain a closed formula for the weights in the case of real zj s this formula makes explicit a series formula of wilf we compare numerically the accuracy of the optimal formula with that of some well known integration formulae for points equidistant on a circle of radius r the formula allows an alternate derivation of a formula obtained by golomb we give also the barycentric formula for optimal evaluation with these points as well as an experimentally stable sequence of radii r for integrating with an increasing number of points

spectrally accurate algorithm for analysis of convection in corrugated conduits abstract an algorithm suitable for analysis of convection problems in corrugated conduits has been developed the algorithm uses the immersed boundary conditions ibc concept to capture the effects of corrugated walls the field equations are discretized on a regular domain surrounding the flow domain using fourier expansions in the stream wise direction and chebyshev expansions in the wall normal direction the boundary conditions are expressed in the form of constraints and the spectrally accurate discretization of these constraints is discussed the solution is carried out using the fix flow rate constraint discretization as well as a method for the direct inclusion of this constraint in the flow solver is presented several tests are used to demonstrate the spectral accuracy of the solution the solution process relies on iterations with an efficient solver taking advantage of the structure of the coefficient matrix used to solve the linear system the limitations of the algorithm are discussed and gains associated with the over determined formulation are presented

optimal boussinesq model for shallow water waves interacting with a microstructure in this paper we consider the propagation of water waves in a long wave asymptotic regime when the bottom topography is periodic on a short length scale we perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated we obtain explicit expressions for the coefficients of the resulting effective korteweg de vries kdv equations we show that it is possible to choose the free parameter of the reduced model so as to match the kdv limits of the full and reduced models hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem which has a microstructure we also discuss the impact of the rough bottom on the effective wave propagation in particular nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case

an iterative method for the conformal mapping of doubly connected regions an iterative method is described for the determination of the conformal mapping of a circular annulus rq z q z 1 onto a doubly connected region g with smooth boundary curves the method is analogous to the method of 7 for simply connected regions it requires in each step the solution of a riemann hilbert problem on rq this problem can be solved explicitly in terms of a generalized conjugation operator k a degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step if the boundary curves of g admit parametrizations by functions with second derivatives which are holder continuous with exponent 0 1 then the method converges locally in the sobolev space w of 2 periodic absolutely continuous functions with square integrable derivatives the order of convergence is at least 1 the convergence is quadratic if the have lipschitz continuous second derivatives for the numerical implementation of the method the operator k can be approximated by a wittich type method which can be performed very effectively using fft a calculation with n 2m grid points on the computer requires storage of the order o n and computing time o mn the results of some test calculations are reported and compared with results of calculations with the method of theodorsen garrick

a curvilinear coordinate based split step parabolic equation method for propagation predictions over terrain propagation of radiowaves over irregular terrain and in an inhomogeneous atmosphere is solved by the parabolic equation method using the split step fourier algorithm on a terrain conformal mesh a piecewise continuous coordinate system is generated by the specification of 1 the terrain profile shape at discrete points and 2 an upper height the resulting mesh is conformal to the terrain at the lower boundary and gradually flattens off at the maximum height in addition to preserving the number of points on any vertical line between the terrain and the maximum height from one range step to another the coordinate transformation used in the paper produces a correction term in the refractive index whose gradient diminishes with height as a result the sampling requirements over steep terrain are relaxed when compared to the beilis tappert transformation formulation and results are given both for the horizontal and vertical polarizations

conformal mapping based coordinate generation method for channel flows abstract a conformal mapping based coordinate generation method for channel type configurations has been developed a channel of an arbitrary shape in the physical plane is mapped into a straight channel in the computational plane the parameters of the transformation have to be determined through a method of successive approximations a simple iteration scheme that converges quite rapidly even with a poor initial guess is presented solutions for different configurations are displayed to illustrate the capabilities of the method results of the tests show higher order accuracy of the method in the case of curved channels

convergence proofs and error estimates for an iterative method for conformal mapping the iterative method as introduced in 8 and 9 for the determination of the conformal mapping of the unit disc onto a domaing is here described explicitly in terms of the operatork which assigns to a periodic functionu its periodic conjugate functionk u it is shown that whenever the boundary curve ofg is parametrized by a function with lipschitz continuous derivative dot eta ne 0 then the method converges locally in the sobolev spacew of 2 periodic absolutely continuous functions with square integrable derivative if is in a holder classc 2 the order of convergence is at least 1 if is inc l 1 withl 1 0 1 then the iteration converges inc l for analytic boundary curves the convergence takes place in a space of analytic functions for the numerical implementation of the method the operatork can be approximated by wittich s method which can be applied very effectively using fast fourier transform the sobolev norm of the numerical error can be estimated in terms of the numbern of grid points it iso n 1 l if is inc l 1 ando exp n 2 if is an analytic curve the number in the latter formula is bounded by logr wherer is the radius of the largest circle into which can be extended analytically such that z 0 for z r the results of some test calculations are reported

the evaluation of the conjugate function of a periodic spline on a uniform mesh abstract we present an o n log n algorithm for the simultaneous evaluation at all n points of a uniform mesh of a function conjugate to a periodic spline the algorithm is based on the attenuation factor theory and makes use of the fft it can be applied to other function classes with known attenuation factors also for the approximation by functions conjugate to splines of odd degree we state several optimality criterions and error bounds

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

a characterization of nonchaotic continuous maps of the interval stable under small perturbations recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles some times this is possible even in the case when the trajectory is not asymptotically periodic or is e chaotic for some e 0 in certain cases the first property is stable under small perturbations this means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small in other words there are nonchaotic maps without chaos explosions in the paper we give a characterization of these maps along with some consequences using the main result it is possible to prove that generically the nonchaotic maps are stable asymptotic properties of trajectories have been intensively considered since the middle of the sixties when excellent papers by a n sarkovskii appeared the main reason is applicability of these results in mathematical modelling important are maps with trajectories having regular asymptotical behavior like asymptotical periodicity however this property can be easily destroyed by ar bitrarily small perturbations of the maps and the corresponding unstable models in connection with this there appeared some results guaranteeing the stability in 15 it is shown that every map having no periodic orbits of period dif ferent from 1 2 2 for some n with per nowhere dense is stable this result was generalized 16 to the maps with per nowhere dense which have all trajectories asymptotically periodic these maps can have cycles of period 2 for every n but no other cycles and are characterized e g by the property that per is closed 12 cf also 10

on the fast solution of a linear system arising in numerical conformal mapping abstract one step of the newton method for the discretized theodorsen equation in conformal mapping requires the solution of a certain 2n 2n system application of the gaussian algorithm costs o n3 arithmetic operations a o we present an algorithm which reduces the problem to the solution of three n n linear toeplitz systems these systems can be solved in o n log2n a o this is also the amount of work required by the whole algorithm

peruse a numerical treatment of rough surface scattering for the parabolic wave equation scattering of underwater acoustic signals from real ocean surfaces often does not fit into any of the classical theoretical approaches to the problem thus the need for a numerical approach is clear a novel method is presented that uses a sequence of conformal mappings to locally flatten successive segments of the surface which is assumed piecewise linear and frozen in time each conformal mapping preserves the elliptic form of the reduced wave equation so that in each transformed space the parabolic approximation can be made and the solution advanced one range step using the split step fourier algorithm this peruse pe rough surface numerical method is validated by exhibiting classically predicted bragg angle peaks for single scatter of a plane wave from a sinusoidal surface for multiple rough surface scattering in surface ducts peruse results are analyzed by resolution in depth into the virtual modes of labianca j acoust soc am 53 1137 1147 1973 the resulting modal intensities are then

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

numerical conformal mapping onto a rectangle with applications to the solution of laplacian problems abstract let f be the function which maps conformally a simply connected domain onto a rectangle r so that four specified points on are mapped respectively onto the four vertices of r in this paper we consider the problem of approximating the conformal map f and present a survey of the available numerical methods we also illustrate the practical significance of the conformal map by presenting a number of applications involving the solution of laplacian boundary value problems

schwarz christoffel methods for conformal mapping of regions with a periodic boundary abstract numerical conformal mapping methods for regions with a periodic boundary have been developed these methods are based on the generalized schwarz christoffel equation and can deal with boundary curves of arbitrary forms i e made up of one or more rectifiable jordan curves high order quadrature rules have been implemented in order to increase accuracy of the mapping this is of particular relevance to highly accurate grid generation techniques required by for example implementation of high order compact finite difference discretization schemes

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

a new rough surface parabolic equation program for computing low frequency acoustic forward scattering from the ocean surface the parabolic equation acoustic propagation program ram has been extended to handle a rough air water interface treated as a series of stair steps because much finer vertical spacing may be needed to resolve the interface than to propagate a low frequency acoustic wave accurately in the rest of the domain an option to use one vertical spacing near the surface and a coarser one in the rest of the ocean has been implemented the necessary galerkin approximations of derivatives on an unequally spaced grid have been worked out using computer algebra the new program rrsfc is efficient enough to compute the frequency spectrum of the field scattered off a moving surface by treating the surface as a sequence of frozen surfaces a comparison of one such spectrum with actual ocean data is provided in order to make a more detailed assessment of the program both the depth dependent pressure for individual surfaces and the frequency spectrum for sequences of surfaces are compared with the results of numerically exact integral equation calculations for a few constant sound speed test cases