stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

the overlapped radial basis function finite difference rbf fd method a generalization of rbf fd abstract we present a generalization of the rbf fd method that computes rbf fd weights in finite sized neighborhoods around the centers of rbf fd stencils by introducing an overlap parameter 0 1 such that 1 recovers the standard rbf fd method and 0 results in a full decoupling of stencils we provide experimental evidence to support this generalization and develop an automatic stabilization procedure based on local lebesgue functions for the stable selection of stencil weights over a wide range of values we provide an a priori estimate for the speedup of our method over rbf fd that serves as a good predictor for the true speedup we apply our method to parabolic partial differential equations with time dependent inhomogeneous boundary conditions neumann in 2d and dirichlet in 3d our results show that our method can achieve as high as a 60 speedup in 3d over existing rbf fd methods in the task of forming differentiation matrices

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

long time behaviour for a class of low regularity solutions of the camassa holm equation in this paper we investigate the long time behaviour for a class of low regularity solutions of the camasssa holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks

rbf fd formulas and convergence properties the local rbf is becoming increasingly popular as an alternative to the global version that suffers from ill conditioning in this paper we study analytically the convergence behavior of the local rbf method as a function of the number of nodes employed in the scheme the nodal distance and the shape parameter we derive exact formulas for the first and second derivatives in one dimension and for the laplacian in two dimensions using these formulas we compute taylor expansions for the error from this analysis we find that there is an optimal value of the shape parameter for which the error is minimum this optimal parameter is independent of the nodal distance our theoretical results are corroborated by numerical experiments

meshfree methods on manifolds for hydrodynamic flows on curved surfaces a generalized moving least squares gmls approach abstract we utilize generalized moving least squares gmls to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds we use exterior calculus to formulate incompressible hydrodynamic equations in the stokesian regime and handle the divergence free constraints via a generalized vector potential this provides less coordinate centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth order governing equations in terms of a system of second order elliptic operators using a hodge decomposition we develop methods for manifolds having spherical topology we show the methods exhibit high order convergence rates for solving hydrodynamic flows on curved surfaces the methods also provide general high order approximations for the metric curvature and other geometric quantities of the manifold and associated exterior calculus operators the approaches also can be utilized to develop high order solvers for other scalar valued and vector valued problems on manifolds

fredholm determinants and the camassa holm hierarchy the equation of camassa and holm 2 2 is an approximate description of long waves in shallow water

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

a conservative semi lagrangian multi tracer transport scheme cslam on the cubed sphere grid a conservative multi tracer transport algorithm on the cubed sphere based on the semi lagrangian approach cslam has been developed the scheme relies on backward trajectories and the resulting upstream cells polygons are approximated with great circle arcs biquadratic polynomial functions are used for approximating the density distribution in the cubed sphere grid cells the upstream surface integrals associated with the conservative semi lagrangian scheme are computed as line integrals by employing the gauss green theorem the line integrals are evaluated using a combination of exact integrals and high order gaussian quadrature the upstream cell trajectories information and computation of weights of integrals can be reused for each additional tracer the cslam scheme is extensively tested with various standard benchmark test cases of solid body rotation and deformational flow in both cartesian and spherical geometry and the results are compared with those of other published schemes the cslam scheme is accurate robust and moreover the edges and vertices of the cubed sphere discontinuities do not affect the overall accuracy of the scheme the cslam scheme exhibits excellent convergence properties and has an option for enforcing monotonicity the advantages of introducing cross terms in the fully two dimensional biquadratic density distribution functions are also examined in the context of cartesian as well as the cubed sphere grid which has six local sub domains with discontinuous edges and corners

stochastic collocation methods on unstructured grids in high dimensions via interpolation in this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes to accomplish this we present the framework of least orthogonal interpolation which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions these interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space this feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes we present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm numerical benchmark problems are also presented to demonstrate the efficacy of the method

factorization problem on the hilbert schmidt group and the camassa holm equation in this paper we solve the camassa holm equation for a relatively large class of initial data by using a factorization problem on the hilbert schmidt group 2007 wiley periodicals inc

global well posedness in l 2 for the periodic benjamin ono equation we prove that the benjamin ono equation is globally well posed in h s bbb t for s ge 0 moreover we show that the associated flow map is lipschitz on every bounded set of h s 0 bbb t s ge 0 and even real analytic in this space for small times this result is sharp in the sense that the flow map if it can be defined and coincides with the standard flow map on h 0 infty bbb t cannot be of class c 1 alpha alpha 0 from h 0 s bbb t into h 0 s bbb t as soon as s 0

on choosing a radial basis function and a shape parameter when solving a convective pde on a sphere radial basis function rbf approximations have been used for some time to interpolate data on a sphere as well as on many other types of domains their ability to solve to spectral accuracy convection type pdes over a sphere has been demonstrated only very recently in such applications there are two main choices that have to be made i which type of radial function to use and ii what value to choose for their shape parameter denoted by e and with flat basis functions stretched out in the radial direction corresponding to e 0 the recent rbf qr algorithm has made it practical to compute stably also for small values of e results from solving a convective type pde on a sphere are compared here for many choices of radial functions over the complete range of e values from very large down to the limit of e 0 the results are analyzed with a methodology that has similarities to the customary fourier analysis in equispaced 1 d periodic settings in particular we find that high accuracy can be maintained also over very long time integrations we furthermore gain insights into why rbfs sometimes offer higher accuracy than spherical harmonics since the latter arise as an often non optimal special case of the former anticipated future application areas for rbf based methods in spherical geometries include weather and climate modeling

a radial basis function hermite finite difference approach to tackle cash or nothing and asset or nothing options abstract a novel local meshfree method is proposed for simulating cash or nothing and asset or nothing options at which the initial condition is discontinuous the novelty of the scheme is in the use of hermite radial basis function rbf interpolation on set of local nodes we also construct the rbf generated hermite finite difference hfd formulas and subsequently obtain their associated differentiation matrices dms the developed accuracy equipped with a shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

a semi lagrangian double fourier method for the shallow water equations on the sphere we describe a numerical method based on the semi lagrangian semi implicit approach for solving the shallow water equations swes in spherical coordinates the most popular spatial discretization method used in global atmospheric models is currently the spectral transform method which generates high order numerical solutions and provides an elegant solution to the pole problems induced by a spherical coordinate system however the standard spherical harmonic spectral transform method requires associated legendre transforms which for problems with resolutions of current interest have a computational complexity of o n3 where n is the number of spatial subintervals in one dimension thus the double fourier spectral method which uses trigonometric series may be a viable alternative the advantage of the double fourier method is that fast fourier transforms which have a computational complexity of o n2 log n can be used in both the longitudinal and latitudinal directions in this implementation the swes are discretized in time by means of the three time level semi lagrangian semi implicit method which integrates along fluid trajectories and allows large time steps while maintaining stability numerical results for the standard swes test suite are presented to demonstrate the stability and accuracy of the method

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

generalized affine lie algebras and the solution of a class of flows associated with the qr eigenvalue algorithm in 3 the authors show that the qr algorithm to compute the eigenvalues of a matrix is the integer time evaluation of a completely integrable hamiltonian system here the authors show that all the associated commuting integrals generate flows that can be solved explicitly via a factorization procedure on a suitable finite or infinite dimensional lie algebra

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies

an insight into rbf fd approximations augmented with polynomials abstract radial basis function generated finite differences rbf fd based on the combination of polyharmonic splines phs with high degree polynomials have recently emerged as a powerful and robust numerical approach for the local interpolation and derivative approximation of functions over scattered node layouts among the key features i high orders of accuracy can be achieved without the need of selecting a shape parameter or the issues related to numerical ill conditioning and ii the harmful edge effects associated to the use of high order polynomials better known as runge s phenomenon can be overcome by simply increasing the stencil size for a fixed polynomial degree the present study complements our previous results providing an analytical insight into rbf fd approximations augmented with polynomials it is based on a closed form expression for the interpolant which reveals the mechanisms underlying these features including the role of polynomials and rbfs in the interpolant the approximation error and the behavior of the cardinal functions near boundaries numerical examples are included for illustration

multiscale analysis in sobolev spaces on the sphere we consider a multiscale approximation scheme at scattered sites for functions in sobolev spaces on the unit sphere mathbb s n the approximation is constructed using a sequence of scaled compactly supported radial basis functions restricted to mathbb s n a convergence theorem for the scheme is proved and the condition number of the linear system is shown to stay bounded by a constant from level to level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points

short note a simple mass conserving semi lagrangian scheme for transport problems a mass conserving semi lagrangian sl scheme is achieved with a combination of a simple explicit smoothness based mass correction and a standard non conservative interpolating sl scheme the resulting mass correction can be incorporated into any existing sl scheme with negligible extra cost a more selective and less damping monotonicity filter by comparison to traditional filters is also presented results from various tests from the literature show that in addition to mass conservation the proposed scheme has negligible impact on the overall accuracy of the standard non conservative sl scheme

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

fast and accurate interpolation of large scattered data sets on the sphere in this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented the solution method is local and involves a modified spherical shepard s interpolant which uses zonal basis functions as local approximants the associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere specifically this technique is mainly based on the partition of the sphere in a suitable number of spherical zones the construction of spherical caps as local neighbourhoods for each node and finally the employment of a spherical zone searching procedure computational cost and storage requirements of the spherical algorithm are analyzed moreover several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

the orthogonal gradients method a radial basis functions method for solving partial differential equations on arbitrary surfaces much work has been done on reconstructing arbitrary surfaces using the radial basis function rbf method but one can hardly find any work done on the use of rbfs to solve partial differential equations pdes on arbitrary surfaces in this paper we investigate methods to solve pdes on arbitrary stationary surfaces embedded in r 3 using the rbf method we present three rbf based methods that easily discretize surface differential operators we take advantage of the meshfree character of rbfs which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension two out of the three methods which we call the orthogonal gradients ogr methods are the result of our work and are hereby presented for the first time

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

a semi lagrangian discontinuous galerkin method for scalar advection by incompressible flows a new conservative semi lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form the approach combines the accuracy and conservation properties of the discontinuous galerkin dg method with the computational efficiency and robustness of semi lagrangian sl techniques unconditional stability in the von neumann sense is proved for the proposed discretization in the one dimensional case a monotonization technique is then introduced based on the flux corrected transport approach this yields a multi dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard dg methods the accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests in the case of incompressible flows the comparison with results obtained by standard sl and dg methods highlights several advantages of the new technique

well posedness in energy space for the periodic modified benjamin ono equation abstract we prove that the periodic modified benjamin ono equation is locally well posed in the energy space h 1 2 this ensures the global well posedness in the defocusing case the proof is based on an x s b analysis of the system after a gauge transform

vorticity divergence semi lagrangian shallow water model of the sphere based on compact finite differences abstract the semi lagrangian representation of advection allows circumventing of the cfl restriction on time steps which is especially severe for finite difference models on the regular latitude longitude grid the distinct features of the presented semi lagrangian model are the use of vorticity and divergence as prognostic variables in conjunction with the fourth order compact finite differences on the unstaggered regular latitude longitude grid the key point of this approach is the solution of the poisson equations on the sphere which is necessary for reconstructing the velocity field from vorticity and divergence the accurate and efficient direct solver for this problem is described the results of the standard test set for shallow water equations on the sphere demonstrate the accuracy and computational efficiency of the model with the time steps several times greater than in the eulerian model

transport schemes in spherical geometries using spline based rbf fd with polynomials abstract this work presents a numerical algorithm for using radial basis function generated finite differences rbf fd to solve partial differential equations pdes on s 2 using polyharmonic splines with added polynomials defined in a 2d plane phs poly we introduce a novel method for calculating rbf fd phs poly differentiation weights on s 2 using first a householder reflection and then a projection onto the tangent plane the new phs poly rbf fd method is implemented on two standard test cases 1 solid body rotation on s 2 and 2 3d tracer transport within earth s atmosphere compared to existing methods including those in the rbf literature at similar resolutions this approach requires fewer degrees of freedom and is algorithmically much simpler a matlab code to implement the method is included in the appendix

a radial basis function rbf compact finite difference fd scheme for reaction diffusion equations on surfaces we present a new high order local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in mathbb r d the novelty of the method is in the approximation of the laplace beltrami operator for a given surface using hermite radial basis function rbf interpolation over local node sets on the surface this leads to compact or implicit rbf generated finite difference rbf fd formulas for the laplace beltrami operator which gives rise to sparse differentiation matrices the method only requires a set of scattered nodes on the surface and an approximation to the surface normal vectors at these nodes additionally the method is based on cartesian coordinates and thus does not suffer from any coordinate singularities we also present an algorithm for selecting the nodes used to construct the compact rbf fd formulas that can guarantee the resulting differentiation matrices have desirable stability properties the improved accuracy and computa

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

a class of deformational flow test cases for linear transport problems on the sphere a class of new benchmark deformational flow test cases for the two dimensional horizontal linear transport problems on the sphere is proposed the scalar field follows complex trajectories and undergoes severe deformation during the simulation however the flow reverses its course at half time and the scalar field returns to its initial position and shape this process makes the exact solution available at the end of the simulation and facilitates assessment of the accuracy of the underlying transport scheme a procedure to eliminate possible cancellations of errors when the flow reverses is proposed the test suite consists of four cases three are based on non divergent flow fields and one on a divergent flow the initial conditions are prescribed in terms of regular latitude longitude spherical coordinates and are easy to implement the divergent flow is specifically aimed for conservative global transport schemes to test for conservation consistency and monotonicity or positivity of limiters filters in a challenging flow environment in the context of semi lagrangian schemes the time varying flow fields can be used to test trajectory algorithms where the exact trajectories do not follow great circle arcs the characteristics of the test cases are demonstrated with two different transport schemes

equidistributed icosahedral configurations on the sphere abstract we present and analyze a new sequence of equidistributed icosahedral configurations these configurations are created by combining the m n icosahedral nodes of caspar and klug and adapting the azimuthal projection method of snyder to create a low deformation equal area mapping from the regular icosahedron to s 2 we prove an upper bound on the mesh ratio of these configurations and analyze numerically the riesz potential energies of the point sets of size n 50 000

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

hyperviscosity based stabilization for radial basis function finite difference rbf fd discretizations of advection diffusion equations abstract we present a novel hyperviscosity formulation for stabilizing rbf fd discretizations of the advection diffusion equation the amount of hyperviscosity is determined quasi analytically for commonly used explicit implicit and implicit explicit imex time integrators by using a simple 1d semi discrete von neumann analysis the analysis is applied to an analytical model of spurious growth in rbf fd solutions that uses auxiliary differential operators mimicking the undesirable properties of rbf fd differentiation matrices the resulting hyperviscosity formulation is a generalization of existing ones in the literature but is free of any tuning parameters and can be computed efficiently to further improve robustness we introduce a simple new scaling law for polynomial augmented rbf fd that relates the degree of polyharmonic spline phs rbfs to the degree of the appended polynomial when used in a novel ghost node formulation in conjunction with the recently developed overlapped rbf fd method the resulting method is robust and free of stagnation errors we validate the high order convergence rates of our method on 2d and 3d test cases over a wide range of peclet numbers 1 1000 we then use our method to solve a 3d coupled problem motivated by models of platelet aggregation and coagulation again demonstrating high order convergence rates

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

global well posedness in the energy space for the benjamin ono equation on the circle we prove that the benjamin ono equation is locally well posed in h 1 2 mathbb t this leads to a global well posedeness result in h 1 2 mathbb t thanks to the energy conservation

grid free adaptive semi lagrangian advection using radial basis functions this paper proposes a new grid free adaptive advection scheme the resulting algorithm is a combination of the semi lagrangian method slm and the grid free radial basis function interpolation rbf the set of scattered interpolation nodes is subject to dynamic changes at run time based on a posteriori local error estimates a self adaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs due to well known features of slm and rbf the method is guaranteed to be stable it has good approximation behaviour and it works for arbitrary space dimension numerical examples in two dimensions illustrate the performance of the method in comparison with existing grid based advection schemes

enhancing finite differences with radial basis functions polynomials are used together with polyharmonic spline phs radial basis functions rbfs to create local rbf finite difference rbf fd weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences fd the presented method replicates the convergence properties of fd but for arbitrary node layouts it is tested on the 2d compressible navier stokes equations at low mach number relevant to atmospheric flows test cases are taken from the numerical weather prediction community and solved on bounded domains thus attention is given on how to handle boundaries with the rbf fd method as well as a novel implementation for hyperviscosity comparisons are done on cartesian hexagonal and quasi uniform node layouts consideration and guidelines are given on phs order polynomial degree and stencil size the main advantages of the present method are 1 capturing the basic physics of the problem surprisingly well even at very coarse resolutions 2 high order accuracy without the need of tuning a shape parameter and 3 the inclusion of polynomials eliminates stagnation saturation errors a matlab code is given to calculate the differentiation weights for this novel approach

sharp ill posedness result for the periodic benjamin ono equation abstract we prove the discontinuity for the weak l 2 t topology of the flow map associated with the periodic benjamin ono equation this ensures that this equation is ill posed in h s t as soon as s 0 and thus completes exactly the well posedness result obtained in molinet 2008 12

icosahedral discretization of the two sphere we present an almost uniform triangulation of the two sphere derived from the icosahedron and describe a procedure for discretization of a partial differential equation using this triangular grid the accuracy of our procedure is described by a strong theoretical estimate and verified by large scale numerical experiments we also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer

aspects of the dynamical core of a nonhydrostatic deep atmosphere unified weather and climate prediction model the dynamical core which governs the evolution of resolved fluid dynamical processes is a critical element of any atmospheric model its governing equations must include all relevant dynamical terms and the numerical formulae used to approximate them must be accurate stable and efficient this is particularly so in a unified modeling environment in which the same dynamical core is used for both operational weather prediction and long term climate simulations recent research at the met office on unified dynamical core issues is reviewed aspects covered include properties of various equation sets vertical coordinates semi lagrangian advection and conservation trajectory computation and dynamical equivalence horizontal and vertical discretization and coupling of physical parameterizations to a dynamical core

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method

matrix factorizations and integrable systems on montre que les algorithmes qr lij et de cholesky pour calculer les valeurs propres de matrices reelles sont les evaluations a temps entier de flots hamiltoniens completement integrables