combining the radial basis function eulerian and lagrangian schemes with geostatistics for modeling of radionuclide migration through the geosphere to assess the long term safety of a radioactive waste disposal system mathematical models are used to describe groundwater flow chemistry and potential radionuclide migration through geological formations a number of processes need to be considered when predicting the movement of radionuclides through the geosphere the most important input data are obtained from field measurements which are not available for all regions of interest for example the hydraulic conductivity as an input parameter varies from place to place in such cases geostatistical science offers a variety of spatial estimation procedures methods for solving the solute transport equation can also be classified as eulerian lagrangian and mixed the numerical solution of partial differential equations pde has usually been obtained by finite difference methods fdm finite element methods fem or finite volume methods fvm kansa introduced the concept of solving partial differential equations using radial basis functions rbf for hyperbolic parabolic and elliptic pdes the aim of this study was to present a relatively new approach to the modeling of radionuclide migration through the geosphere using radial basis function methods in eulerian and lagrangian coordinates in this study we determine the average and standard deviation of radionuclide concentration with regard to variable hydraulic conductivity which was modelled by a geostatistical approach radionuclide concentrations will also be calculated in heterogeneous and partly heterogeneous 2d porous media

two numerical meshless techniques based on radial basis functions rbfs and the method of generalized moving least squares gmls for simulation of coupled klein gordon schrodinger kgs equations in the present study three numerical meshless methods are being considered to solve coupled klein gordon schrodinger equations in one two and three dimensions first the time derivative of the mentioned equation will be approximated using an implicit method based on crank nicolson scheme then kansa s approach rbfs pseudo spectral ps method and generalized moving least squares gmls method will be used to approximate the spatial derivatives the proposed methods do not require any background mesh or cell structures so they are based on a meshless approach applying three techniques reduces the solution of the one two and three dimensional partial differential equations to the solution of linear system of algebraic equations as is well known the use of kansa s approach makes the coefficients matrix in the above linear system of algebraic equations to be ill conditioned and we applied lu decomposition technique but when we employ ps method fasshauer 2007 the matrix of coefficients in the obtained linear system of algebraic equations is well conditioned also the gmls technique yields a well conditioned linear system because a shifted and scaled polynomial basis will be used at the end of this paper we provide some examples on one two and three dimensions for obtaining numerical simulations also the obtained numerical results show the applicability of the proposed three methods to find the numerical solution of the kgs equations

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

multiscale rbf collocation for solving pdes on spheres in this paper we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere the approximate solution is constructed in a multi level fashion each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh two variants of the collocation method are considered sometimes called symmetric and unsymmetric although here both are symmetric a convergence theory is given which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions

a stabilized least squares radial point collocation method ls rpcm for adaptive analysis abstract in this paper a stabilized radial point collocation method rpcm that uses locally supported nodes is proposed using least squares stabilization technique the focus of this work is to stabilize the solution of the rpcm in order to perform adaptive analysis good stiffness matrix properties such as symmetric and positive definite spd are gained via a least squares procedure and yet the formulation procedure of the stabilized ls rpcm is still kept simple and straightforward adaptive analysis study has then been successfully carried out using the stabilized ls rpcm error indicator based on interpolation error is adopted in the adaptive scheme for nodal refinement thanks for the meshfree features of the rpcm refinement process can be easily done by inserting additional nodes based on the voronoi diagram without worrying about nodal connectivity in the formulation of system equations good numerical performance has been shown in the numerical examples presented in this paper

a scalable and implicit meshless rbf method for the 3d unsteady nonlinear richards equation with single and multi zone domains the local hermitian interpolation lhi method is a strong form meshless numerical technique which uses radial basis function rbf interpolants to satisfy linear and nonlinear governing equations and boundary operators recent developments have shown that for linear transport problems applying the pde governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions in this work the lhi formulation with local pde interpolation is extended to the nonlinear gravity driven richards equation in order to solve unsteady problems involving flow in unsaturated porous media the application of the linearised pde operator to the basis functions incorporates information such as the effective velocity field directly into the solution construction this results in a form of analytical upwinding which helps to stabilise the solution in addition the local interpolation itself satisfies the linearised governing equation allowing for more accurate reconstruction of partial derivatives and hence a more accurate solution the procedure is tested using a 3d infiltration problem with a known analytical solution the performance of the lhi method both with and without local pde interpolation is compared to the finite element method via the femwater software the lhi formulation with pde data centres shows consistent improvement over the femwater solutions reducing errors by several orders of magnitude in addition a procedure is introduced to model strongly heterogeneous and layered soils the physically correct matching conditions are applied over layer interfaces i e continuity of pressure and mass flux the double collocation property of the hermitian rbf method is exploited to enforce both matching conditions at the same set of locations on the layer interface this procedure allows the accurate capture of solutions across such interfaces replicating the required discontinuities in the first derivatives of the pressure profile the multi layer formulation is validated using a transient two layer infiltration problem with the analytical solution replicated to a high precision in a variety of configurations copyright 2010 john wiley sons ltd

some observations on unsymmetric radial basis function collocation methods for convection diffusion problems in this paper we re investigate the unsymmetric radial basis function rbf collocation method for solving convection diffusion problems with high peclet number as in power and barraco computers and mathematics with applications 2002 43 551 by testing different rbfs and different numbers of nodes we found that the unsymmetric method can still solve high peclet number problems reasonably well by using more nodes and domain decomposition techniques compared to solving one large problem the domain decomposition method is shown to be very efficient and can improve the accuracy especially when the peclet number is not that high from our tests it seems that the rbfs r4 ln r and r8 ln r are not very stable while r6 ln r r5 and r7 perform very well copyright 2003 john wiley sons ltd

a double boundary collocation hermitian approach for the solution of steady state convection diffusion problems in this article a double boundary collocation approach based on the meshless radial basis function hermitian method symmetric method is proposed and compared with the conventional single collocation in the double boundary collocation approach at the boundary collocation points the boundary condition and the governing partial differential equation are required to be satisfied simultaneously instead of only the boundary condition as required in the single collocation we are able to carry out this type of algorithm due to the robustness of the proposed hermite interpolation scheme in which the resulting matrix will be non singular as long as the partial differential operators applied to each point are linearly independent even if in a single node we impose two different differential conditions the results obtained with this new method are characterized by a higher precision especially for the prediction of the fluxes at the boundaries this is due to the higher order of continuity of the approximation at the boundary points imposed by the double collocation

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

the uniform convergence of thin plate spline interpolation in two dimensions let f be a function from cal r 2 to cal r that has square integrable second derivatives and let s be the thin plate spline interpolant to f at the points underline v i i 1 2 ldots n in cal r 2 we seek bounds on the error f underline x s underline x when underline x is in the convex hull of the interpolation points or when underline x is close to at least one of the interpolation points but need not be in the convex hull we find for example that if underline x is inside a triangle whose vertices are any three of the interpolation points then f underline x s underline x is bounded above by a multiple of h where h is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points further if cal d is any bounded set in cal r 2 that is not a subset of a single straight line then we prove that a sequence of thin plate spline interpolants converges to f uniformly on cal d specifically we require h rightarrow 0 where h is now the least upper bound on the numbers d underline x cal v underline x in cal d and where d underline x cal v underline x in cal r 2 is the least euclidean distance from underline x to an interpolation point our method of analysis applies integration by parts and the cauchy schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation

solving time dependent differential equations by multiquadric trigonometric quasi interpolation multiquadric mq quasi interpolation is a popular method for the numerical solution of differential equations however mq quasi interpolation is not well suited for the equations with periodic solutions this is mainly due to the fact that its kernel the mq function is not a periodic function a reasonable way of overcoming the difficulty is to use a quasi interpolant whose kernel itself is also periodic in these cases the paper constructs such a quasi interpolant error estimates of the quasi interpolant are also provided the quasi interpolant possesses many fair properties of the mq quasi interpolant i e simplicity efficiency stability etc moreover it is more suitable than the mq quasi interpolant for periodic problems since the quasi interpolant as well as its derivatives are periodic examples of solving both linear and nonlinear partial differential equations whose solutions are periodic by the quasi interpolant and the mq quasi interpolant are compared at the end of the paper numerical results show that the quasi interpolant outperforms the mq quasi interpolant for periodic problems

a comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations abstract in this article we present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations a series of test examples was solved with these two schemes different problems with different type of governing equations and boundary conditions particular emphasis was paid to the ability of these schemes to solve the steady state convection diffusion equation at high values of the peclet number from the examples tested in this work it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better however the unsymmetric method has the advantage of being simpler to implement two main features about the results obtained in this work are worthy of special attention first with the symmetric method it was possible to solve convection diffusion problems at a very high peclet number without the need of any artificial damping term and second with these two approaches symmetric and unsymmetric it is possible to impose free boundary conditions for problems in unbounded domains

application of radial basis functions to linear and nonlinear structural analysis problems the basic characteristic of the techniques generally known as meshless methods is the attempt to reduce or even to eliminate the need for a discretization at least not in the way normally associated with traditional finite element techniques in the context of numerical solutions for boundary and or initial value problems the interest in meshless methods is relatively new and this is why despite the existence of various applications of meshless techniques to several problems of mechanics as well as to other fields these techniques are still relatively unknown to engineers furthermore and compared to traditional finite elements it may be difficult to understand the physical meaning of the variables involved in the formulations as an attempt to clarify some aspects of the meshless techniques and simultaneously to highlight the ease of use and the ease of implementation of the algorithms applications are made in this work to structural analysis problems the technique used here consists of the definition of a global approximation for a given variable of interest in this case components of the displacement field by means of a superposition of a set of conveniently placed in the domain and on the boundary radial basis functions rbfs in this work various types of one dimensional problems are analyzed ranging from the static linear elastic case free vibration and linear stability analysis for a beam on elastic foundation to physically nonlinear damage models problems to further complement the range of problems analysed the static analysis of a plate on elastic foundation was also addressed several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global rbfs the results obtained show that rbf collocation leads to good approximations and very high convergence rates

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

radial point interpolation collocation method rpicm for partial differential equations this paper presents a truly meshfree method referred to as radial point interpolation collocation method rpicm for solving partial differential equations this method is different from the existing point interpolation method pim that is based on the galerkin weak form because it is based on the collocation scheme no background cells are required for numerical integration radial basis functions are used in the work to create shape functions a series of test examples were numerically analysed using the present method including 1 d and 2 d partial differential equations in order to test the accuracy and efficiency of the proposed schemes several aspects have been numerically investigated including the choice of shape parameter c with can greatly affect the accuracy of the approximation the enforcement of additional polynomial terms and the application of the hermite type interpolation which makes use of the normal gradient on neumann boundary for the solution of pdes with neumann boundary conditions particular emphasis was on an efficient scheme namely hermite type interpolation for dealing with neumann boundary conditions the numerical results demonstrate that good improvement on accuracy can be obtained after using hermite type interpolation the h convergence rates are also studied for rpicm with different forms of basis functions and different additional terms

a new bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis in this paper a new matrix approach for solving second order linear partial differential equations pdes under given initial conditions has been proposed the basic idea includes integrating from the considered pdes and transforming them to the associated integro differential equations with partial derivatives therefore bernoulli operational matrices of differentiation and integration together with the completeness of bernoulli polynomials can be used for transforming integro differential equations to the corresponding algebraic equations a rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded a numerical example is included to demonstrate the validity and the applicability of the technique the results confirm the theoretical prediction

multicloud multigrid convergence with a meshless operator the primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud multicloud is well founded in the mathematical basis of multigrid but relies on a meshless operator on coarse levels the meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes the performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the euler equations on two dimensional test cases results indicate comparable convergence rates per unit work for multicloud and multigrid however because of its mesh and scheme transparency multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques the implication is that multicloud can be implemented in a completely modular fashion allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three dimensional problems

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

a numerical study of compact approximations based on flat integrated radial basis functions for second order differential equations abstract in this paper we propose a simple but effective preconditioning technique to improve the numerical stability of integrated radial basis function irbf methods the proposed preconditioner is simply the inverse of a well conditioned matrix that is constructed using non flat irbfs much larger values of the free shape parameter of irbfs can thus be employed and better accuracy for smooth solution problems can be achieved furthermore to improve the accuracy of local irbf methods we propose a new stencil namely combined compact irbf ccirbf in which i the starting point is the fourth order derivative and ii nodal values of first and second order derivatives at side nodes of the stencil are included in the computation of first and second order derivatives at the middle node in a natural way the proposed stencil can be employed in uniform nonuniform cartesian grids the preconditioning technique in combination with the ccirbf scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by poisson burgers convection diffusion and navier stokes equations highly accurate and stable solutions are obtained in some cases the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning

meshfree explicit local radial basis function collocation method for diffusion problems this paper formulates a simple explicit local version of the classical meshless radial basis function collocation kansa method the formulation copes with the diffusion equation applicable in the solution of a broad spectrum of scientific and engineering problems the method is structured on multiquadrics radial basis functions instead of global the collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node the computational effort thus grows roughly linearly with the number of the nodes the developed approach thus overcomes the principal large scale problem bottleneck of the original kansa method two test cases are elaborated the first is the boundary value problem nafems test associated with the steady temperature field with simultaneous involvement of the dirichlet neumann and robin boundary conditions on a rectangle the second is the initial value problem associated with the dirichlet jump problem on a square the accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes number of nodes in the domain of influence multiquadrics free parameter and timestep length on uniform and nonuniform node arrangements the developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the dirichlet jump where the approximation properties appear similar