The RBF-FD method: developments and applications

Radial Basis Function (RBF) methods have become a truly meshless alternative for the interpolation of multidimensional scattered data and the solution of partial differencial equations (PDEs) on irregular domains. The dependence on the distance between centers makes RBF methods conceptually simple and easy to implement in anys dimension or shape of the domain. There are two different formulations for the solution of PDEs: the global RBF method and the local RBF method. In the global RBF formulation, the approximate solution is computed in the functional space spanned by a set of translated RBFs. The coordinates of the solution in this space are obtained by collocation. This formulation yields dense differentiation matrices which are spectrally convergent independently of the distribution of RBF centers. The principal drawback is that, as the overall number of centers increases, the condition number of the collocation matrices also increases, what restricts the applicability of the method in practical problems ....

Propagation of premixed laminar flames in 3D narrow open ducts using RBF-generated finite differences

Laminar flame propagation is an important problem in combustion modelling for which great advances have been achieved both in its theoretical understanding and in the numerical solution of the governing equations in 2D and 3D. Most of these numerical simulations use finite difference techniques on simple geometries (channels, ducts, ...) with equispaced nodes. The objective of this work is to explore the applicability of the radial basis function generated finite difference (RBF-FD) method to laminar flame propagation modelling. This method is specially well suited for the solution of problems with complex geometries and irregular boundaries. Another important advantage is that the method is independent of the dimension of the problem and, therefore, it is very easy to apply in 3D problems with complex geometries. In this work we use the RBF-FD method to compute 2D and 3D numerical results that simulate premixed laminar flames with different Lewis numbers propagating in open ducts.This work has been supported by Spanish MICINN grants FIS2010-18473, FIS2011-28838 and CSD2010-00011

Optimal shape parameter for the solution of elastostatic problems with the RBF method

Radial basis functions (RBFs) have become a popular method for the solution of partial differential equations. In this paper we analyze the applicability of both the global and the local versions of the method for elastostatic problems. We use multiquadrics as RBFs and describe how to select an optimal value of the shape parameter to minimize approximation errors. The selection of the optimal shape parameter is based on analytical approximations to the local error using either the same shape parameter at all nodes or a node-dependent shape parameter. We show through several examples using both equispaced and nonequispaced nodes that significant gains in accuracy result from a proper selection of the shape parameter.This work was supported by Spanish MICINN Grants FIS2011-28838 and CSD2010-00011 and by Madrid Autonomous Region Grant S2009-1597. M.K. acknowledges Fundación Caja Madrid for its financial support

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

A numerical model based on radial basis functiongenerated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linearelliptic partial differential equation (PDE) in a sphericallyshaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To ourknowledge, this is (1) the first numerical model of the GECto combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this,(2) the first RBF-FD method to use irregular 3-D stencils fordiscretization to handle the topography. It benefits from themesh-free nature of RBF-FD, which is especially suitable formodeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes nolimiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the righthand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.This work was supported by NSF awards AGS-1135446 and DMS-094581. The National Center for Atmospheric Research is sponsored by the NSF.Publicad

On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

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.The presented research was supported by the NSF grants DMS-0934317, OCI-0904599 and by Shell International Exploration and Production, Inc. Victor Bayona was a post-doctoral fellow funded by the Advanced Study Program at the National Center for Atmospheric Research (NCAR) during the development of this research. NCAR is sponsored by the National Science Foundation

Optimal constant shape parameter for multiquadric based RBF-FD method

Radial basis functions (RBFs) have become a popular method for interpolation and solution of partial differential equations (PDEs). Many types of RBFs used in these problems contain a shape parameter, and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter. In this paper, we focus on PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method. We propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error. The algorithm is based on analytical approximations to the local RBF-FD error derived in [1]. We show through several examples in 1D and 2D, both with structured and unstructured nodes, that very accurate solutions (compared to finite differences) can be achieved using the optimal value of the constant shape parameter.This work has been supported by Spanish MICINN grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region grant S2009-1597

Gaussian RBF-FD weights and its corresponding local truncation errors

In this work we derive analytical expressions for the weights of Gaussian RBF-FD and Gaussian RBF-HFD formulas for some differential operators. These weights are used to derive analytical expressions for the leading order approximations to the local truncation error in powers of the inter-node distance h and the shape parameter є. We show that for each differential operator, there is a range of values of the shape parameter for which RBF-FD formulas and RBF-HFD formulas are significantly more accurate than the corresponding standard FD formulas. In fact, very often there is an optimal value of the shape parameter є+ for which the local error is zero to leading order. This value can be easily computed from the analytical expressions for the leading order approximations to the local error. Contrary to what is generally believed, this value is, to leading order, independent of the internodal distance and only dependent on the value of the function and its derivatives at the node.This work has been supported by Spanish MICINN Grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region Grant S2009-1597. M.K. acknowledges Fundacion Caja Madrid for its financial support

Optimal variable shape parameter for multiquadric based RBF-FD method

In this follow up paper to our previous study in Bayona et al. (2011) [2], we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) [2] we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) [2], both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter.This work has been supported by Spanish MICINN Grants FIS2010-18473, CSD2010-00011 and by Madrid Autonomous Region grant S2009-1597. M.K. acknowledges Fundacion Caja Madrid for its financial support

Improving the inter-hemispheric gradient of total column atmospheric CO2 and CH4 in simulations with the ECMWF semi-Lagrangian atmospheric global model

It is a widely established fact that standard semi-Lagrangian advection schemes are highly efficient numerical techniques for simulating the transport of atmospheric tracers. However, as they are not formally mass conserving, it is essential to use some method for restoring mass conservation in long time range forecasts. A common approach is to use global mass fixers. This is the case of the semi-Lagrangian advection scheme in the Integrated Forecasting System (IFS) model used by the Copernicus Atmosphere Monitoring Service (CAMS) at the European Centre for Medium-Range Weather Forecasts (ECMWF). Mass fixers are algorithms with substantial differences in complexity and sophistication but in general of low computational cost. This paper shows the positive impact mass fixers have on the inter-hemispheric gradient of total atmospheric column-averaged CO2 and CH4, a crucial feature of their spatial distribution. Two algorithms are compared: the simple "proportional" and the more complex Bermejo-Conde schemes. The former is widely used by several Earth system climate models as well the CAMS global forecasts and analysis of atmospheric composition, while the latter has been recently implemented in IFS. Comparisons against total column observations demonstrate that the proportional mass fixer is shown to be suitable for the low-resolution simulations, but for the high-resolution simulations the Bermejo-Conde scheme clearly gives better results. These results have potential repercussions for climate Earth system models using proportional mass fixers as their resolution increases. It also emphasises the importance of benchmarking the tracer mass fixers with the inter-hemispheric gradient of long-lived greenhouse gases using observations

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.This work has been supported by Spanish MECD Grants FIS2007-62673, FIS2008-04921 and by Madrid Autonomous Region Grant S2009-1597