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In this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrodinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. L ∞ and L 2 error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.

Mixed convection flow in a lid-driven cavity in the presence of a uniform partial magnetic field is investigated utilizing the pseudo spectral method based on polyharmonic spline radial basis functions. The two-dimensional problem is concerned in a unit square cavity designed by adiabatic horizontal walls and vertical walls with constant temperatures. The top wall moves either in the positivexor in the negativexdirections. The partial magnetic field is taken into account in the positiveydirection. Working fluid is chosen as air. Efficiency of dimensionless Hartmann number and Richardson number on fluid flow and heat transfer is presented as well as the length and the central location of partial magnetic field. Flow is affected in the direction of vertical partial magnetic field, and is propelled to the other parts of the cavity regarding the central location of magnetic field. In case of middle centered partial magnetic field. The increase in the length of the partial magnetic field and the middle centered partial magnetic field as well reduce the convective heat transfer. The negativexdirected lid-driven cavity has less influence on convection than the positivexdirected lid-driven cavity.

Rough polyharmonic splines (RPS) is a variational method which has recently been developed for linear divergence-form operators with arbitrary rough coefficients. RPS method does not rely on concepts of ergodicity or scale-separation, but on compactness properties of the solution space. In this paper, we extend RPS approach method for the optimal control problem governed by parabolic systems with rough L ∞ coefficients. RPS method is used for the spatial discretization, while the temporal discretization is performed by the finite difference method. As the iterative solution of the optimal control problem requires solving the state and co-state equations many times with different right hand sides, RPS method only requires one-time pre-computation on the fine scale and the following iterations can be done to coarse degrees of freedom. First of all, we extend an approximation method for the multiscale optimal control problem. Secondly, we obtain the error estimates of the multiscale optimal control problem. Finally, numerical experiments are presented to validate the theoretical analysis.

The radial basis function generatedfinite difference (RBFFD) method is applied to the analysis of homogenous waveguides. To this end, the Helmholtz equation and the boundary conditions are collocated on the waveguide cross section. At each collocation node, derivatives are locally approximated by RBFFD formulas based on polyharmonic splines supplemented with highdegree polynomials. As a result, a sparse matrix eigenvalue problem is obtained which allows cutoff wavenumbers and axial fields to be calculated. To illustrate the accuracy of the method, we consider a semicircular and an eccentric circular waveguides. This work was supported in part by the Spanish Government (MCIU/AEI) and the European Commission (FEDER, UE) under Research Projects PGC2018-098350-B-C21and PGC2018-098350-B-C22.

Radial basis function generated finite differences (RBF-FD) represent the latest discretization approach for solving partial differential equations. Their benefits include high geometric flexibility, simple implementation, and opportunity for large-scale parallel computing. Compared to other meshfree methods, typically based upon moving least squares (MLS), the RBF-FD method is able to recover a high order of algebraic accuracy while remaining better conditioned. These features make RBF-FD a promising candidate for kinetic-based fluid simulations such as lattice Boltzmann methods (LB). Pursuant to this approach, we propose a characteristic-based off-lattice Boltzmann method (OLBM) using the strong form of the discrete Boltzmann equation and radial basis function generated finite differences (RBF-FD) for the approximation of spatial derivatives. Decoupling the discretizations of momentum and space enables the use of irregular point cloud, local refinement, and various symmetric velocity sets with higher order isotropy. The accuracy and computational efficiency of the proposed method are studied using the test cases of Taylor–Green vortex flow, lid-driven cavity, and periodic flow over a square array of cylinders. For scattered grids, we find the polyharmonic spline + poly RBF-FD method provides better accuracy compared to MLS. For Cartesian node layouts, the results are the opposite, with MLS offering better accuracy. Altogether, our results suggest that the RBF-FD paradigm can be applied successfully also for kinetic-based fluid simulation with lattice Boltzmann methods.

New micro-devices, such as unmanned aerial vehicles or micro-robots, have increased the demand of a new generation of small-scale combustion power system that go beyond the energy-density limitations of batteries or fuel cells. The characteristics short residence times and intense heat losses reduce the efficiency of combustion-based devices, a key factor that requires of an acute modelling effort to understand the competing physicochemical phenomena that hamper their efficient operation. With this objective in mind, this paper is devoted to the development of a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences (RBF-FD) based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly). In our model, the combustion chamber of a micro-rotary engine is simulated by a system of unsteady reaction-diffusion equations coupled with a steady flow passing a bidimensional stenotic channel of great slenderness. The conversion efficiency is characterized by identifying the different combustion regimes that emerged as a function of the ignition point. We show that PHS+poly based RBF-FD is able to achieve high-order algebraic convergence on scattered node distributions, enabling for node refinement in key regions of the fluid domain. This feature makes it specially well adapted to integrate problems in irregular geometries with front-like solutions, such as reactive fronts or shock waves. Several numerical tests are carried out to demonstrate the accuracy and effectiveness of our approach. VB was supported by Spanish MECD Grant FIS2016-77892-R. MSS and EFT acknowledge the financial support of the Spanish Government under projects ENE2015-65852-C2-1-R and PID2019-108592RB-C41 (MINECO/FEDER,UE).

Polyharmonic splines of order m satisfy the polyharmonic equation of order m in n variables. Moreover, if employed as basis functions for interpolation they are radial functions. We are concerned with the problem of construction of the smooth interpolation formula presented as the minimizer of suitable functionals subject to interpolation constraints for n ≥ 1 . This is the principal motivation of the paper. We show a particular procedure for determining the interpolation formula that in a natural way leads to a linear combination of polyharmonic splines of a fixed order, possibly complemented with lower order polynomial terms. If it is advantageous for the interpolant in the problem solved to be a polyharmonic spline we can construct such an interpolant directly using the multivariate smooth approximation technique. The smoothness of the spline can be a priori chosen. Smooth interpolation can be very useful e.g. in signal processing, computer aided geometric design or construction of geographic information systems. A 1D computational example is presented.

The localized method of approximated particular solutions (LMAPS) was first introduced in 2011 [31]. The method is then modified by employing integrated polyharmonic splines with polynomial basis. However, the current reported results on LMAPS still limited to the evenly distributed data points and Dirichlet boundary conditions. On the other hand, the traditional point-wise moving least square method is improved by piece-wise moving least squares (PMLS) in [19] for scattered data approximation. The paper proved that the PMLS is is an optimal design for data approximation. In this paper, the modified LMAPS is further improved by involving the Hermite-type PMLS to construct shape functions. The improved LMAPS is called piece-wise smoothed LMAPS (PS-LMAPS). Together with the original LMAPS, PS-LMAPS is used to solve elliptic partial differential equations with Dirichlet and Neumann mixed boundary conditions on the scattered data points. Performance of PS-LMAPS in comparison with LMAPS is tested on two Poisson equation with mixed boundary conditions using evenly-spaced nodes and scattered nodes, modified Helmholtz equations and a non-smooth problem. Particularly, PS-LMAPS can avoid some of the ill-conditioning issues of the system as shown in Example 3. The computational complexity ratio and relative error ratio indicate that the PS-LMAPS is much more efficient than that original LMAPS. The conclusion is supported by theoretical analysis of computational complexity and numerical experiments on the error analysis.

The invention and development of Polymerase Chain Reaction (PCR) technology have revolutionised molecular biology and molecular diagnostics. There is an urgent need to optimise the performance of these devices while reducing the total construction and operation costs. This study proposes a CFD-enabled optimisation methodology for continuous flow (CF) PCR devices with serpentine-channel structure, which enables the optimisation of DNA amplification efficiency and pressure drop to be explored while varying the width (W) and height (H) of the microfluidic (μ) channel. This is achieved by using a surrogate-enabled optimisation approach accounting for the geometrical features of a μCFPCR device by performing a series of simulations using COMSOL Multiphysics 5.4®. The values of the objectives are extracted from the CFD solutions, and the response surfaces are created using polyharmonic splines. Genetic algorithms are then used to locate the optimum design parameters. The results indicate that there is the possibility of improving the DNA concentration and the pressure drop in a PCR cycle by ~2.1 % ([W, H] = [400 μm, 50 μm]) and ~95.2 % ([W, H] = [400 μm, 80 μm]) respectively, by modifying its geometry.

Local meshless methods obtain higher convergence rates when RBF approximations are augmented with monomials up to a given order. If the order of the approximation method is spatially variable, the numerical solution is said to be p-refined. In this work, we employ RBF-FD approximation method with polyharmonic splines augmented with monomials and study the numerical properties of p-refined solutions, such as convergence orders and execution time. To fully exploit the refinement advantages, the numerical performance is studied on a Poisson problem with a strong source within the domain.

We develop supporting arguments in favour of kernel-based reconstruction methods in the recovery step of particle simulations.We strongly recommend polyharmonic spline kernels, whose key features and advantages are briefly explained.

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the considered dynamics, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo-Levi spaces. The performances of the method are then tested on functions which are common in the framework of the Laplace transform inversion. Comparisons with Support Vector Regression (SVR) are also carried out and show that the proposed method is effective particularly since it does not require to train complex architecture constructions.

Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since the meshless method does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated. Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have additional shape parameters that have to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy the momentum and continuity equations. Systematic convergence tests have been performed for five model problems with two of them having analytical solutions. We demonstrate fast convergence both with refinement of number of points and degree of appended polynomials. The method is further applied to solve problems such as lid-driven cavity and vortex shedding over circular cylinder. We have also analyzed the performance of this approach for solution of Euler equations. The proposed method shows promise to solve fluid flow and heat transfer problems in complex domains with high accuracy.

Recently, methods that do not rely on the regularity of the solution were introduced: generalized finite element methods [1], the rough polyharmonic splines [22], the variational multiscale method (VMS) [13], and the Localized Orthogonal Decomposition (LOD) [16, 10].

In this work, one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) coupled damped Schrodinger system is solved numerically. A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed. Polyharmonic splines are used as radial basis function with augmented polynomials. The use of the polyharmonic splines saves us from choosing an optimum shape parameter which is not a simple task for infinitely smooth RBFs such as multiquadrics or Gaussians. For time discretization classical fourth-order Runge Kutta method is utilized. L ∞ error norm and conserved quantities are computed to indicate performance of the proposed method. Stability of the proposed method is examined numerically. Some computer codes are devised in Julia programming language for obtaining numerical results. Acquired numerical results and their comparison with other studies available in literature such as cubic B-spline Galerkin method and direct meshless local Petrov–Galerkin (DMLPG) method endorse the performance and reliability of the proposed method.

In this paper, a closed-form particular solution of polyharmonic splines has been obtained for high order partial differential operators. Instead of using complex derivation, the new particular solution is derived simply by adding or subtracting several available particular solutions. The proposed particular solution is further coupled with polynomial basis for numerically solving thin plate problems. The relationship between number of nodes and order of polynomials are fully studied. Numerical examples with irregular domains are presented to demonstrate the effectiveness of the proposed algorithm.

In this paper, we utilized the method of fundamental solutions, which is meshless and integral-free, to analyze the non-linear Berger equation for thin elastic plate. Based on the proposed numerical scheme, the deflection can be expressed as the linear combination of the homogeneous solution and the particular solutions. The particular solution, based on the polyharmonic splines, is derived and then the spatial-dependent loading term of the Berger equation can be approximated by the polyharmonic splines. After the particular solution is obtained, the homogeneous solution, which is governed by the homogeneous partial differential equations, can be solved by the method of fundamental solutions. Several numerical examples are adopted to demonstrate the flexibility and robustness of the proposed meshless scheme, especially the irregular plate with spatial-dependent loading function. Furthermore, we also performed the convergence test for various orders of the polyharmonic splines.

By the Mindlin/Reissner plate theory, the displacement ω and rotations θ x , θ y are interpolated by independent functions, for which only C 0 continuity condition is required. The difficulty for constructing interpolation bases on the quadrilateral element without isoparametric transformation can be overcome by using the spline method. In this paper, two sets of spline interpolation bases are adopted to construct two quadrilateral spline Mindlin plate elements (QSMP1 and QSMP2) with 12 degrees of freedom. The spline elements can be applied for both thick and thin plates, and can converge for the very thin case. Numerical examples are discussed to show that the Mindlin plate element combined with the spline interpolation bases can possess good accuracy.

The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline.

The paper is concerned with the use of polyharmonic splines as basis functions in multivariate data fitting. We present several properties of polyharmonic splines and their mutual links: they are commonly used radial basis functions, they are basis functions resulting from the application of a particular smooth approximation procedure, and the form and coefficients of the approximant can be obtained as a solution of a boundary value differential problem for the polyharmonic equation. The construction of the approximant is based on the least squares approach. Approximation of the kind mentioned is often used in practical computation especially with the data measured in 2D and 3D for geographic information systems or computer aided geometric design. The smooth approximation point of view provides the best description of the properties of polyharmonic splines employed for approximation. We mention the connections to interpolation where appropriate.

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to obtain the solution of two- and three-dimensional coupled Klein–Gordon–Schrodinger (KGS) equations. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the polyharmonic splines are used as the basis functions and to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The aim of this paper is to show that the SMRPI procedure is suitable for the treatment of the KGS equations. The results of numerical experiments are compared with analytical solution and stability analysis is checked to confirm the accuracy and efficiency of the presented scheme.

Cubic splines with equidistant spline sampling points are a common method in atmospheric science, used for the approximation of background conditions by means of filtering superimposed fluctuations from a data series. What is defined as background or superimposed fluctuation depends on the specific research question. The latter also determines whether the spline or the residuals – the subtraction of the spline from the original time series – are further analysed.Based on test data sets, we show that the quality of approximation of the background state does not increase continuously with an increasing number of spline sampling points and/or decreasing distance between two spline sampling points. Splines can generate considerable artificial oscillations in the background and the residuals.We introduce a repeating spline approach which is able to significantly reduce this phenomenon. We apply it not only to the test data but also to TIMED-SABER temperature data and choose the distance between two spline sampling points in a way that is sensitive for a large spectrum of gravity waves.

The best L 1 approximation of the Heaviside function and the best l 1 approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L 1 spline fits which are the best L 1 approximations in an appropriate spline space obtained by the union of L 1 interpolation splines. We prove here the existence of L 1 spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.

International audience; We consider the wide class of all piecewise Chebyshevian splines with connection matricesat the knots. We prove that a spline space of this class is “good for interpolation” if and only ifthe spline space obtained by integration is “good for design”. As a consequence, this providesus with a simple practical description of all such spline spaces which can be used for solvingHermite interpolation problems. These results strongly rely on the properties of blossoms.

The coefficients of a tensor-product spline surface in R d with m × n control points form a tensor of order 3 and dimension ( m , n , d ) . Motivated by applications in isogeometric analysis we analyze the rank of this tensor. In particular, we propose a new construction for low rank tensor-product spline surfaces from given boundary curves. While the results of this construction are generally not affinely invariant, we propose a simple standardization procedure that guarantees affine invariance for d = 2 . In addition we provide a detailed comparison with existing constructions of spline surfaces from boundary data.

In the era of big data, it is very important to study large-scale data fitting methods. In order to ensure the calculation speed and accuracy, we propose a new kind of hierarchical many-knot splines (hereinafter called “hierarchical MK splines,” generally abbreviated as HMK splines) in this paper. The HMK splines method produces a sequence of MK spline functions. These MK spline functions are constructed into one ideal interpolation function by the MK spline refinement. In the case of regular sampling data, HMK splines can achieve the purpose of accurate approximation for the given data points without solving systems of equations. Further, in order to deal with the issues of scattered data fitting, the use of least-squares method will lead to the necessary of solving a linear system of equations. Since the ill-conditioned systems of equations often lead to unacceptable deviation of calculation results, one tries to avoid it as much as possible. The HMK splines algorithm can meet this requirement; it can avoid the intolerable deviation caused by solving systems of equations. Experimental results show that large-scale scattered data fitting can be easily achieved by the HMK splines algorithm and the reconstruction of nonuniform samples has a high accuracy.

In the main result of the paper we prove the decomposition of polyharmonic Bergman spaces over the upper-half plane into spaces of polyanalytic functions. Then, we introduce the decomposition of polyharmonic Bergman spaces into the orthogonal sum of its true polyharmonic Bergman subspaces and we state isometric isomorphisms between the different true polyharmonic Bergman spaces. This allows us to define the k-th harmonic Hilbert component of a polyharmonic Bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic Bergman spaces. The harmonic complex Fourier transform is introduced in order to give an explicit description of the cartesian and the Laguerre harmonic components of the images of a Bargmann type transform for the true polyharmonic Bergman spaces. Finally, it is proved that the polyharmonic Bergman space of order j is isometric isomorphic to 2j copies of the corresponding Hardy space.

The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matern and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017

A stable splitting of 2m-th order elliptic partial differential equations into 2(m− 1) problems of Poisson type and one generalized Stokes problem is established for any space dimension d ≥ 2 and any integer m ≥ 1. This allows a numerical approximation with standard finite elements that are suited for the Poisson equation and the Stokes system, respectively. For some fourthand sixth-order problems in two and three space dimensions, precise finite element formulations along with a priori error estimates and numerical experiments are presented.

In this paper, we analyse the polyharmonic equation in a bounded smooth domain with homogeneous Dirichlet conditions on the boundary and critical exponents. We will obtain a relationship between the topological domain and the multiplicity of solutions. We also give symmetry properties of the solutions and a result about breaking symmetry.

In this paper, we consider shape-preserving properties of the local integro-cubic spline. The sufficient conditions for convex, monotone, or positive approximation are given. Some examples are provided to illustrate shape-preserving properties of the splines.

For accurate cloud ceiling information, a data fusion approach is proposed that utilizes satellite data to extend surface station information to much wider areas. Cloud base height (CBH) retrieved from satellite observations provides for much larger spatial coverage and higher resolution. The direct comparison of GOES-16 CBH with surface station ceiling yields a local bias that has to be corrected for in the initial GOES-16 cloud base information. This sparsely sampled bias correction presents an irregular 2D mesh of control points, which is then interpolated by constructing a continuous smooth field using polyharmonic splines. The influence of remote stations is restricted by grouping the control points into clusters depending on an effective distance. This cluster-based approach allows for constructing separate spline surfaces corresponding to physically different clouds. The obtained continuous bias correction function is then applied to the entire GOES-16 pixel level CBH except for areas far away from surface stations in data sparse regions such as offshore. The described method is currently being tested using daytime-only observations over the central and eastern United States. Overall, this approach has potential to provide more accurate, high spatial resolution cloud ceiling information for the aviation community.

A recently suggested technique for high order meshless approximations is described and analyzed in this paper. It involves constructing ordinary Radial Basis Function-generated finite difference approximations augmented with monomials up to a given order to ensure higher convergence rates. These approximations are used to solve the Poisson’s equation on an annulus to demonstrate the predicted convergence rates. The presented methodology is then applied to a scattering problem, which is described by a coupled system of two complex-valued PDEs on two domains, sharing a common boundary.