Basis Function Finite Difference Research Articles

A local meshless procedure to determine the unknown control parameter in the multi-dimensional inverse problems

This article is devoted to applying a local meshless method for specifying an unknown control parameter in one- and multi-dimensional inverse problems which are considered with a temperature overspecification condition at a specific point or an energy overspecification condition over the computational domain. Finding the unknowns in inverse problems is a challenge because these problems are modeled as non-classical parabolic problems and also have a significant role in describing physical phenomena of the real world. In this study, a combination of the meshless method of radial basis functions and finite difference method (called radial basis function-finite difference method) is used to solve inverse problems because this method has two important features. First it does not require any mesh generation. Consequently, it can be exerted to handle the high-dimensional inverse problems. Secondly, since this method is local, at each time step, a system with a sparse coefficient matrix is solved. Hence, the computational time and cost will be much low. Various numerical examples are examined, and also the accuracy and computational time required are presented. The numerical results indicate that the mentioned procedure is appropriate for the identification of the unknown parameter of inverse problems.

Consistent, non-oscillatory RBF finite difference solutions to boundary layer problems for any degree on uniform grids

Singular perturbation problem cannot be solved without oscillations by the classical finite difference methods on the uniform grid unless sufficient resolution is employed. In this note we use the Gaussian radial basis function (RBF) finite difference method and show that it is possible for the finite difference method to yield non-oscillatory and accurate solutions to the problem for any degree of resolution on uniform grids with the adaptivity of the shape parameter of the RBFs. Even with a highly small number of grid points, non-oscillatory and accurate solutions to the singular perturbation problem can be obtained with the optimal value of the shape parameter. We show that even the error vanishes for some cases. We also provide a possible suggestion to choose the optimal shape parameter that minimizes the error using the total variation norm of the numerical solution.

A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov–Rubenchik equations

In this study, we examine numerical solutions of Zakharov–Rubenchik system which is a coupled nonlinear partial differential equation. The numerical method in the current study is based on radial basis function finite difference (RBF-FD) meshless method and an explicit Runge–Kutta method. As a radial basis function we choose polyharmonic spline augmented with polynomials. The essential motivation for choosing polyharmonic spline is that it is free of shape parameter which has a crucial role in accuracy and stability of meshless methods. The main benefit of the proposed method is the approximation of the differential operators is performed on local-support domain which produces sparse differentiation matrices. This reduces computational cost remarkably. To see performance of the proposed method, some test problems are solved. L∞ error norms and conserved quantities such as mass and energy are calculated. Numerical outcomes are presented and compared with other methods available in the literature. From the comparison it can be deduced that the proposed method gives reliable and precise results with low computational cost. Stability of the proposed method is also discussed.

Simulation flows with multiple phases and components via the radial basis functions-finite difference (RBF-FD) procedure: Shan-Chen model

The current paper describes a rapid and impressive numerical technique to simulate the flows with multiple phases and components based upon the Shan-Chen model. The Shan-Chen model is a generalization of the incompressible Navier-Stokes equation with the variable density. The local radial basis function-generated finite difference method or local RBF-FD approach has been developed to solve the considered model. Furthermore, to get acceptable results, an extra diffusion term has been added to the density equation with a small coefficient. A reduced order technique e.g. proper orthogonal decomposition (POD) idea has been utilized to achieve a fast numerical procedure and to decrease the elapsed CPU time. The proposed numerical experiments show the ability, efficiency and acceptable results of the presented numerical scheme.

Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method

The main propose of this investigation is to introduce a rapid and impressive numerical procedure to simulate the time dependent incompressible Navier–Stokes equation with variable density. The developed formulation is constructed by using the meshfree RBF-FD technique. Also, to improve the numerical results, an extra diffusion term has been added to the density equation with a small coefficient. On the other hand, a reduced order technique e.g. proper orthogonal decomposition (POD) has been employed to get a fast numerical method and to decrease the elapsed CPU time. On the other hand, since the considered problem is fully nonlinear, thus in the reduced order model based on the POD idea, there are some nonlinear terms that they take more computational time. Hence, we employ the discrete empirical interpolation method (DEIM) to overcome the nonlinear terms. Four test problems have been proposed to show the ability of the numerical simulations.

Thermal analysis of heat transfer in pipe cooling concrete structure by a meshless RBF-FD method combined with an indirect model

Numerical simulation of heat conduction in the concrete structure with cooling pipes is meaningful and full of challenges. In this paper, the radial basis function finite difference method (RBF-FD) combined with a self-correcting prediction model (SCM) is proposed for thermal analysis of heat transfer in concrete structures with pipe cooling system. Using the Multiquadric radial basis function (MQ-RBF), the SCM is proposed to give full play to advantages of the localized meshless method. This technique allows use of more number of nodes in the whole domain to solve the ill-conditioning problem associated with multiple regions of rapid variation and increases the accuracy of models. A multiple-scale technique is applied to determine the shape parameter in the MQ-RBF and the fictitious nodes method is modified by using just one fictitious node outside the domain to deal with boundary conditions of the third kind in tiny holes. To show the flexibility and efficiency of the proposed scheme, we consider several examples with different number, size, and positions of pipes. Meanwhile, a guidance of determination of effected area near a tiny hole is provided and a simple optimization design of cooling pipe system is performed. The results demonstrate advantages of the proposed model which can effectively handle heat conduction problem with multiple holes subjected to the third kind boundary conditions.

A Robust Hyperviscosity Formulation for Stable RBF-FD Discretizations of Advection-Diffusion-Reaction Equations on Manifolds

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds $\mathbb{M} \subset \mathbb{R}^3$ of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and surface advection-diffusion. Finally, we demonstrate the applicability of our formulation to advection-diffusion-reaction equations on manifolds described purely as point clouds. Our surface discretizations use the recently developed RBF-least orthogonal interpolation method and, with the addition of hyperviscosity, are now empirically high-order accurate, stable, and free of stagnation errors.

Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

An Upwind Local Radial Basis Functions-finite Difference (RBF-FD) Method for Solving Compressible Euler Equation with Application in Finite-rate Chemistry

The main aim of the current paper is to propose an upwind local radial basis functions-finite difference (RBF-FD) method for solving compressible Euler equation. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations and finite-rate chemistry is studied. The presented technique is based on the developed idea in [58]. For checking the ability of the new procedure, the compressible Euler equation is solved. This equation has been classified in category of system of advection-diffusion equations. The solutions of advection equations have some shock, thus, special numerical methods should be applied for example discontinuous Galerkin and finite volume methods. Moreover, two problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify implementation and benchmarking of new methods. Some of the examples are chosen within a subset of problems where collocation is traditionally known to perform sub-par, namely when the solution sought is non-smooth, i.e. contains discontinuities, singularities or sharp gradients. For such problems and other simpler ones with smooth solutions, we study in depth the influence of the weight function, correction function, and the number of nodes in a given support. We also propose new stabilization approaches to improve the accuracy of the numerical methods. In particular, we experiment with the use of a Voronoi diagram for weight computation, collocation method stabilization approaches, and support node selection for problems with singular solutions. With an appropriate selection of the above-mentioned parameters, the resulting collocation methods are compared to the moving least-squares method (and variations thereof), the radial basis function finite difference method and the finite element method. Extensive tests involving two and three dimensional problems indicate that the methods perform well in terms of efficiency (accuracy versus computational time), even for non-smooth solutions.

An RBF-FD sparse scheme to simulate high-dimensional Black–Scholes partial differential equations

The objective of this research is to investigate the numerical solution of high-dimensional Black–Scholes partial differential equations (PDEs). To do this, the weights of the radial basis function-finite difference (RBF-FD) scheme using inverse multi-quadric function are used with an emphasis on the hotzone. The proposed RBF-FD technique is capable to price multi-asset options efficiently.

A high-order RBF-FD method for option pricing under regime-switching stochastic volatility models with jumps

In this paper, we develop a high-order radial basis function finite difference (RBF-FD) approximation on a five-point stencil for pricing options under the regime-switching stochastic volatility models with log-normal and contemporaneous jumps (SVCJ). We fully exploit the capabilities of the RBF-FD to perform the interpolation, differentiation and integration approximations tasks required in the numerical solution of the SVCJ pricing partial integro-differential equation (PIDE). The resulting systems of equations are sparse and our treatment of the non-local integro discretisation allows an efficient implementation based on the fast Fourier transform (FFT) algorithm. We show that a local mesh refinement strategy gives high-order convergent solutions for both the European and the path dependent barrier option prices.

A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.

Improved numerical solution of multi-asset option pricing problem: A localized RBF-FD approach

The objective of this work is to present a novel procedure for tackling European multi-asset option problems, which are modeled mathematically in terms of time-dependent parabolic partial differential equations with variable coefficients. To use as low as possible of number computational grid points, a non-uniform grid is generated while a radial basis function-finite difference scheme with the Gaussian function is applied on such a grid to discretize the model as efficiently as possible. To reduce the burdensome for tackling the resulting set of ordinary differential equations, a Krylov method, which is due to the application of exponential matrix function on a vector, is taken into account. The combination of these techniques reduces the computational effort and the elapsed time. Several experiments are brought froward to illustrate the superiority of the new improved approach. In fact, the contributed procedure is capable to tackle even 6D PDEs on a normally-equipped computer quickly and efficiently.

Development and validation of a 3D RBF-spectral model for coastal wave simulation

With the objective of simulating wave propagation in the nearshore zone for engineering-scale applications, a two dimensional (2DV) model based on the Euler–Zakharov equations [73,54] is extended to three dimensions (3D). To maintain the flexibility of the approach with the goal of applying the model to irregularly shaped domains, the horizontal plane is discretized with scattered nodes. The horizontal derivatives are then estimated using the Radial Basis Function-Finite Difference (RBF-FD) method, while a spectral approach is used in the vertical dimension. A sensitivity analysis examined the robustness of the RBF-FD approach as a function of RBF parameters when estimating the derivatives of a representative function. For a targeted stencil size between 20 and 30 nodes, Piecewise-Smooth (PS) polyharmonic spline (PHS) functions are recommended, avoiding the use of Infinitely-Smooth (IS) RBFs, which are less appropriate for the desired applications because of their dependence on a shape parameter. Comparisons of simulation results to observations from two wave basin experiments show that nonlinear effects induced by complex bottom bathymetries are reproduced well by the model with the recommended RBF approach, validating the use of this method for 3D simulations of wave propagation.

Implementation of a scalable, performance portable shallow water equation solver using radial basis function-generated finite difference methods

In this article, we describe and analyze the computational performance of a parallel shallow water equation (SWE) solver for atmospheric simulation using radial basis function-finite difference (RBF-FD) methods. The inherent “meshless” nature of RBF-FD methods provides significant numerical benefits over standard pseudospectral and traditional FD methods, but there are many challenges in terms of their performance and parallel implementation, due to RBF-FDs use of relatively large halos and unstructured indexing. With the use of reverse Cuthill–McKee node ordering and tiled transposition of the state variable matrices and RBF-FD differentiation matrices, these challenges were overcome. The RBF-FD solver was implemented for the SWE on the rotating sphere using message passing interface plus OpenMP/OpenACC to demonstrate scalability and performance portability on the three currently dominant high performance computing (HPC) architectures, namely, Intel Xeon multicore, Intel Xeon Phi manycore, and NVIDIA graphics processing unit systems.

An RBF–FD method for pricing American options under jump–diffusion models

American option prices under jump–diffusion models are determined as solutions to partial integro-differential equations (PIDE). In this paper a new combination of a time and spatial discretization applied to a linear complementary formulation (LCP) of the free boundary PIDE is proposed. First a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain. An implicit–explicit time discretization applied to the reformulated LCP on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system. The radial basis function (RBF) finite difference method is a local method resulting in a sparse linear system in contrast to global RBF-methods which lead to ill-conditioned dense matrix systems. For the corresponding European option we prove consistency, stability and second-order convergence in a discrete L2-norm. We derive mild conditions for the model parameters under which these results hold. Numerical experiments are performed with European and American options, and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm.

A Gaussian radial basis function-finite difference technique to simulate the HCIR equation

Localized meshless radial basis functions (RBFs), which are due to the application of RBFs and finite difference (FD) methods at the same time, are popular well-resulted computational tools for tackling higher dimension partial differential equations (PDEs). In this research, the objective is not only to study a spatial discretization of the financial Heston–Cox–Ingresoll–Ross (HCIR) PDE, but also to apply a non-uniform distribution of nodes for the application of Gaussian RBFs. The combination of these ideas simulate the HCIR PDE quickly and efficiently.

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

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.

RBF-FD meshless optimization using direct search (GLODS) in the analysis of composite plates

An optimization technique, Global and Local Optimization using Direct Search (GLODS) is used to optimize stencil size and shape parameter of the multiquadric function used in the Radial Basis Function-Finite Difference (RBF-FD) meshless numerical method. The mechanical problem to be solved by the meshless method is a composite plate in bending, under sinusoidal load. Results show GLODS is capable of finding values for stencil size and shape parameter that produce excellent numerical solutions when compared with analytical solutions.