Guidelines for RBF-FD Discretization: Numerical Experiments on the Interplay of a Multitude of Parameter Choices

Abstract

There exist several discretization techniques for the numerical solution of partial differential equations. In addition to classical finite difference, finite element and finite volume techniques, a more recent approach employs radial basis functions to generate differentiation stencils on unstructured point sets. This approach, abbreviated by RBF-FD (radial basis function-finite difference), has gained in popularity since it enjoys several advantages: It is (relatively) straightforward, does not require a mesh and generalizes easily to higher spatial dimensions. However, its application is not quite as blackbox as it may appear at first sight. The computed solution might suffer severely from various sources of errors if RBF-FD parameters are not selected carefully. Through comprehensive numerical experiments, we study the influence of several of these parameters on the condition numbers of intermediate (local) weight matrices, on the condition number of the resulting (global) stiffness matrix and ultimately on the approximation error of the computed discrete solution to the partial differential equation. The parameters of investigation include the type of RBF (and its shape or other parameters if applicable), the degree of polynomial augmentation, the discretization stencil size, the underlying type of point set (structured/unstructured), and the total number of (interior and boundary) points to discretize the PDE, here chosen as a three-dimensional Poisson’s problem with Dirichlet boundary conditions. Numerical tests on a sphere as well as tests for the convection-diffusion equation are included in a supplement and demonstrate that the results obtained for the Laplace problem on a cube generalize to wider problem classes. The purpose of this paper is to provide a comprehensive survey on the various components of the basic algorithms for RBF-FD discretization and steer away from potential pitfalls such as computationally more expensive setups which not always lead to more accurate numerical solutions. We guide toward a compatible selection of the multitude of RBF-FD parameters in the basic version of RBF-FD. For many of its components we refer to the literature for more advanced versions.

A superconvergence result in the RBF-FD method

“…As our discretisation is irregular, we have computed these errors over 30 different discretisation sets, varied by changing the random seed in our previously mentioned discretisation algorithm. We can see that for each monomial augmentation degree m, the errors scale as approximately IOP Publishing doi:10.1088/1742-6596/2766/1/012161 4 ∝ h m−1 , which is in agreement with the theory -operator discretisation error should scale as ∝ h m+1−l , where l is the derivative order, in our case l = 2 [14]. Next we consider the solution error, expressing the error of the whole solution procedure when solving our chosen Poisson problem.…”

“…As this paper deals with a specific property of the RBF-FD method, we will assume the reader already has some familiarity with it and not delve into the details. If that is not the case, [14] should fill in the missing details. We then obtain an approximation of ∇ 2 in the following form:…”

Oscillatory behaviour of the RBF-FD approximation accuracy under increasing stencil size

“…We discretise the domain with the discretisation distance h = 0.01, first discretising the boundary and then the interior using the algorithm proposed in [4]. The Laplacian is then discretised using the RBF-FD algorithm as described in [5], where we choose the radial cubics as our PHS (φ(r) = r 3 ) augmented with monomials up to degree m = 3, inclusive. This requires us to associate to each discretisation point x i its stencil, which we take to consist of its n nearest neighbours.…”

Stable meshless discretization of two-dimensional Fisher-type equations by local multiquadric method over non-rectangular domains

“…In fact, the presence of distance function (r) and shape parameter (c) in the RBFs make them unique and a preferred choice over polynomial basis functions. The successful application of global (MQ) RBF methods to solve various challenging PDEs can be seen in [3,6,10,[18][19][20]30]. Despite these, the strong-form global RBF meshless methods have the inherited drawback of solving large ill-conditioned linear system of equations when either the number of data points is large or a bad shape parameter choice is made [21].…”

“…Also, it is known that global RBF methods have theoretically proven spectral convergence, yet in applications, they are hard to achieve due to the limitations of machine double precision (which can be mitigated upon the use of variable-precision arithmetic at expense of increasing computational cost) [21]. This is where the local RBF-based (a.k.a RBF-FD) meshless methods come in demand to improve upon various aspects of global RBF methods [6]. These methods were independently introduced and investigated by Tolstykh et al [39,40] and by Shu et al [38]; and have recently witnessed popularity amongst scientists for solution of various challenging PDE problems [7,12,31,33,36].…”