Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers’ equation

Abstract

An efficient and accurate hybrid radial basis function (HRBF) method is proposed to numerically solve the quasi-linear viscous Burgers' equation. Two different RBFs are combined: an infinite smooth RBF defined by a shape parameter and a piecewise smooth RBF independent of the shape parameter. This combination formulates a family of HRBF methods, which is then used to approximate the spatial operator of the viscous Burgers equation. An efficient high-order solver is then used to integrate in time the resulting initial value problem. The developed numerical scheme is tested on some benchmark problems varying the Reynolds number. Accuracy and efficiency is validated using L ∞ , L 2 and L rms error norms, as well as the number of nodes N over the domain of influence. A rigorous comparison is conducted against well-known numerical schemes to manifest improved accuracy of the proposed scheme. Eigenvalue stability analysis of the proposed technique is thoroughly discussed and confirmed by numerical examples. The proposed scheme is found a well-behaved alternative to RBF (Kansa) and RBF-PS methods with improved accuracy and good conditioning properties. Keywords Burgers' equation• Hybrid RBF • Method of lines • Eigenvalues stability • Reynolds number • RBF-PS and Kansa method

A Hybrid RBF Collocation Method and Its Application in the Elastostatic Symmetric Problems

“…As presented in [24], by coupling the infinite smoothing RBFs (φ 1 ), such as MQ, IMQ, and GA, with the piecewise smooth RBFs Sn and TPS RBFs (φ 2 ), the HRBF is constructed and can be defined as…”

“…It has been dictated in Mishra's article [33] that a small value of the weight parameter w usually gives acceptable accuracy with a stable spectrum (when all the eigenvalues of the differentiation matrix lie in the left half-plane). In Manzoor's article [24], the authors calculated many numerical examples and concluded that in most of the test examples, for the three types of hybrid radial basis functions, MQ + S3, GS + S3, and IMQ + S3, the weight parameter w = 10 −3 works best. On the other hand, for the three types of hybrid radial basis functions, MQ + TPS, GS + TPS, and IMQ + TPS, the weight parameter w = 10 −9 .…”

“…Satisfying the patch test requires the same linear function to give the displacement of any interior node and requires the strains and stresses to be constant in the patch. In this problem, we randomly choose the shape parameter c = 1, the weight parameter referring to Manzoor's article [24], the variant related to TPS, the weight parameter w = 10 −9 , the variant associated with S3, and the weight parameter w = 10 −3 . The HRBF passes the patch test with precision, as well as the traditional RBF.…”

“…Different radial basis functions have different accuracies. The exact results of the internal node calculations for the MQ-HRBF and the traditional MQ-RBF are shown in Table 3 when the parameters are taken as E = 1 and v = 0.3. the weight parameter referring to Manzoor's article [24], the variant related to TPS, the weight parameter 𝑤 = 10 , the variant associated with S3, and the weight parameter 𝑤 = 10 . The HRBF passes the patch test with precision, as well as the traditional RBF.…”

“…However, there are still some deficiencies in the calculation accuracy, and the operation time is too long. Recently, Manzoor [24] used this similar idea to propose a hybrid radial basis function (HRBF) method, which combines two different RBFs: an infinite smooth RBF defined by shape parameters and a piecewise smooth RBF that is independent of shape parameters. This method has better accuracy when solving Burgers' equations.…”