A comparison of kansa and hermitian RBF interpolation techniques for the solution of convection-diffusion problems

Mesh free modeling techniques are a promising alternative to traditional meshed methods for solving computational fluid dynamics problems. These techniques aim to solve for the field variable using solely the values of nodes and therefore do not require the generation of a mesh. This results in a process that can be much more reliably automated and is therefore attractive. Radial basis functions (RBFs) are one type of meshless method that has shown considerable growth in the past 50 years. Using these RBFs to directly solve a partial differential equation is known as Kansa\u27s method and has been used to successfully solve many flow problems. The problem with Kansa\u27s method is that there is no formal guarantee that its solution matrix will be non-singular. More recently, an expansion on Kansa\u27s method was proposed that incorporates the boundary and PDE operators into the solution of the field variable. This method, known as Hermitian method, has been shown to be non-singular provided certain nodal criteria are met. This work aims to perform a comparison between Kansa and Hermitian methods to aid in future selection of a method. These two methods were used to solve steady and transient one-dimensional convection-diffusion problems. The methods are compared in terms of accuracy (error) and computational complexity (conditioning number) in order to evaluate overall performance. Results suggest that the Hermitian method does slightly outperform Kansa method at the cost of a more ill-conditioned collocation matrix