Most traditional numerical methods for approximating the solutions of problems in science, engineering, and mathematics require the data to be arranged in a structured pattern and to be contained in a simply shaped region, such as a rectangle or circle. In many important applications, this severe restriction on structure cannot be met, and traditional numerical methods cannot be applied. In the 1970s, radial basis function (RBF) methods were developed to overcome the structure requirements of existing numerical methods. RBF methods are applicable with scattered data locations. As a result, the shape of the domain may be determined by the application and not the numerical method. Radial basis function methods can be implemented both globally and locally. Comparisons between these two techniques are made in this work to illustrate how the local method can obtain very similar accuracy to the global method while only using a small subset of available points, and thus using substantially less computer memory. Finally, radial basis function methods are applied to solve systems of nonlinear partial differential equations (PDEs) that model pattern formation in mathematical biology. The local RBF method will be used to evaluate Turing pattern and chemotaxis models that are both modeled by advection-reaction-di↵usion type PDEs
