A LOCAL RADIAL BASIS FUNCTION METHOD FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

The method approximates the spatial derivatives by RBF interpolation using a small set of nodes in the neighborhood of any data center. This approach yields the generation of a small interpolation matrix for each data center and hence advancing solutions in time will be of comparatively lower cost. 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. Radial basis function (RBF) methods were developed to overcome the structure requirements of existing numerical methods. Radial basis function methods can be implemented both globally and locally.

The method approximates the spatial derivatives by RBF interpolation using a small set of nodes in the neighborhood of any data center. This approach yields the generation of a small interpolation matrix for each data center and hence advancing solutions in time will be of comparatively lower cost. 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. Radial basis function (RBF) methods were developed to overcome the structure requirements of existing numerical methods. Radial basis function methods can be implemented both globally and locally.