Pseudospectral Methods and Radial Basis Functions

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Both pseudospectral (PS) methods and radial basis functions (RBFs) were first proposed in the early 1970's, in connection with turbulence modeling and cartography respectively. Only in the last few years have the two topics begun to merge, in the area of high-accuracy solutions of partial differential equations (PDEs). Superior performance of PS methods was demonstrated early on in applications such as accurate long term integrations of wave- type equations in very simple geometries. In contrast to usual spectral basis functions, different RBFs are not distinguished by how rapidly they oscillate, but instead by being different translates of one single function. Although certain orthogonalities are lost, spectral accuracy not only remains, but can now also be reached on complex domains with arbitrary node distributions. A new (and counterintuitive) result tells that, in the limit of RBFs becoming increasingly flat, the classical PS methods are recovered. Recently great progress has been made in overcoming the high computer cost and numerical ill-conditioning that earlier were thought to severely limit this approach. The observations above point towards extensions and generalizations of almost all numerical methods which, by tradition, have been polynomial-based. The challenge has changed from exploring the potential of RBFs for arbitrary-geometry spectral methods for PDEs, into exploiting it.Most phenomena in science, engineering, sciences, and society can be approximated by some kinds of mathematical models. These often involve a construct known as partial differential equations (PDEs). Equations of this kind can only rarely be solved without resorting to computational methods. During the last century, a few main classes of such solution methods have evolved. A quite new class - pseudospectral (PS) methods - emerged some 30 years ago. Around the same time, radial basis functions (RBFs) were invented in a quite different context. The latter have still seen only exploratory use for PDEs. However, during the last three-year period, the PI demonstrated that PS methods can be seen as a special case of RBFs, with the latter approach greatly extending the PS methods' scope and generality. Continuing research will expand on this fundamental result, with the long-term goal of producing practical algorithms which are capable of becoming routine tools for high-precision solutions of PDEs in irregular multidimensional domains.