Finite integration method with radial basis function for solving stiff problems

We combine in this paper the recently developed finite integration method (FIM) with radial basis function (FIM-RBF) to solve stiff problems. The idea of FIM is to transform partial differential equations (PDEs) into integral equations whose approximations can be stably and accurately obtained from standard numerical quadratures. This contributes to one distinct benefit: smoothing out the instability of solution process in solving stiff problems modelled by PDEs. The other distinct benefit comes from the truly meshfree advantage of radial basis function (RBF) method for solving various kinds of well-posed PDEs with superior accuracy. Since the RBF method gives an intrinsic full, and hence ill-conditioned, resultant matrix, it fails to tackle stiff problems. The combination of FIM and RBF method enjoys not only the distinct benefit of smoothing out stiffness but also gives a much less ill-conditioned resultant matrix. As a result, the headache problem on choosing critical value of shape parameter for spectral convergence in RBF does not persist. Numerical results indicated that the accuracy of the FIM-RBF has been increased by approximately two order of magnitude with only one third of CPU time in comparing to some other spectral methods such as wavelet adaptive scheme.