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discussed in the previous chapter has provided a number of new insights. We can apply many standard pseudospectral procedures to RBF solvers. In particular, we now have “standard ” procedures for solving time-dependent PDEs with RBFs. In this chapter we illustrate how the RBF pseudospectral approach can be applied in a way very similar to standard polynomial pseudospectral methods. Among our numerical illustrations are several examples taken from the book [Trefethen (2000)] (see Programs 17, 35 and 36 there). We will also use the 1D transport equation from the previous chapter to compare the RBF and polynomial PS methods. fasshauer@iit.edu MATH 590 – Chapter 43 3Computing the RBF-Differentiation Matrix in MATLAB How to compute the discretized differential operators In order to compute, for example, a first-order differentiation matrix we need to remember that — by the chain rule — the derivative of an RBF will be of the general form ∂ x d ϕ(‖x‖) = ∂x r dr ϕ(r). We require both the distances, r, and differences in x, where x is the first component of x. In our first MATLAB subroutine DRBF.m we compute these distance and difference matrices on lines 5 and 6. The differentiation matrix is then given by (see lines 8–10) D = AxA −1. Note the use of the matrix right division operator / or mrdivide i

Year: 2013

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