On a high order numerical method for functions with singularities

Abstract

Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

Convergence acceleration of modified Fourier series in one or more dimensions

“…Usually the r th cardinal function p [i] r is specified to be a polynomial of degree 2(r + 1) − i [4,10,22], in which case q [i] r = x 2(r+1)−i and we refer to {p [i] r } as cardinal polynomials. This explains the name "polynomial subtraction".…”

“…As noted in [10], the previous lack of robust methods for the approximation of jump values is the central reason why the polynomial subtraction technique has not been more extensively utilised (see also [23, p. 101]). In this paper, to circumvent the aforementioned problem, we use Eckhoff's method for this task [8,9,10].…”

“…In this paper, to circumvent the aforementioned problem, we use Eckhoff's method for this task [8,9,10]. This approach is based on the well-known observation that the (modified) Fourier coefficients themselves contain sufficient information to reconstruct the jump values (see [8] for further references).…”

“…In standard implementations [4,10,20,22] such a function is constructed from a certain set of polynomials (the possibility of using other functions was suggested in [10]). Although this is the most convenient choice, it leads to poor numerical stability.…”

“…First, as we demonstrate in this paper, the combination of the multivariate version of this technique and hyperbolic cross index sets facilitates the construction of accurate approximations comprising only a small number of terms. Second, Eckhoff's method can be incorporated into spectral approximations of boundary value problems (see [10] and the references therein for hyperbolic problems and [1,2] for applications of modified Fourier expansions to second order boundary value problems). Finally, Eckhoff's technique is not restricted to Cartesian product domains.…”