An iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities

In [J.-H. Jung, Appl. Numer. Math. 57 (2007) 213–229], an adaptive multiquadric radial basis function method has been proposed for the reconstruction of discontinuous functions. Utilizing the vanishing shape parameters near the local jump discontinuity, the adaptive method considerably reduces the Gibbs oscillations and enhances convergence. In this paper, a new jump discontinuity detection method is developed based on the adaptive method. The global maximum of the expansion coefficients, λi, exists at the strongest jump discontinuity and its magnitude increases exponentially with the number of the center points, N. The adaptive method, however, dynamically reduces its magnitude to O(N) once applied. The global maximum of λi then exists at the next strongest jump discontinuity and its magnitude is exponentially large. In this way, the local jump discontinues are successively detected with the adaptive method applied iteratively. Numerical examples are provided using the piecewise analytic functions and the numerical solution of the shock interaction equations. Numerical results verify that the proposed method is efficient and accurate in finding local jump discontinuities.