Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws

The theory of optimal recovery was applied to be reconstruction problem in finite volume methods for hyperbolic conservation laws. Polynomial recovery -the almost exclusively used reconstruction - was shown to be optimal only in trivial cases. Splines were identified as optimal recovery functions in certain associated semi-Hilbert spaces. As a particular spline we constructed the Beppo-Levi spline (or thin plate spline) in Beppo-Levi spaces of order m, where m>n/2, n denoting space dimension, has to be assumed to keep the Dirac functional continuous. An algorithm of ENO type was derived and its superiority over piecewise linear polynomial recovery was demonstrated by means of a linear model problem. In the case of the Euler equations the Beppo-Levi spline recovery lead to results well comparable with polynomial recovery, but drastic improvements as in the case of the linear model problem could not be observed. Two questions arise naturally. It is possible to generalise the theory of Beppo-Levi spline recovery and find other semi-Hilbert spaces (hence, find other recovery splines) useful for the recovery problem and, is it possible to find local functions with good approximation properties in order to avoid the oscillatory behaviour of globally defined splines? The first question is already answered through the work on radial basis functions which turn out to be splines in certain semi-Hilbert spaces. The inner structure of such semi-Hilbert spaces as well as the construction of other splines with regard to recovery in finite volume methods is content of a future paper. The second question was treated quite recently in a diploma thesis by Wendland and in work by Wu. They constrcuted positive definite radial basis functions with compact support on a differentiability scale. First numerical experiments indicate the superiority of these functions in the recovery problem in finite volume methods. (orig.)SIGLEAvailable from TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman