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

The theory of optimal recovery is applied to finite volume methods for the numerical solution of conservation laws in multiple space dimensions. Classical polynomial ENO (essentially non-oscillatory) algorithms can be interpreted as trivial recovery operators for the point functional in the light of this theory. Thin plate splines are identified as optimal recovery functions in Beppo-Levi spaces and can be seen as multi-dimensional analogues of cubic splines. Recovery algorithms of ENO-type based on thin plate splines are developed and applied to test problems including the Euler equations of compressible gas dynamics.