An overview on the multidimensional optimal order detection method

Abstract

Finite volume method is the usual framework to deal with numerical approximations for hyperbolic systems such as Shallow-Water or Euler equations due to its natural built-in conservation property. Since the first-order method produces too much numerical diffusion, popular second-order techniques, based on the MUSCL methodology, have been widely developed in the ’80s to provide both accurate solutions and robust schemes, avoiding non-physical oscillations in the vicinity of the discontinuities. Although second-order schemes are accurate enough for the major industrial applications, they still generate too much numerical diffusion for particular situations (acoustic, aeronautic, long time simulation for Tsunami) and very high-order methods i.e. larger than third-order, are required to provide an excellent approximation for local smooth solution as well as an efficient control on the spurious oscillations deriving from the Gibbs’ phenomenon. During the ’90s and up to nowadays, two main techniques have been developed to tackle the accuracy issue. The ENO/WENO which can cast in the finite volume context mainly concerns structured grids since the unstructured case turns to be very complex with a huge computational cost. The Discontinuous Galerkin method handles very well accurate approximations but the computational cost and implementation effort are also very high. In 2010 was published a seminal paper that proposed a radically different method. The philosophy consists to use an a posteriori approach to prevent from creating oscillations whereas the traditional methods employ an a priori method which dramatically cuts the accuracy order. In this document, I shall briefly present the MOOD method, show its main advantages and give an overview of the current applications.