High order finite volume methods using multi-moments or multi-moment constraints:basic idea, numerical formulations & applications to geophysical fluid dynamics
Seminar Room 1, Newton Institute
Discrete quantities, such as cell integrated average, point value and derivatives, which are appellatively called moments in our context, reveal different respects of a physical field. Using more than two kinds of these quantities simultaneously as the predicted variables or the constraints to derive the evolution equations for the predicted variables leads to a class of schemes that are different from the conventional finite difference or finite volume methods. Rather than the Galerkin inner product procedure, the moments in a high order multi-moment finite volume (MV) or multi-moment constrained finite volume (MCV) scheme can be chosen through a more intuitive and physically motivated way, which allows greater flexibility in defining the computational variables and in deriving the corresponding prognostic equations to update the unknowns. Different moments are connected by a local (cell-wise) reconstruction, and time marching is based on a set of equations which can be of different forms but consistent to the original governing equation(s). For example, a semi-Lagrangian scheme which maps the point values, can be combined with a finite volume formulation, which predicts the cell integrated values through a flux form, to devise a conservative scheme. The moments can be either used directly as the prognostic variables as in an MV scheme, which can be interpreted as a modal type method, or used as the constraints to generate the equations to update the unknowns. A representative formulation of the latter is the nodal type MCV method, in which the unknowns are the point values defined at the solution points, and the prognostic equations to predict these unknowns are derived from the constraint conditions in terms of different moments. Multi-moment constraint concept also applies to the flux reconstruction formulation for conservation laws, which gives a more general platform to accommodate many existing high order scheme, including discontinuous Galerkin method and spectral element method. Using multi-moment constraints when reconstructing the numerical flux function makes the present method distinguished from other high order schemes. High order multi-moment methods have attractive properties for practical applications, such as algorithmic simplicity, flexibility and computational efficiency, and have been applied to various problems in computational fluid dynamics, such as compressible and incompressible flows, interfacial multi-phase flows. Efforts have also been made to develop numerical models for geophysical flows. This talk will present the underlying idea of the methods, typical schemes and the major differences compared to other existing methods. Progress of using the multi-moment approach to develop high order numerical models for geophysical fluid dynamics will be reported as well.