WorkshopsTutorial Title: Construct high-order schemes using multi-moments or multi-moment constraints
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Discrete quantities, such as cell integrated average, point value and derivatives, which are appellatively called moments in our context, reveal different respects of the 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 local (cell-wise) reconstructions, and time marching is based on a set of equations which can be of different forms but consistent to the original governing equation(s). 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 other alternatively defined 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 provides a more general framework to accommodate many existing high order scheme, including discontinuous Galerkin method and spectral element method. Using multi-moment constraints to reconstruct the numerical flux function distinguishes the present methodology from other existing ones. 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. This talk will present the underlying idea of the methods, typical schemes and applications to fluid dynamics.
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Feng Xiao
2012-11-06
14:10:00 - 15:20:00
308 , Mathematics Research Center Building (ori. New Math. Bldg.)
Discrete quantities, such as cell integrated average, point value and derivatives, which are appellatively called moments in our context, reveal different respects of the 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 local (cell-wise) reconstructions, and time marching is based on a set of equations which can be of different forms but consistent to the original governing equation(s). 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 other alternatively defined 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 provides a more general framework to accommodate many existing high order scheme, including discontinuous Galerkin method and spectral element method. Using multi-moment constraints to reconstruct the numerical flux function distinguishes the present methodology from other existing ones. 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. This talk will present the underlying idea of the methods, typical schemes and applications to fluid dynamics.