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discretization, while being accurate. Recently, Huynh (2007) introduced a new approach to high-order accuracy called flux reconstruction (FR), where equations are solved in differential form. An interesting feature of the FR approach is that it unifies several existing element-based high-order methods such as DG, spectral difference, and spectral volume methods into a common framework. The discontinuous fluxes at the element edges are corrected by correction functions , which maintain flux continuity
discretization, while being accurate. Recently, Huynh (2007) introduced a new approach to high-order accuracy called flux reconstruction (FR), where equations are solved in differential form. An interesting feature of the FR approach is that it unifies several existing element-based high-order methods such as DG, spectral difference, and spectral volume methods into a common framework. The discontinuous fluxes at the element edges are corrected by correction functions , which maintain flux continuity
means the material time derivative d / dt = ∂/∂ t + v · ∇ . The second and third equations, (2) and (3) , provide a simple thermodynamic model for ice growth. These are continuity equations for the ice concentration and effective thickness, respectively. The terms S A and S h are thermodynamic source terms. A full description of the model can be found in the references ( Hibler 1979 ; Gray and Morland 1994 ). To complete the model, the differential equations are augmented by initial
means the material time derivative d / dt = ∂/∂ t + v · ∇ . The second and third equations, (2) and (3) , provide a simple thermodynamic model for ice growth. These are continuity equations for the ice concentration and effective thickness, respectively. The terms S A and S h are thermodynamic source terms. A full description of the model can be found in the references ( Hibler 1979 ; Gray and Morland 1994 ). To complete the model, the differential equations are augmented by initial
n +1 = Y n , s +1 . Thus, the update and the internal stages are treated in a unique way. The coefficient matrices ( α ) ij = α ij , ( β ) ij = β ij , ( γ ) ij = γ ij are strictly lower triangular. We have s computational stages where the differential equation for Z n , i has to be solved because the first stage simply sets Y n ,1 = y n . The length of the integration interval in each stage is given by d i . In an implementation the exact integration procedure for z ′ = f c
n +1 = Y n , s +1 . Thus, the update and the internal stages are treated in a unique way. The coefficient matrices ( α ) ij = α ij , ( β ) ij = β ij , ( γ ) ij = γ ij are strictly lower triangular. We have s computational stages where the differential equation for Z n , i has to be solved because the first stage simply sets Y n ,1 = y n . The length of the integration interval in each stage is given by d i . In an implementation the exact integration procedure for z ′ = f c
multiple of the forcing period; (ii) arranged as n cycles, such that the sum of every n periods was an integer multiple of the forcing period; or (iii) continuous, with the average QBO period being an irrational multiple of the forcing period (or at least a rational multiple with a large denominator). In this paper we attempt to interpret the abovementioned observations by studying a class of equations that we call descent rate models . These models are simple ordinary differential equations that
multiple of the forcing period; (ii) arranged as n cycles, such that the sum of every n periods was an integer multiple of the forcing period; or (iii) continuous, with the average QBO period being an irrational multiple of the forcing period (or at least a rational multiple with a large denominator). In this paper we attempt to interpret the abovementioned observations by studying a class of equations that we call descent rate models . These models are simple ordinary differential equations that
1. Introduction The critical importance of the scheme used for integrating stochastic differential equations (SDEs) is clearly demonstrated in Ewald et al. (2004 , hereafter EPT04 ). In EPT04 the authors contrast an implicit stochastic integration scheme of Ewald and Temam (2003) with a “naive” implementation of a similar implicit scheme (see their Fig. 2). The naive technique is unable to replicate the system’s climatological probability distribution function (PDF), and the authors state
1. Introduction The critical importance of the scheme used for integrating stochastic differential equations (SDEs) is clearly demonstrated in Ewald et al. (2004 , hereafter EPT04 ). In EPT04 the authors contrast an implicit stochastic integration scheme of Ewald and Temam (2003) with a “naive” implementation of a similar implicit scheme (see their Fig. 2). The naive technique is unable to replicate the system’s climatological probability distribution function (PDF), and the authors state
channel) is simply the derivative of the channel radius In practice, one solves (10) numerically for the channel radius given appropriate initial conditions, and then a numerical derivative of the solution is taken to obtain the radial velocity. Details of the solution process are provided in the next subsection. f. Numerical solution Equation (10) is a first-degree, second-order, nonlinear ordinary differential equation (ODE) for the radius as a function of time, with the altitude viewed as a
channel) is simply the derivative of the channel radius In practice, one solves (10) numerically for the channel radius given appropriate initial conditions, and then a numerical derivative of the solution is taken to obtain the radial velocity. Details of the solution process are provided in the next subsection. f. Numerical solution Equation (10) is a first-degree, second-order, nonlinear ordinary differential equation (ODE) for the radius as a function of time, with the altitude viewed as a
, and ρ [ x , y , f ( x , y , t )] = ρ f ( x , y , t ) is the mass density, on the surface z = f ( x , y , t ). Thus, the term on the right-hand side of Eq. (14) may be kinematically interpreted as the differential flux of mass across the moving surface z = f ( x , y , t ). The general theory of mass and volume transport across isosurfaces of a balanced fluid property, that is, a fluid property obeying a balance equation, may be used to derive other balance configurations in the
, and ρ [ x , y , f ( x , y , t )] = ρ f ( x , y , t ) is the mass density, on the surface z = f ( x , y , t ). Thus, the term on the right-hand side of Eq. (14) may be kinematically interpreted as the differential flux of mass across the moving surface z = f ( x , y , t ). The general theory of mass and volume transport across isosurfaces of a balanced fluid property, that is, a fluid property obeying a balance equation, may be used to derive other balance configurations in the
through the definition of Ψ. However, the implicitness affects only the irrotational part of the wind. If the normal velocity is known at the boundaries of the domain, we may determine Ψ at time τ as the solution of the elliptic partial differential equation obtained by substituting the velocity formula into the continuity equation [ (55) ] ( Hunt and Hussain 1991 ). From the curl of (71) or, alternatively, (68) , we obtain the following explicit formula of the form (5) for the absolute
through the definition of Ψ. However, the implicitness affects only the irrotational part of the wind. If the normal velocity is known at the boundaries of the domain, we may determine Ψ at time τ as the solution of the elliptic partial differential equation obtained by substituting the velocity formula into the continuity equation [ (55) ] ( Hunt and Hussain 1991 ). From the curl of (71) or, alternatively, (68) , we obtain the following explicit formula of the form (5) for the absolute
DECEMBER 1993 SWA RZTRA UB ER 3415The Vector Harmonic Transform Method for SolvingPartial Differential Equations in Spherical Geometry PAUL N. SWARZTRAUBERNational Center for Atmospheric Research, * Boulder, Colorado(Manuscript received 9 February 1993, in final form 24 May 1993)ABSTRACT The development of computational methods for solving partial differential equations in spherical
DECEMBER 1993 SWA RZTRA UB ER 3415The Vector Harmonic Transform Method for SolvingPartial Differential Equations in Spherical Geometry PAUL N. SWARZTRAUBERNational Center for Atmospheric Research, * Boulder, Colorado(Manuscript received 9 February 1993, in final form 24 May 1993)ABSTRACT The development of computational methods for solving partial differential equations in spherical
known that in some cases [see (3.10) below] even splitting a continuous system gives rise to errors. In such cases, obviously, splitting errors of the discretized system will contain additional errors. It is not clear what kind of splitting errors are responsible in CLZ98 and C98. Considering the framework of ODEs, which could be thought of as arising from a semidiscretization of evolutionary partial differential equations (PDEs) we try to distinguish these errors and suggest some remedial
known that in some cases [see (3.10) below] even splitting a continuous system gives rise to errors. In such cases, obviously, splitting errors of the discretized system will contain additional errors. It is not clear what kind of splitting errors are responsible in CLZ98 and C98. Considering the framework of ODEs, which could be thought of as arising from a semidiscretization of evolutionary partial differential equations (PDEs) we try to distinguish these errors and suggest some remedial
