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Courant Number (courant + number)
Selected AbstractsA second order discontinuous Galerkin method for advection on unstructured triangular meshesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2003H. J. M. Geijselaers Abstract In this paper the advection of element data which are linearly distributed inside the elements is addressed. Across element boundaries the data are assumed discontinuous. The equations are discretized by the Discontinuous Galerkin method. For stability and accuracy at large step sizes (large values of the Courant number), the method is extended to second order. Furthermore the equations are enriched with selective implicit terms. This results in an explicit and local advection scheme, which is stable and accurate for Courant numbers less than .95 on unstructured triangle meshes. Results are shown of some pure advection test problems. Copyright © 2003 John Wiley & Sons, Ltd. [source] Dispersion and stability analyses of the linearized two-dimensional shallow water equations in boundary-fitted co-ordinatesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003S. Sankaranarayanan Abstract In the present investigation, a Fourier analysis is used to study the phase and group speeds of a linearized, two-dimensional shallow water equations, in a non-orthogonal boundary-fitted co-ordinate system. The phase and group speeds for the spatially discretized equations, using the second-order scheme in an Arakawa C grid, are calculated for grids with varying degrees of non-orthogonality and compared with those obtained from the continuous case. The spatially discrete system is seen to be slightly dispersive, with the degree of dispersivity increasing with an decrease in the grid non-orthogonality angle or decrease in grid resolution and this is in agreement with the conclusions reached by Sankaranarayanan and Spaulding (J. Comput. Phys., 2003; 184: 299,320). The stability condition for the non-orthogonal case is satisfied even when the grid non-orthogonality angle, is as low as 30° for the Crank Nicolson and three-time level schemes. A two-dimensional wave deformation analysis, based on complex propagation factor developed by Leendertse (Report RM-5294-PR, The Rand Corp., Santa Monica, CA, 1967), is used to estimate the amplitude and phase errors of the two-time level Crank,Nicolson scheme. There is no dissipation in the amplitude of the solution. However, the phase error is found to increase, as the grid angle decreases for a constant Courant number, and increases as Courant number increases. Copyright © 2003 John Wiley & Sons, Ltd. [source] A Petrov,Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2001Seung-Buhm Woo Abstract A new finite element method is presented to solve one-dimensional depth-integrated equations for fully non-linear and weakly dispersive waves. For spatial integration, the Petrov,Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C2 -continuity. For the time integration an implicit predictor,corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth-order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd. [source] Tracking accuracy of a semi-Lagrangian method for advection,dispersion modelling in riversINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2007S. Néelz Abstract There is an increasing need to improve the computational efficiency of river water quality models because: (1) Monte-Carlo-type multi-simulation methods, that return solutions with statistical distributions or confidence intervals, are becoming the norm, and (2) the systems modelled are increasingly large and complex. So far, most models are based on Eulerian numerical schemes for advection, but these do not meet the requirement of efficiency, being restricted to Courant numbers below unity. The alternative of using semi-Lagrangian methods, consisting of modelling advection by the method of characteristics, is free from any inherent Courant number restriction. However, it is subject to errors of tracking that result in potential phase errors in the solutions. The aim of this article is primarily to understand and estimate these tracking errors, assuming the use of a cell-based backward method of characteristics, and considering conditions that would prevail in practical applications in rivers. This is achieved separately for non-uniform flows and unsteady flows, either via theoretical considerations or using numerical experiments. The main conclusion is that, tracking errors are expected to be negligible in practical applications in both unsteady flows and non-uniform flows. Also, a very significant computational time saving compared to Eulerian schemes is achievable. Copyright © 2006 John Wiley & Sons, Ltd. [source] A 2D implicit time-marching algorithm for shallow water models based on the generalized wave continuity equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2004Kendra M. Dresback Abstract This paper builds upon earlier work that developed and evaluated a 1D predictor,corrector time-marching algorithm for wave equation models and extends it to 2D. Typically, the generalized wave continuity equation (GWCE) utilizes a three time-level semi-implicit scheme centred at k, and the momentum equation uses a two time-level scheme centred at k+12. It has been shown that in highly non-linear applications, the algorithm becomes unstable at even moderate Courant numbers. This work implements and analyses an implicit treatment of the non-linear terms through the use of an iterative time-marching algorithm in the two-dimensional framework. Stability results show at least an eight-fold increase in the maximum time step, depending on the domain. Studies also examined the sensitivity of the G parameter (a numerical weighting parameter in the GWCE) with results showing the greatest increase in stability occurs when 1,G/,max,10, a range that coincides with the recommended range to minimize errors. Convergence studies indicate an increase in temporal accuracy from first order to second order, while overall error is less than the original algorithm, even at higher time steps. Finally, a parallel implementation of the new algorithm shows that it scales well. Copyright © 2004 John Wiley & Sons, Ltd. [source] SLICE-S: A Semi-Lagrangian Inherently Conserving and Efficient scheme for transport problems on the SphereTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 602 2004Mohamed Zerroukat Abstract The Semi-Lagrangian Inherently Conserving and Efficient (SLICE) scheme developed for Cartesian geometry is generalized to spherical geometry. The spherical version, SLICE-S, is similarly based on a Control Volume approach and multiple sweeps of a one-dimensional O(,s4) (where s is the spherical distance) conservative remapping algorithm along Eulerian latitudes, then along Lagrangian longitudes. The resulting conservative scheme requires no restriction on either the polar meridional or zonal Courant numbers. SLICE-S is applied to the standard problems of solid-body rotation and deformational flow, and results are compared with those of a standard non-conservative and other published conservative semi-Lagrangian schemes. In addition to mass conservation, and consistent with the performance of SLICE, the present scheme is competitive in terms of accuracy and efficiency. © Crown copyright, 2004. Royal Meteorological Society [source] | ||||||||||