Interpretation of Complex-Step-Finite-Difference Method and It's Implementation with Spectral Element Method

2017 ◽  
Author(s):  
Su Zeming* ◽  
Gao Jinghuai
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Element Method

scholarly journals Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

2014 ◽  
Vol 7 (6) ◽  
pp. 2717-2731 ◽  
Author(s):  
S.-J. Choi ◽  
F. X. Giraldo ◽  
J. Kim ◽  
S. Shin
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Gravity Wave ◽  
Euler Equations ◽  
Difference Method ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Basis Functions ◽  
Third Order ◽  
Vertical Coordinate

Abstract. The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge–Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma–pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.

Weighted Finite Difference and Boundary Element Methods Applied to Groundwater Pollution Problems

1991 ◽  
Vol 23 (1-3) ◽  
pp. 517-524
Author(s):  
M. Kanoh ◽  
T. Kuroki ◽  
K. Fujino ◽  
T. Ueda
Keyword(s):  
Boundary Element Method ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Element ◽  
Difference Method ◽  
Groundwater Pollution ◽  
Small Region ◽  
Boundary Element Methods ◽  
Computational Time ◽  
Element Method

The purpose of the paper is to apply two methods to groundwater pollution in porous media. The methods are the weighted finite difference method and the boundary element method, which were proposed or developed by Kanoh et al. (1986,1988) for advective diffusion problems. Numerical modeling of groundwater pollution is also investigated in this paper. By subdividing the domain into subdomains, the nonlinearity is localized to a small region. Computational time for groundwater pollution problems can be saved by the boundary element method; accurate numerical results can be obtained by the weighted finite difference method. The computational solutions to the problem of seawater intrusion into coastal aquifers are compared with experimental results.

A Hybrid Finite-difference and Spectral-element Method for Free-surface Topography Modeling

2018 ◽  
Author(s):  
P.P. Moghaddam ◽  
A. Rahimi Dalkhani ◽  
N. Keshavarz ◽  
H. Beyrami ◽  
M. Sharifi ◽  
...  
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Element Method

A novel green element method by mixing the idea of the finite difference method

2018 ◽  
Vol 95 ◽  
pp. 238-247 ◽  
Author(s):  
Xiang Rao ◽  
Linsong Cheng ◽  
Renyi Cao ◽  
Jun Jiang ◽  
Ning Li ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Green Element Method ◽  
The Finite Difference Method ◽  
Element Method

GALERKIN FINITE ELEMENT METHOD AND FINITE DIFFERENCE METHOD FOR SOLVING CONVECTIVE NON-LINEAR EQUATION

2010 ◽  
Vol 9 (1-2) ◽  
pp. 69
Author(s):  
E. C. Romão ◽  
M. D. De Campos ◽  
L. F. M. De Moura
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Non Linear ◽  
Element Method

The fast progress has been observed in the development of numerical and analytical techniques for solving convection-diffusion and fluid mechanics problems. Here, a numerical approach, based in Galerkin Finite Element Method with Finite Difference Method is presented for the solution of a class of non-linear transient convection-diffusion problems. Using the analytical solutions and the L2 and L∞ error norms, some applications is carried and valuated with the literature.

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