scholarly journals A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Yaw Kyei ◽  
John Paul Roop ◽  
Guoqing Tang
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Three Dimensional ◽  
Difference Schemes ◽  
Poisson’S Equation ◽  
Sixth Order ◽  
Finite Difference Schemes ◽  
Poisson's Equation ◽  
Compact Finite Difference ◽  
Compact Finite Difference Schemes

We derive a family of sixth-order compact finite-difference schemes for the three-dimensional Poisson's equation. As opposed to other research regarding higher-order compact difference schemes, our approach includes consideration of the discretization of the source function on a compact finite-difference stencil. The schemes derived approximate the solution to Poisson's equation on a compact stencil, and thus the schemes can be easily implemented and resulting linear systems are solved in a high-performance computing environment. The resulting discretization is a one-parameter family of finite-difference schemes which may be further optimized for accuracy and stability. Computational experiments are implemented which illustrate the theoretically demonstrated truncation errors.

Prefactored optimized compact finite-difference schemes for second spatial derivatives

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB87-WB95 ◽  
Author(s):  
Hongbo Zhou ◽  
Guanquan Zhang
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Compact Scheme ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Compact Schemes ◽  
Compact Finite Difference ◽  
Wide Range ◽  
Compact Finite Difference Schemes ◽  
Spatial Derivatives

We have described systematically the processes of developing prefactored optimized compact schemes for second spatial derivatives. First, instead of emphasizing high resolution of a single monochromatic wave, we focus on improving the representation of the compact finite difference schemes over a wide range of wavenumbers. This leads to the development of the optimized compact schemes whose coefficients will be determined by Fourier analysis and the least-squares optimization in the wavenumber domain. The resulted optimized compact schemes provide the maximum resolution in spatial directions for the simulation of wave propagations. However, solving for each spatial derivative using these compact schemes requires the inversion of a band matrix. To resolve this issue, we propose a prefactorization strategy that decomposes the original optimized compact scheme into forward and backward biased schemes, which can be solved explicitly. We achieve this by ensuring a property that the real numerical wavenumbers of both the forward and backward biased stencils are the same as that of the original central compact scheme, and their imaginary numerical wavenumbers have the same values but with opposite signs. This property guarantees that the original optimized compact scheme can be completely recovered after the summation of the forward and backward finite difference operators. These prefactored optimized compact schemes have smaller stencil sizes than even those of the original compact schemes, and hence, they can take full advantage of the computer caches without sacrificing their resolving power. Comparisons were made throughout with other well-known schemes.

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