Research Article
On the Accuracy and Stability of the Meshless RBF Collocation Method for Neutron Diffusion Calculations
Abstract
Accuracy and stability are the main properties that make an algorithm preferable to its counterparts in modelling of physical phenomena. The radial basis function collocation method is a novel meshless technique, which exhibits an exponential convergence rate for the numerical solution of partial differential equations. However, it is a global approximation scheme and the ill-conditioning of the collocation matrix may become a serious issue if dense sets of interpolation nodes or high values of shape parameters are utilized. This study discusses four strategies to improve the accuracy and stability of the radial basis function collocation method for the numerical solution of the multigroup neutron diffusion equation. These strategies include using a higher precision value for computations, utilizing higher exponents for the generalized multiquadric, decreasing the value of the shape parameter with the number of nodes and singular value decomposition filtering. The results have shown that by using a higher precision value, choosing a variable shape parameter strategy and filtering the smallest singular values of the collocation matrix it is possible to improve the performance of the meshless collocation method, while increasing the exponent of the multiquadric results in a more accurate but less stable algorithm.
References
- [1] Cheng, A.H.D., Golberg, M.A., Kansa, E.J., Zammito, G. (2003). Exponential convergence and H-cmultiquadric collocation method for partial differential equations. Numerical Methods for Partial Differential Equations, 19:571-594.
- [2] Liu, G.R., Gu, Y.T. (2005). An Introduction to Meshfree Methods and Their Programming, Springer, Netherlands.
- [3] Kansa, E.J. (1990). Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-I. Computers and Mathematics with Applications, 19:127-145.
- [4] Chunyu, Z., Gong, C. (2018). Fast solution of neutron diffusion problem by reduced basis finite element method. Annals of Nuclear Energy, 111:702-708.
- [5] Tanbay, T, Ozgener, B. (2013). Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Mathematical and Computational Applications, 18:399-407.
- [6] Tanbay, T, Ozgener, B. (2014). A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations, Engineering Analysis with Boundary Elements, 46:30-40.
- [7] Fasshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB. World Scientific Publishing, Singapore.
- [8] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
- [9] Micchelli, C.A. (1986). Interpolation of scattered data-distance matrices and conditionally positive definite functions. Constructive Approximation, 2:11-22.
- [10] Fedoseyev, A.I., Friedman, M.J., Kansa, E.J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications, 43:439-455.
- [11] Cheng, A.H.D. (2012). Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation. Engineering Analysis with Boundary Elements, 36:220-239.
- [12] Fasshauer, G.E., Zhang, J.G. (2007). On choosing “optimal” shape parameters for RBF approximation. Numerical Algorithms, 45:345-368.
- [13] Wertz, J., Kansa, E.J., Ling, L. (2006). The role of the multiquadric shape parameters in solving elliptic partial differential equations. Computers and Mathematics with Applications, 51:1335-1348.
- [14] Fornberg, B., Zuev, J. (2007). The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers and Mathematics with Applications, 54:379-398.
- [15] Boyd, J.P. (2010). Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval. Computers and Mathematics with Applications, 60:3108-3122.
- [16] Golub, G.H. and Van Loan, C.F. (2013). Matrix Computations (4th ed.), The John Hopkins University Press, Baltimore, MD.
- [17] Akritas, A.G. and Malaschonok, G.I. (2004). Applications of singular value decomposition (SVD). Mathematics and Computers in Simulation, 67:15-31.
Year 2018,
Volume: 2 Issue: 1, 8 - 18, 20.06.2018
Abstract
References
- [1] Cheng, A.H.D., Golberg, M.A., Kansa, E.J., Zammito, G. (2003). Exponential convergence and H-cmultiquadric collocation method for partial differential equations. Numerical Methods for Partial Differential Equations, 19:571-594.
- [2] Liu, G.R., Gu, Y.T. (2005). An Introduction to Meshfree Methods and Their Programming, Springer, Netherlands.
- [3] Kansa, E.J. (1990). Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics-I. Computers and Mathematics with Applications, 19:127-145.
- [4] Chunyu, Z., Gong, C. (2018). Fast solution of neutron diffusion problem by reduced basis finite element method. Annals of Nuclear Energy, 111:702-708.
- [5] Tanbay, T, Ozgener, B. (2013). Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Mathematical and Computational Applications, 18:399-407.
- [6] Tanbay, T, Ozgener, B. (2014). A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations, Engineering Analysis with Boundary Elements, 46:30-40.
- [7] Fasshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB. World Scientific Publishing, Singapore.
- [8] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
- [9] Micchelli, C.A. (1986). Interpolation of scattered data-distance matrices and conditionally positive definite functions. Constructive Approximation, 2:11-22.
- [10] Fedoseyev, A.I., Friedman, M.J., Kansa, E.J. (2002). Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications, 43:439-455.
- [11] Cheng, A.H.D. (2012). Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation. Engineering Analysis with Boundary Elements, 36:220-239.
- [12] Fasshauer, G.E., Zhang, J.G. (2007). On choosing “optimal” shape parameters for RBF approximation. Numerical Algorithms, 45:345-368.
- [13] Wertz, J., Kansa, E.J., Ling, L. (2006). The role of the multiquadric shape parameters in solving elliptic partial differential equations. Computers and Mathematics with Applications, 51:1335-1348.
- [14] Fornberg, B., Zuev, J. (2007). The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Computers and Mathematics with Applications, 54:379-398.
- [15] Boyd, J.P. (2010). Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval. Computers and Mathematics with Applications, 60:3108-3122.
- [16] Golub, G.H. and Van Loan, C.F. (2013). Matrix Computations (4th ed.), The John Hopkins University Press, Baltimore, MD.
- [17] Akritas, A.G. and Malaschonok, G.I. (2004). Applications of singular value decomposition (SVD). Mathematics and Computers in Simulation, 67:15-31.
There are 17 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | June 20, 2018 |
| Published in Issue | Year 2018Volume: 2 Issue: 1 |
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