Radial Basis Functions Alternative Solutions to ShallowWater Equations
DOI:
https://doi.org/10.34874/IMIST.PRSM/fsejournal-v1i1.26867Keywords:
Water resources, Burger’s, Shallow-water equations, Radial Basis Functions (RBFs), Floods, Ourika valleyAbstract
In this paper, shallow water equations (SWE) are solved through a variety of meshless methods
known as radial basis functions (RBF) methods. RBF based Meshless methods have gained much
attention in recent years for both the mathematics as well as the engineering community. They have
been extensively popularized owing to their flexibility, power and simplicity in solving partial differential
equations. The technique is of quasi-analytical type and based on the collocation formulation
and does not require the generation of a grid or integrals evaluation. A bundle of techniques such as
MQRBF, IMQRBF, GSSRBF, CSRBF are formulated for the specific case of shallow water equations
(SWE), with adequate parameters, these techniques show robustness and low computational costs. For
validation purposes, two applications are presented. One deals with Burger’s equation (2- D), and the
second with a linear two-dimensional hydrodynamics model of rectangular channel shape. Finally a
third application concerning a real case study of the hydrodynamic of the Ourika valley in Morocco
is investigated aiming at rebuilding the 1995 historical flood wave propagation, therefore delineate
inundated areas, estimate velocities and wave time arrivals.
References
Alhuri. Y, A. Naji, D. Ouazar, A. Taik, Numerical Simulation of Saint-Venant Equations with Turbulence
using Radial Basis Functios: Application to Lake Booregreg. International Journal of Computer Applications
in Technology IJCAT (2011), (to appear).
Ahmed. S. G , A collocation method using new combined radial basis functions of thin plate and multiquadraic
types, Eng. Anal. Bound. Elem, 30, (2006), pp 697-701.
Belhaj. H., A Taik, D. Ouazar, Computing two dimensional flood wave propagation using unstructured
finite volume method: Application to the Ourika valley. International Journal of Finite Volume, (IJFV)
volume 3, No 1 (2006).
Belytschko. T, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: an overview and
recent developments, Computer Methods in Applied Mechanics and Engineering, special issue on Meshless
Methods, 139, (1996), pp 3-47.
Buhmann M. D, Radial basis functions : theory and implementations, Cambridge university press, (2003).
Cao.W,W. Huang and R.D. Russell, An r−adaptive finite element method based upon moving mesh PDEs,
J.Comput phys, 44, (1999), No. pp 149- 221.
Chen. C. S, M. D. Marcozzi and S. Choi, The method of fundamental solution and compactly supported
radial basis function: a meshless approach to 3D problems, International series on advances in boundary
elements, (1999), pp 1460-1419 .
Dubal. M. R, Construction of three-dimensional black-hole initial data via multiquadrics, Phys. Rev, D.45,
(1992), pp 1178-1187,
Fasshauer. G. E, On the numerical solution of differential equations with radial basis functions, Boundary
Element Technology XIII, (C. S. Chen, C. A. Brebbia, and D. W. Pepper, Eds), Wit Press, (1999), pp
-300,
Ferreira. A. J. M, E. J. Kansa G. E. Fasshauer and V. M. A. Leito, Progress on Meshless Methods, (2009),
Springer Science.
Fornberg. B. and C. Piret, On choosing a radial basis function and a shape parameter when solving a
convective PDE on a sphere. Journal of Computational Physics 227, (2008), pp 2758 - 2780.
Golberg. M. A. and C.S. Chen, The theory of radial basis basis function applied to the BEM for inhomogeneous
partial differential equations, Boundary element communications, 5, (1994), pp 57-61.
Golberg. M. A, and C.S. Chen, Improved multiquadric approximation for partial differential equations,
Engr. Anal. with Bound. Elem, 18, (1996), pp 9-17.
LI. J, Y. C. Hon and C. S. Chen, Numerical comparisons of two meshless methods using radial basis
functions, Engineering analysis with biundary elements, (2002), pp 205-225.
Kansa. E. J. and Y. C. Hon, Circumventing the ill-Conditioning problem with multiquadric radial basis
functions: applications to elliptic partial Differential Equations, Computers and mathematics with applications,
, (2000), pp 123-137.
Kansa. E. J. and R. E. Carlson, Improved accuracy of Multiquadric interpolation using variable shape
parameters, Computers math. Applic. Vol. 24, NO.12, (1992), pp.99-120.
Kansa. E. J, Multiquadrics a scattered data approximation scheme with application to computational fluid
dynamics, part I, Computers and Mathematics with Applications, Vol. 19, (1990), pp.127-145.
Kansa. E. J, Multiquadrics a scattered data approximation scheme with application to computational fluid
dynamics, part I, Computers and Mathematics with Applications, Vol. 19, (1990), pp.127-145.
Kansa. E. J, H. Power, G.E. Fasshauer and L. Ling, A volumetric integral radial basis function method
for time-dependent partial differential equations. I. Formulation, Engineering Analysis with Boundary Elements
, (2004), pp 1191-1206.
Hardy. R. L, Multiquadric equations of topography and other irregular surfaces, J. Geophy. Res., vol. 76,
(1971), pp 1905-1915.
Hon. Y. C, K. F. Cheung, X. Z. Mao and E. J. Kansa, A Multiquadric Solution for the shallow water
Equations, ASCE J. of Hydrodlic Engineering, vol.125, (1999), No.5, pp 524-533.
Micchelli. C. A, Interpolation of scattered data: distance matrices and conditionally positive definite functions.
Constr Approx 2, (1986), pp 1122.
Powell. M. J. D, The theory of radial basis function approximation in 1990, in advances in numerical
analysis, Vol. II: Wavelets, subdivision algoritms and radial functions, W. Light, ed., Oxford university
press, UK,(1992), pp 105-210.
Schaback. R. , Creating surfaces from scattered data using radial basis functions in: Mathematical methods
forcurves and surfaces, Vanderbilt university press, Nashville, (1995), pp 477-496.
Wendland. H, Piecewise polynomial, positive definite and compactly supported radial functions of minimal
degree, Adv. Comput Math. 4, (1995), pp 389-396.
Wendland. H, Scattered data Approximation, Cambridge university press, (2005).
Wong. S. M, Y. C. Hon and M. A. Golberg, Compactly supported radial basis functions shallow water
equations. J. Appl. Sci. Comput, 127, (2002), pp 79-101.
Wong. S.M, Meshless methods for large-scale computations by radial basis functions, Ph.D. Thesis, City
Univ. Hong Kong, China, (2001).
Wu. Z, Multivariate compactly supported positive definite radial basis functions, Advances in computational
mathematics, 4, (1995), pp 283-292.
