Y.-C. Hon, K. F. Cheung, X.-Z. Mao, and E. J. Kansa. Multiquadric solution for shallow water equations. Journal of Hydraulic Engineering, 125(5):524--533, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....In this section, the three collocation methods for elliptic PDEs that we are comparing are presented in the chronological order of their introductions. Note that the basic solution approach is not limited to elliptic problems. Applications to other types of problems are found, e.g. in [5] 11] [12]. Method 1. Straight collocation A straightforward RBF based collocation method for elliptic problems was introduced by Kansa, 1990 [4] 5] Let the RBF approximation to the solution u(x)be ,#) Collocation with the boundary data at the boundary points and with the PDE at the interior ....

Y. C. Hon, K. F. Cheung, X. Z. Mao, and E. J. Kansa, Multiquadric solution for shallow water equations. ASCE J. Hydr. Engrg. 125 (5) 524--533 (1999).

....three collocation methods In this section, the three collocation methods for elliptic PDEs that we are comparing are presented in chronological order. Note that the basic solution approach is not limited to elliptic problems. Applications to other types of problems are found, e.g. in [3] 9] [10]. 4 Elisabeth Larsson and Bengt Fornberg Method 1. Straight collocation A straightforward RBF based collocation method for elliptic problems was introduced by Kansa, 1990 [2] 3] Let the RBF approximation to the solution u(x)be s(x,#) N X j=1 # j #( x x j ,#) Collocation with ....

Y. C. Hon, K. F. Cheung, X. Z. Mao, and E. J. Kansa, Multiquadric solution for shallow water equations. ASCE J. Hydr. Engrg. 125 (5) 524--533 (1999).

.... or a given class of linear functionals, and is even less trivial to derive bounds for the error u s between an exact solution u of (1) and an approximate solution s of the form (5) to the discretized system (4) We refer the reader to the literature on collocation with radial basis functions [2, 4, 5, 7, 8, 9, 10]. In this contribution, we focus on the numerical techniques to solve very large systems like (4) eciently, making use of the background of radial basis functions. In addition, we concentrate on approximate solutions with only few nonzero coecients j . The reason is that the evaluation of a ....

Hon, Y. C., K. F. Cheung, X. Z. Mao, and E. J. Kansa, A multiquadric solution for the shallow water equations, ASCE J. Hydraulic Engineering, Volume 125, Issue 5, 1999, 524-533.

.... Burgers equation with shock wave [24] complicated biphasic model for tissue 3 engineering problem [25] free boundary value di usion type problems like American option pricing [26] sti boundary value problems [27] and real physical problems like tide, current, wind, and water quality models [28 30]. Multiphasic Mixture Theory The multiphasic model developed by Lai et al. 14] for charged and hydrated soft tissues consists of three phases: 1) a solid phase (superscribed by s) 2) a water phase (superscribed by w) and (3) an ion phase composed of anion and cation (superscribed by and ....

Y.C.Hon, K.F.Cheung, X.Z.Mao, and E.J. Kansa, "A multiquadric solution for shallow water equation", ASCE Journal of Hydraulic Engineering, Vol. 125, No. 5, pp. 524-533, 1999.

....to ensure the uniqueness. This condition is given as below p d m (x j ) 2 m and p d m (x j ) 0 j j = 1; 2; N ) p d m 0: 3. 31) Generally if the interpolation points are properly posed, the unique solutions can be achieved without adding the polynomial as discussed in [14]. The numerical results in this study indicated that adding the polynomial term did not a ect the overall accuracy. Applying the RBFs method to solve PDEs may be viewed as a multivariate interpolation problem. The functions and the derivatives are interpolated by using the collocation approach. To ....

....hydrodynamics model. The numerical discretization for the time derivative of the governing equation is obtained using a nite di erence method and the spatial derivatives are approximated by the RBFs in each time steps. This example only concentrates on the MQ functions. As indicated in [14] to [16] the MQ function performs well in dealing with hydrodynamics equations. Following the same augmented RBFs approximation proposed by Golbery et al. [29] and [30] the MQ interpolant as in (3.28) taking a polynomial p d 2 (x; y) b 1 b 2 x b 3 y is generalized by writing s(x; y) in ....

Y.C. Hon, K.F. Cheung, X.Z. Mao, E.J. Kansa, \A multiquadric solution for shallow water equation", ASCE J. of Hydraulic Engineering, Vol. 125, No. 5, pp. 524-533, 1999.

....in the multiquadric (MQ) function given by (17) is called a shape parameter whose magnitude of value a ects the accuracy of the approximation. In most applications of using the MQ for scattered data interpolation, a constant shape parameter is assumed for simplicity. Kansa [18] Hon et al. [15], and Golberg et al. 10] have shown that the use of the MQ for solving partial di erential equations is highly e ective and accurate. However, the accuracy of the MQ is greatly a ected by the choice of the shape parameter c 2 whose optimal value is still unknown. Hickernell and Hon [13] and ....

Hon Y.C., K. F. Cheung, X. Z. Mao, and E. J. Kansa, "Multiquadric Solution for Shallow Water Equations", ASCE J. Hydraulic Engineering, Vol. 125, No. 5, pp. 524-533, 1999.

....a three dimensional Poisson equation could be solved with only 60 randomly distributed knots to the same degree of accuracy as a FEM solution 2 with 71,000 linear elements. Golberg and Chen [9] also successfully combined the TPS into the dual reciprocity method (DRM) for solving PDEs. Hon et al. [13, 14, 15, 16, 17] further extended the use of the MQ RBFs on the numerical solutions of various ordinary and partial di erential equations including general initial value problems [13] a complicated biphasic mixture model for tissue engineering problems [14] nonlinear Burgers equation with shock wave [15] the ....

.... and partial di erential equations including general initial value problems [13] a complicated biphasic mixture model for tissue engineering problems [14] nonlinear Burgers equation with shock wave [15] the shallow water equation for tide and currents simulation under irregular boundary [16], and free boundary problems like American option pricing [17] The computations showed the de nite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The proof of convergence rate of the RBFs method for the solution of PDEs have recently ....

Hon, Y.C., K.F. Cheung, X.Z. Mao, and E.J. Kansa, A multiquadric solution for shallow water equation, ASCE Journal of Hydraulic Engineering, 125(5), 524-533, (1999).

....a class of RBFs called multiquadric function. Hardy s multiquadric function was then widely adopted to solve hyperbolic, parabolic and elliptic partial di erential equations in various scienti c and engineering disciplines due to its ease of use and the super convergence by Kansa [2] and Hon et al. [3]. In order to improve the computational eciency and the ill conditioned results, we couple a domain decomposition algorithm with RBFs to reduce the size of the full resultant matrix. The paper is organized as follows: Section 2 introduces the mathematical description of the multilayer model. ....

Y.C. Hon, K.F. Cheung, X.Z. Mao, E.J. Kansa "A multiquadric solution for shallow water equation", ASCE J. of Hydraulic Engineering, Vol. 125, No. 5, pp. 524-533, 1999.

....(1.1) about time t. A threshold of the mass N , which is the L 2 norm of wave function , is then derived for global existence of equation (1.1) about time t. Finally, for the purpose of numerical veri cation, we apply a newly developed meshless computational radial basis function method (see [6] and [3] to compute the threshold of the mass N in R 3 . From this we also obtain the numerical value of the best constant for the Gagliardo Nirenberg inequality in three dimensional space. It is remarked here that the methodology used in this paper is inspired by the work of Weinstein [14] who ....

....n (x i ) 2 3 u n (x i ) u n 1 (x i ) j u n 1 (x i ) j 4=3 : 4:8) From the boundary equations (4.5) we have the remaining 2 equations du n (0) dx = 0 (4:9) and u n (1) 0 (4:10) to obtain a system of M 2 linear equations for the M 2 unknown coecients 0 s. Refer to [6] and [3] for more details on the theoretical foundation and applications of this RBF method. In this paper, the numerical computation is taken by choosing x j = j , x i = i where = 1= M 1) and M = 99. The initial guess u 0 can be taken to be any positive constant. The iteration will stop ....

Y. C. Hon, K. F. Cheung, X. Z. Mao and E. J. Kansa, Multiquadric solution for shallow water equations, ASCE J. Hydraulic Engineering, 125, 1999, 524-533.

.... radial basis functions (RBFs) RBFs were originally devised for scattered geographical data interpolation by Hardy who introduced a class of functions called multiquadrics [1] These functions have been widely adopted to solve hyperbolic, parabolic and elliptic partial di erential equations [2] to [8]. Since the computational cost of using global RBFs is a major factor to be considered when solving large scale problems with many collocation points, various techniques have been suggested to improve computational eciency. A common approach is to use domain decomposition. In [9] Dubal used domain ....

....with a full matrix. However, as shown by Madych and Nelson [15] the MQs can be exponentially convergent so one can often use a relatively small number of basis elements in which can be computationally ecient. As a consequence, the MQ method has been progressively re ned and widely used in [3] to [8] for solving partial di erential equations in various scienti c and engineering disciplines. For example Hon et al. [8] used the MQ method to solve a system of time4 dependent hydrodynamics equations and this was extended by Wong et al. to study the coupled hydrodynamics and mass transport ....

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Y.C. Hon, K.F. Cheung, X.Z. Mao, E.J. Kansa "A multiquadric solution for shallow water equation", ASCE J. of Hydraulic Engineering, Vol. 125, No. 5, pp. 524-533, 1999.

.... further extended the use of the MQ RBFs on the numerical solutions of various ordinary and partial di erential equations including general initial value problems [23] nonlinear Burgers equation with shock wave [24] shallow water equation for tide and currents simulation under irregular boundary [25], and free boundary problems like American option pricing [26] The computations showed the de nite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The existence, uniqueness, and convergence proofs in applying the RBFs were given by ....

Y.C. Hon, K.F. Cheung, X.Z. Mao, and E.J. Kansa, 'A multiquadric solution for shallow water equation', ASCE Journal of Hydraulic Engineering, 125, 524-533, (1999).

....independence and data structure is exible. This function space can also be used for discretizing the partial di erential and integral equations which provides a truly meshfree numerical method for the solutions of partial di erential and integral equations (for examples, see [6] 8] 9] 10] [11] and [12] These numerical computations indicated the de nite advantages in using this truly mesh free RBFs for solving various initial and boundary values problems. 3 The existence, uniqueness, and convergence proofs in applying the RBFs to solving partial di erential equations were recently ....

Y.C. Hon, K.F. Cheung, X.Z. Mao & E.J. Kansa, Multiquadric solution for shallow water equations, ASCE J. Hydraulic Engineering, pp. 524533, 1999.

.... and Golberg and Chen [7] Hon et al. further extended the use of the RBFs on the numerical solutions of various ordinary and partial di erential equations including nonlinear Burgers equation with shock wave [13] shallow water equation for tide and currents simulation under irregular boundary [14], and complicated biphasic and triphasic models for tissue engineering [15] 16] The computations showed the de nite advantages in using this truly meshless RBFs method for solving various initial and boundary values problems. The existence, uniqueness, and convergence proofs in applying the ....

Hon Y.C., Cheung K.F., Mao X.Z., and Kansa E.J., A Multiquadric Solution for the Shallow Water Equations, ASCE J. Hydraulic Engineering, Vol. 125, pp. 524-533, 1999.

....differential equations (PDEs) of elliptic, parabolic, and hyperbolic types. Hon et al. further extended this MQ to solve various nonlinear initial and boundary value problems including biphasic mixture model with solid and fluid phases [6] and shallow water model under irregular coastal line [7]. Golberg and Chen [4] improved the numerical accuracy by using the MQ in their dual reciprocity method for approximating the particular solution of partial differential equations. The proof of convergence rate of the RBFs method for the solution of PDEs have just recently been given by Franke and ....

Hon Y.C., Cheung K.F., Mao X.Z., and Kansa E.J.,A Multiquadric Solution for the Shallow Water Equations, ASCE J. Hydraulic Engineering, Vol. 125, pp. 524-533, 1999.

No context found.

Hon, Y.C., K.F. Cheung, X.Z. Mao, and E.J. Kansa, 'A multiquadric solution for shallow water equation, ASCE J. Hydraulic Engineering, 125(5), 524-533, (1999).

.... ordinary and partial di erential equations including general initial value problems [13] complicated biphasic mixture model for tissue engineering problems [14] nonlinear Burgers equation with shock wave [15] shallow water equation for tide and currents simulation under irregular boundary [16], and free boundary problems like American option pricing [17] The computations showed the de nite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The existence, uniqueness, and convergence proofs in applying the RBFs were given by ....

....polynomial degree. The resultant matrices related to the interpolation or solving PDEs by using the CSRBFs are then sparse. Recently, Wong et al. 30] successfully applied the CSRBFs to improve the eciency in the solving of the shallow water equation for tide and currents simulation given by [16]. This paper further combines the CSRBFs with those existing advanced numerical techniques like multigrid; preconditioning; and Schwarz iterative scheme from FEM in the hope that the meshfree RBFs can be extended to solve large scale problems. In general, the well known convergence results in ....

Y.C. Hon, K.F. Cheung, X.Z. Mao, and E.J. Kansa, Multiquadric solution for shallow water equation, ASCE J. Hydraulic Engineering, 125(5), pp. 524-533, 1999.

.... including general initial value problems [9] nonlinear Burgers equation with shock wave [10] surface wind field computation from scattered data [5] complicated biphasic and triphasic models of mixtures [7] 8] shallow water equation for tide and currents simulation under irregular boundary [6], and free boundary problems like American option pricing [11] The computations showed the definite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The corresponding (N q) Theta (N q) matrix was always found to be nonsingular, but ....

Hon, Y.C., K.F. Cheung, X.Z. Mao, and E.J. Kansa, Multiquadric solution for shallow water equations, ASCE Journal of Hydraulic Engineering, 125 (1999), 524-533.

.... including general initial value problems [9] nonlinear Burgers equation with shock wave [10] surface wind eld computation from scattered data [5] complicated biphasic and triphasic models of mixtures [7] 8] shallow water equation for tide and currents simulation under irregular boundary [6], and free boundary problems like American option pricing [11] The computations showed the de nite advantages in using this truly mesh free MQ RBFs for solving various initial and boundary values problems. The corresponding (N q) N q) matrix was always found to be nonsingular, but there was no ....

Hon, Y.C., K.F. Cheung, X.Z. Mao, and E.J. Kansa, A multiquadric solution for shallow water equation, ASCE Journal of Hydraulic Engineering, May 1999, to appear.

No context found.

Y.-C. Hon, K. F. Cheung, X.-Z. Mao, and E. J. Kansa. Multiquadric solution for shallow water equations. Journal of Hydraulic Engineering, 125(5):524--533, 1999.

No context found.

Y.C. Hon, K.F. Cheung, X.Z. Mao & E.J. Kansa, A multiquadric solution for shallow water equation, to appear at ASCE Journal of Hydraulic Engineering, 1999.

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