RIIT Vol.X. Num.3. 2009 197-206, ISSN1405-7743 FI-UNAM (artículo arbitrado)
Exponential Convergence of Multiquadric Collocation Method: a Numerical Study Convergencia exponencial del esquema de colocación con multicuádricos: un estudio numérico J.A. Muñoz-Gómez Departamento de Ingenierías, CUCSUR. Universidad de Guadalajara, Autlán, Jalisco, México. E-mail:
[email protected]
P. González-Casanova UICA-DGSCA, Universidad Nacional Autónoma de México. E-mail:
[email protected]
G. Rodríguez-Gómez Ciencias Computacionales. Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, México. E-mail:
[email protected] (Recibido: noviembre de 2006; aceptado: noviembre de 2008) Abstract Recent numerical studies have proved that multiquadric collocation methods can achieve exponential rate of convergence for elliptic problems. Although some investigations has been performed for time dependent problems, the influence of the shape parameter of the multiquadric kernel on the convergence rate of these schemes has not been studied. In this article, we investigate this issue and the influence of the Péclet number on the rate of convergence for a convection diffusion problem by using both an explicit and implicit multiquadric collocation techniques. We found that for low to moderate Péclet number an exponential rate of convergence can be attained. In addition, we found that increasing the value of the Péclet number produces a value reduction of the coefficient that determines the exponential rate of convergence. Moreover, we numerically showed that the optimal value of the shape parameter decreases monotonically when the diffusive coefficient is reduced. Keywords: Radial basis functions, multiquadric, convection-diffusion, partial differential equation. Resumen Experimentos numéricos recientes sobre los métodos de colocación con mulitcuádricos han demostrado que éstos pueden alcanzar razones de convergencia exponencial en problemas de tipo elípticos. Si bien, algunas investigaciones se han realizado para problemas dependientes del tiempo, la influencia del parámetro c del núcleo multicuádrico en la razón de convergencia de éstos esquemas no ha sido estudiada. En la presente investigación se analiza este tópico y la influencia del número de Péclet en la razón de convergencia para un problema convectivo difusivo, considerando un esquema de discretización implícito y
Expo nen tial Conver gence of Multi qua dric Collo ca tion Method: a Nume rical Study
explicito con técnicas de colocación con mulitcuádricos. Demostramos numéricamente que para valores bajos a moderados del coeficiente de Péclet se obtiene una razón de convergencia exponencial. Además, encontramos que al aumentar el número de Péclet origina una reducción en valor del coeficiente que determina la razón de convergencia exponencial. Adicionalmente, de ter mi na mos que el va lor óp ti mo del parámetro c decrece monótonicamente cuando el coeficiente difusivo es disminuido. Desciptores: Funciones de base radial, multicuádrico, convección-difusión, ecuación diferencial parcial.
Intro duction It is well known that within the study of the numerical multiquadric unsymmetric collocation methods for partial differential equations (Kansa, 1990), a major problem is the determination of the shape parameter. Several authors have avoided the use of this radial basis function due to this difficulty, see; (Boztosun et al., 2002; Zerroukat et al., 2000; Li et al., 2003). Some recent studies has been done to investigate this problem, (Fornberg et al., 2004; Larsson et al., 2005). In particular (Cheng et al., 2003), has treated an elliptic problem showing that an exponential rate of convergence can be attained for h-c refinement multiquadric collocation schemes. In the case of evolutionary convection-diffusion problems, the number of articles are even fewer. In (Sarra, 2005), the author recently studied the behavior of adaptive multiquadric methods concluding that their performance is comparable to spectral Chebyshev techniques. However, among the several open problems in this field, the influence of reducing the diffusion coefficient on the behavior of the shape parameter of the partial differential equation, is up to the authors knowledge, an open problem which has not been studied. In this article, by using both an explicit and implicit multiquadric collocation techniques applied to a convective diffusive problem, we numerical study the effect of the shape parameter on the spectral rate of convergence of these methods. We found that increasing the value of the Péclet number produces a value reduction of the coefficient which determines the exponential rate of convergence. Moreover, we numerically showed that the optimal value of the shape parameter decreases monotonically when the diffusive coefficient is reduced. This suggest that convection dominated problems may be efficiently solved by means of h-c multiquadric collocation methods at an exponential rate of convergence. We also numerically verified that the implicit scheme is unconditionally stable and that the increase
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RIIT Vol.X. Num.3. 2009 197-206, ISSN1405-7743 FI-UNAM
of the time step parameter reduces the exponential rate of convergence, i.e. decreases the slope of the corresponding straight lines in a semi-log scale. We also found that for parabolic dominated problems the range of the shape parameter that leads to spectral rate of convergence is larger. This paper is organized as follows. In section 2, we introduce the continuous advection-diffusion problem. Section 3 is devoted to introduce a meshless collocation method for time dependent problems. In section 4, we conduct a series of experiments to determine the spectral convergences for both explicit and implicit schemes with multiquadrics, we further explore how does the shape parameter influence the convergence behavior. Finally, conclusions are given at the end of the paper. Advec tion-diffu sion equa tion Although in this article we shall be concerned with a 1D problem, we shall state the continuous problem in 3D. This is due to the fact that the code of the algorithm is essentially the same in one or three dimensions. The three-dimensional advection-diffusion can be written as ¶u(x , t) = bÑ 2 u(x , t) + m ×Ñu( x, t) x Î W Ì Â3 , ¶t
(1)
together with the boundary and initial conditions c1 u(x , t) + c 2 Ñu(x , t) = f (x , t) u(x , t) = u 0 (x )
t = 0,
x Î dW , t > 0 ,
(2) (3)
where u(x,t) is the unknown function at the position x at time t, Ñ the gradient differential operator, W a bounded domain in Â3, ¶ W the boundary of W, b the diffusion coefficient, m =[m x , m y , m z ]T the advection coefficient (or velocity) vector, c1 and c 2 are real constants, f(x,t) and u 0 (x ) are know functions.
J.A. Muñoz-Gómez, P. González-Casa nova and G. Rodrí guez-Gómez
A large number of real problems can be modeled by the advection-diffusion equation. For example, dispersion of polluting agents, transport of multiple reacting chemicals, variation of asset prices on stock-market and heat transfer. Implicit and explicit schemes We discretize equation (1) with respect to time and space by means of the standard q scheme, 0 £ q £ 1, which can be expressed as follows u(x , t + Dt) - u(x , t) = Dtq(bÑ 2 u t + Dt + m ×Ñu t + Dt ) + Dt(1 - q)(bÑ 2 u ,+m ×Ñu t ),
(4)
where Dt is the time step. Using the notation u n = u(x , t n ) and t n = t n -1 + Dt, equation (4) can be reformulated as u n + 1 + aÑ 2 u n + 1 + h ×Ñu n + 1 = u n + vÑ 2 u n + x ×Ñu n , (5) where a = -bqDt, h = [ nx , ny , nz ]T = -qDtm , v = bDt(1 - q) y x = [ x x , x y , x z ]T = Dt(1 - q)m.. Moreover, this last equation (5), can be expressed in a more compact form H + u n +1 = H _ u n ,
(6)
where H + = 1 + aÑ 2 + h ×Ñ and H _ = 1 + vÑ 2 + x ×Ñ . The right hand side of (6) represent the know solution at time tk , while left term is the unknown solution at the time tk +1 . The selection of q in equation (4) determines whether the method is an implicit scheme, q=1/2, or an explicit scheme q=0. Radial basis func tions method In this section, we depict how to apply Kansa’s unsymmetric collocation method to the initial value problem, defined by (1), (2) and (3). Let {x i } Ni =1 Ì W be N collocation nodes, and we assume that these nodes can be divided in {x i } Ni =1I interior nodes and {x i } Ni = N I + 1 Î ¶W boundary nodes. In order to obtain the approximate solution ~ u (x , t) to the exact solution u(x , t) of the initial value problem, we first define the radial anzat ~ u ( x , t) given by N
~ u (x , t) = å l j (t)f( x - x j ),
(7)
where l j (t) are the unknown time dependent coefficients to be determine at each time step. Here f( × ), where × is the Euclidean norm, is any sufficiently differentiable semi-positive definite radial basis function (RBF), see table 1. Substituting (7) in (5) and applying the boundary condition (2) with c 2 =0 and c1 =1, we obtain [ F + aF' ' +h × F' ]ln + 1 = [ F + vF' ' + x × F' ]ln ,
(8)
where the matrix F is defined as é f( x1 - x1 ) L f( x1 - x N ê f( x 2 - x1 ) L f( x 2 - x N F=ê ê M O M ê f ( x x ) M f ( x - xN êë N 1 N
)ù ú )ú , ú ú )úû
which is a symmetric matrix F T = F. The RBFs approximations of the gradient and Laplacian are expressed as follows: é f x ( x1 - x1 ) ê M ê êfx ( x N - x1 ) 1 F' = ê f ( x - x1 ) ê N1 ê M ê êë f( x N - x1 )
L O L L O L
f x ( x1 - x N ) ù ú M ú fx ( x N 1 - x N ) ú ú fx ( x N 1 + 1 - x N ) ú ú M ú fx ( x N - x N ) úû
and é fxx ( x1 - x1 ) ê M ê êfxx ( x N - x1 ) 1 F' ' = ê ê f( x N 1 + 1 - x1 ) ê M ê êë f( x N - x1 )
L O L L O L
fxx ( x1 - x N ) ù ú M ú fxx ( x N 1 - x N )ú ú, f( x N 1 + 1 - x N ) ú ú M ú f( x N - x N ) úû
where the matrices F, F' and F' ' Î Â N ´ N . The spatial derivatives are applied over the radial basis function f, and only affects the interior nodes. If we consider the explicit scheme; that is q=0, equation (8) can be written as Fln + 1 = [ F + DtbF' ' + Dtm × F' ]ln ,
(9)
j =1
where Dt is the time step length.
RIIT Vol.X. Num.3. 2009 197-206 ISSN1405-7743 FI-UNAM
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Expo nen tial Conver gence of Multi qua dric Collo ca tion Method: a Nume rical Study
Table 1. Global, infi ni tely smooth RBFs RBF
Definition f( r , c ) = r2 + c2
Multiquadric Inverse Quadratic
f( r , c ) = 1 / ( r2 + c2 )
Inverse Multiquadric
f( r , c ) = 1 / r2 + c2 f( r , c ) = e-( cr
Gaussian
2
)
The RBFs are insensitive to spatial dimensions and it is easier to prepare the code to solve partial differential equations (PDEs) in comparison to meshes based methods. To illustrate this problem more clearly, we show the pseudo-code to solve the general time dependent advection-diffusion problem with an explicit method (9); see Algorithm 1. The output of the Algorithm 1 is the vector l(t), that is used to compute the numerical solution ~ u ( x , t) by the interpolation equation (7), which it has a complexity O(N) for each interpolation node. At each time iteration of Algorithm 1, we solve an algebraic linear system of equations Fln + 1 = H by Gaussian elimination with partial pivoting, with complexity O(tN 3 ). The Algorithm 1 has t = tmax / Dt iterations, in consequence the complexities are O(tN 3 ) in time and O(N 2 ) in space. Algo rithm 1 Explicit method – Compute F and his derivatives F' , F' ' – Approximate the initial condition Fl0 = u 0 (x ) and initialize t=0 while t < tmax do
x i Î ¶W
– Solution: Fln + 1 = H t = t + Dt. end while
200
[F -
Dt Dt bF' ' - × mF' ]ln + 1 = H . 2 2
The implicit method is unconditionally stable and has second-order accurate in time, the explicit method is conditionally stable and it is only first order accurate in time. The stability analysis based on algebraic system’s eigenvalues can be found in (Zerroukat et al., 2000). It is known that the accuracy of the multiquadric interpolant depends on the selection of c (Carlson et al., 1991; Rippa, 1999; Fornberg et al., 2004; Larsson et al., 2005). When c ® ¥ the multiquadric interpolate becomes more accurate but simultaneously the condition number increases in magnitude; see (Schaback, 1995). A fundamental, and by no means an easy problem, is to find the best value c before the algebraic system become numerically unstable. In our numerical examples, we have used the multiquadric (MQ) function shown in table 1, where the shape parameter is selected by using the following inequality u(x , tmax , c i + 1 ) - ~ u (x , tmax , c i + 1 )
2
