DOMAIN DECOMPOSITION BY RADIAL BASIS FUNCTIONS FOR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS Jos´e Antonio Mu˜noz-G´omez Ciencias Computacionales Instituto Nacional de Astrof´ısica, ´ Optica y Electr´onica (INAOE) Puebla, Tonantzintla, M´exico email:
[email protected]
Pedro Gonz´alez-Casanova UICA-DGSCA Universidad Nacional Aut´onoma de M´exico D.F., M´exico email:
[email protected]
ABSTRACT In the last years, there has been an increased investigation of efficient algorithms to solve problems of great scale. The main restriction of the traditional methods, like finite difference methods and finite element methods, is the mesh generation. In this work, we investigate the overlapping domain decomposition method applied to time dependent partial differential equations with unsymmetric radial basis function collocation method. Numerical experiments performed with thin-plate splines as kernel function, for an evolutionary problem in two dimensions, show a drastic time reduction as we increase the number of subdomains, high numerical accuracy and lower numerical diffusion. The numerical results suggest that the scheme proposed can be useful to tackle large scale time dependent problems. KEY WORDS radial basis function, domain decomposition, partial differential equations.
1 Introduction In the last decade, there has been great interest in using meshless methods to find the numeric solution of partial differential equations [1]. In 1990, Kansa introduces a new approach for this kind of problems, where the true solution is approximated for a linear combination of radial basis functions [2]. This method has shown to be more efficient than the traditional methods like, Finite Differences Methods and Finite Element Methods [3, 4]. Owing to the independence of the dimension of radial basis functions, this strategy is very attractive to resolve high dimensional problems. The Kansa’s approach is truly meshless and does not demand any connectivity requirements as needed with the traditional techniques like Finite Differences Methods, Finite Elements Methods and Boundary Element Methods. Such meshing complication has recently been described in [5] as “..., a very time-consuming portion of overall computing is the mesh generation from CAD input data. Typically, more than 70 percent of overall computing time is spent by mesh generators.” On the other hand, a big obstacle for radial basis function collocation method is that
Gustavo Rodr´ıguez-G´omez Ciencias Computacionales Instituto Nacional de Astrof´ısica, ´ Optica y Electr´onica (INAOE) Puebla, Tonantzintla, M´exico email:
[email protected]
the companion matrix is generally ill-conditioned, nonsymmetric and full dense matrix, which constrains the applicability of RBFs method to solve large scale problems. Hence, domain decomposition method can provide a way to reduce the computational time and the ill-conditioning of the matrix. As far as to the authors knowledge, there is scarce literature dedicated to the application of Kansa’s strategy for time dependent partial differential equations using domain decomposition methods for uniform grid and random data. Thus, it might be of clear interest to investigate domain decomposition method for evolutionary partial differential equations. In addition, we investigate the effect in the computational effort and the accuracy in the numerical solution when increasing the number of subdomains. From the numerical experiments and with a single computer, we observed a drastic time reduction of the computational effort. This paper is organized as follows: Section 2 is devoted to introduce meshless collocation methods for time dependent problems. In section 3, we describe in a general way the domain decomposition method. The numerical results are given in Section 4. Finally, in Section 5 conclusions are given.
2 Radial Basis Function Method In this section, we describe how to apply Kansa’s unsymmetric collocation method to a general linear time dependent PDE problem. Consider the following time-dependent problem: ∂u + Lu = 0 x ∈ Ω ⊂ Rd , (1) ∂t where L is some linear differential operator, with u = u(x, t), together with the boundary and initial conditions u(x, t) = g(x, t) x ∈ ∂Ω t > 0,
(2)
u(x, t) = u0 (x) t = 0,
(3)
where u(x, t) is the unknown function at the position x at time t, ∂Ω the boundary of Ω, g(x, t) and u 0 (x) are known functions.
Let {xi }N i=1 ⊂ Ω be N collocation nodes. Let us further assume that these nodes can be divided in interior N I {xi }N i=1 nodes and {x i }i=NI +1 ∈ ∂Ω boundary. In order to obtain the approximate solution u ˜(x, t) to the exact solution u(x, t) of the initial value problem defined by (1), (2) and (3), we first define the radial approximation u ˜(x, t) given by u ˜(x, t) =
N
λj (t)φ(x − xj ),
RBF Multiquadric (MQ) Inverse Multiquadric (IMQ) Gaussian (GA) Thin-Plate Splines (TPS) Smooth Splines (SS)
Definition√ 2 + c2 φ(r, c) = r√ φ(r, c) = 1/ r2 + c2 2 φ(r, c) = e−(cr ) r m log r, m = 2, 4, 6, . . . r m , m = 1, 3, 4, . . .,
Table 1. Global Radial Basis Functions
(4)
j=1
where λj (t) are the unknown time dependent coefficients to be determined at each time step. Here φ( · ), where · is the Euclidean norm, is any sufficiently differentiable semi-positive radial basis function. Substituting (4) in (1) and applying the boundary conditions (2), we obtain N dλj j=1
dt
φ(x − xj ) + λj Lφ(x − xj ) = 0, N
λj φ(x − xj ) = g(xi , t),
(5)
(6)
j=1
where L denote the application of the spatial derivatives on the kernel function (see Table 1), with the indices i = 1, . . . , NI for (5) and i = N I + 1, NI + 2, . . . , N for (6). The time derivative is approximate by a first-order time difference scheme, obtaining d˜ u(x, t) 1 t+∆t = (λj − λtj )φ(x − xj ), dt ∆t j=1 N
(7)
where ∆t is the time step. Substituting (7) in the left side of (5), we can write the equations (5) and (6) in a compact form Φλn+1 = [Φ − ∆tLφ]λn , (8) where Φ is the Gram matrix. The initial condition of (8) is determined by solving the linear algebraic system of N equations Φλ0 = u0 (x). At each step time, n, the righthand term of (8) is known, so the equation (8) is similar to solve the interpolation problem Φλ n+1 = H(x, t), where H(x, t) = [Φ − ∆tLφ]λn . The boundary conditions (2) are applied over H(x, t) at each time step. Finally, the numerical solution u˜(x, t) is obtained by the interpolation equation (4) where the time dependent coefficients λ(t) are provided by (8). The stability criterion based on the eigenvalues of the iterative system (8) are shown in [6]. The most widely used RBFs are shown in Table 1. In our numerical examples we have used the Thin-Plate Splines (TPS) with m = 4, which does not have the coefficient c called the shape parameter.
many small size problems, and solve each subproblem to obtain the global approximate solution. In what follows, we only concentrate on the overlapping domain decomposition method with Schwarz additive technique and matching nodes [7]. Without generality loss, let Ω partitioned into two subdomains Ω1 and Ω2 , where Ω1 Ω2 = ∅; see Figure 1. We denote Γi as the artificial boundary of Ω i that is interior of Ω, the rest of the boundaries are denoted by ∂Ω i \Γi . The original initial value problem on Ω defined by (1), (2) and (3), can be solved by the classical additive Schwarz algorithm, which can be written as: ⎧ ∂un ⎨ ∂t1 + Lun1 = 0 in Ω1 on ∂Ω1 \Γ1 un1 = g ⎩ n on Γ1 u1 = un−1 2 and ⎧ ⎨ ⎩
∂un 2 ∂t n u2 un2
+ Lun2
= 0 = g = un−1 1
(9) in Ω2 on ∂Ω2 \Γ2 on Γ2
.
In the above notation, u ni denotes the solution of the subproblem Ω i . At each subdomain, we approximate the solution of the initial value problem by the Kansa’s unsymmetric collocation method; see Section 2, at each iteration the transmission of the numerical approximation is performed over the overlapping regions; that is, throughout the artificial boundaries.
Figure 1. Partitioning Ω in two overlapping subdomains.
3 Domain Decomposition Method The idea behind of domain decomposition method (DDM) is to decompose the original large global PDE problem into
In our numerical examples we partitioning the domain Ω = [0, 1] × [0, 1] into P = M × M , M = 1, 2, . . . , 14
overlapping subdomains, with two overlapping nodes. Figure 2 shows the case of partitioning Ω into four subdomains, 4 = 2 × 2. In general, to update the nodes conforming the artificial boundaries for each subdomain, requires the knowledge of the nodes that conform the overlapping zones for eight’s surrounding subdomains. 1.2
1
subdomain 1 subdomain 2 subdomain 3 subdomain 4
was solved for the time t ∈ [0, 0.4], using ∆t = 0.0001. The uniform grid employed was 5625 = 75 × 75, where NI = 5329 corresponds to interior nodes and N F = 296 to boundary nodes. It is readily to show that the analytical solution of the initial value problem defined by (10)-(12) can be expressed as: (13) u(x, t) = uo (x − t). As time evolves, the initial data propagates unchanged to the right with constant velocity.
0.8
Emax 0.008063 0.006528 0.007274 0.006914 0.005699 0.005730 0.006836 0.007833 0.008213 0.009314 0.009247 0.023595 0.022367 0.014810
The additive Schwarz algorithm shown above, require the building of the overlapping zone. This task is straightforward for uniform grid. However, for truly random data this task is not trivial and has not been tackled in the literature.
Subdomains 1 2×2 3×3 4×4 5×5 6×6 7×7 8×8 9×9 10 × 10 11 × 11 12 × 12 13 × 13 14 × 14
4 Numerical Results
Table 2. Increasing subdomains vs Time and Error.
0.4
0.2
0
−0.2 −0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 2. Partitioning Ω in four overlapping subdomains.
In this section we aim to obtain the numerical solution of a time dependent partial differential equation in two dimensions by means of overlapping domain decomposition method with radial basis functions. Besides, we investigate the accuracy of the numerical solution and the computational time when increasing the number of subdomains. We choose a uniform grid distribution, which is a first step to find out the issue of random data with DDM and RBF. The program was written in C, and compiled with gcc version 3.3.1 without any special flag. The computations were carried out in a dedicated computer Intel Xeon (3 GHz, 1G MB, running Linux). The time reported is obtained by the command ‘time /ddm.out’, which includes data generation , data partitioning and the numerical solution. Consider the following linear advection equation in two-dimension: ∂u(x, t) + v · ∇u(x, t) = 0, ∂t
(10)
together with initial and boundary conditions u0 (x) = u(x, t) =
2 2 − (x−0.3) +(y−0.3) σ
e 0,
x ∈ ∂Ω t > 0,
RMS 0.000866 0.000711 0.000922 0.000850 0.000863 0.000928 0.000895 0.001032 0.001249 0.001330 0.001257 0.002028 0.002054 0.001986
Time secs. 6335 739 298 181 121 101 71 69 56 49 45 43 41 39
Table 2 demonstrates the influence of the number of subdomains for solving the linear advection equation with overlapping domain decomposition method and radial basis functions. In the first column, we show the quadratic increment of the subdomains, in the second and third columns correspond to the max error E max and Root Mean Square (RMS) respectively, both measures of error are determined over the whole domain. −3
x 10
1.3
1.2490e−03 1.2
RMS
0.6
1.1
1.0320e−03 1
9.2200e−04 0.9
0.8
0.7
0
10
20
30
40
50
60
70
80
90
100
Subdomains
t = 0,
(11) (12)
where Ω = [0, 1] × [0, 1], ∇ denote the gradient operator, and v = [1, 1]T is the velocity vector. The problem
Figure 3. Behavior error vs subdomains.
The first row in Table 2 corresponds to solution of the
We observed that as we increased the number of subdomains the error grows; see Figure 3 from left to right. We would like to remark that this feature has been recently observed for an elliptic problem with DDM and RBF [8]. Dividing Ω until in 49=7 × 7 subdomains, the error stays in the same magnitude, which indicates that the selection of the number of subdomains is not critical.
Figure 4 shows the numerical solution u ˜(x, y, t) at different times t = {0, 0.2, 0.4} with 5 × 5 subdomains, it was obtained a RM S=8.63e-04 and E max =5.69e-03 at the time 0.4, which are slightly more accurate than the errors determined without partitions; see Table 2, rows 1 and 5. We can observe from Figure 4 that the domain [0, 1]× [0, 1] is divided in boxes marked with orthogonal dark lines. Each box corresponds to the subdomain Ω i , at each one we approximate the true solution of the analyzed PDE. 1 Finite Differences Method Radial Basis Function TPS Analytical Solution 0.8
0.6 u(x,t)
PDE without partitions, the time required was 1 hour and 45 minutes, obtaining RM S=0.000866, which represents a high accuracy between the true and the numerical solution. It can be seen from Table 2 that the time decreases drastically as we increase the subdomains. In the case of 7 × 7 subdomains, the time decreases two-orders of magnitude in comparison with the domain without partitions. This fact indicates that the DDM could be applied to solve large scale problems.
0.4
0.2
0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Diagonal Elements of Ω = [0,1] x [0,1]
0.8
0.9
1
Figure 5. Comparison between FDM and unsymmetric collocation method, at time t max = 0.4 We now investigate the numerical accuracy of this 2D problem with Finite Difference Method (FDM) and compare the previous results obtained with RBF. Our results are summarized in Figure 5, which corresponds to the cross section of Figure 4 at time 0.4. The results obtained with unsymmetric collocation method are in excellent agreement with the true solution. The dashed line shows the solution with FDM and the straight line in conjunction with the symbol “+” corresponds to the solution given by DDM with radial basis function; see Figure 5 . It should be observed, that the numerical approximation obtained with FDM display a great numerical diffusion, which was not observed when using the unsymmetric collocation method. The above result provide us cues that in the analyzed method, RBF with DDM, there is little numerical diffusion. However, there is not a mathematically study to determine the amount of the numerical diffusion and dispersion for time dependent problems with Kansa’s approach.
5 Conclusions
Figure 4. Distribution u ˜(x, y, t) at different times t = 0, 0.2 and 0.4 using DDM with 5×5 subdomains and ThinPlate Spline.
This work proposed an overlapping domain decomposition method for time dependent problems with unsymmetric radial basis function collocation method. The performance of the algorithm was proved with a time dependent partial differential equation in two dimensions, and we proved the effectiveness of the strategy proposed since we reduced drastically the computational time as we increased the number of subdomains.
A possible extension of this research is to consider a random data nodes. In this case, the determination of overlapping zones and his programming in parallel are tasks that we are developing.
6 Acknowledgements This work was partially supported by the Consejo Nacional de Ciencia y Tecnolog´ıa of M´exico (CONACYT) under the project CONACYT-2002-C01-4022. The authors wish to thank the support provide from INAOE and UNAM to develop this research.
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