Quasi-uniform Grids and. Ad Hoc Finite Dierence Schemes for BVPs on Innite Intervals. Alessandra Jannelli and Riccardo Fazio

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1 Quasi-uniform Grids and Ad Hoc Finite Dierence Schemes for BVPs on Innite Intervals Alessandra Jannelli and Riccardo Fazio Page 1 of 22 Department of Mathematics and Computer Science, University of Messina Salita Sperone 31, Messina, Italy ajannelli@unime.it rfazio@unime.it Preprint available from the URL: First Prev Next Last

2 Quasi-uniform Grids and Ad Hoc Finite Dierence Schemes for BVPs on Innite Intervals BVPs on innite intervals Page 2 of 22 Finite dierences on quasi-uniform grids Numerical Tests Concluding Remarks First Prev Next Last

3 BVPs on innite intervals Let us consider the class of BVPs dened on an innite interval Page 3 of 22 du dx = f (x, u) x [0, ) (1) g (u(0), u( )) = 0 where u(x) is a d dimensional vector with u l (x) for l = 1,..., d as components. Moreover, f : [0, ) IR d IR d g : IR d IR d IR d First Prev Next Last

4 Quasi-uniform grids Let us consider the smooth strict monotone quasi-uniform maps x = x(ξ), the so-called grid generating functions, x = c ln(1 ξ), (2) Page 4 of 22 and ξ x = c 1 ξ, (3) where ξ [0, 1], x [0, ), and c > 0 is a control parameter. A family of uniform grids ξ n = n/n dened on interval [0, 1] generates one parameter family of quasi-uniform grids x n = x(ξ n ) on the interval [0, ). First Prev Next Last

5 x 0 x 19 x Page 5 of 22 Figure 1 Quasi-uniform mesh for (2) with c = 5 and N = 20. We notice that the last mesh-point is x N =. Half of the intervals are in the domain with length approximately equal to c and x N 1 c ln N for (2). First Prev Next Last

6 Page 6 of 22 The problem (1) can be discretized by introducing a uniform grid ξ n of N + 1 nodes in [0, 1] with ξ 0 = 0 and ξ n+1 = ξ n + h with h = 1/N, so that x n is a quasi-uniform grid in [0, ). The last interval in (2) and (3), namely [x N 1, x N ], is innite but the point x N 1/2 is nite, because the non integer nodes are dened by ( x n+α = x ξ = n + α ), (4) N with n = 0, 1,..., N 1 and 0 < α < 1. These maps describe the innite domain by a nite number of intervals. The last node is placed on innity so right boundary condition is taken into account correctly. First Prev Next Last

7 Page 7 of 22 We dene the values of u(x) on the mid-points of the grid u n+1/2 x n+1 x n+1/2 u n + x n+1/2 x n u n+1. (5) x n+1 x n x n+1 x n As far as the derivative, by considering u = u(ξ(x)), we write du dx = du dξ n+1/2 dξ n+1/2 dx (u n+1 u n ) 2 ( ) ξ n+3/4 ξ n+1/4 n+1/2 (ξ n+1 ξ n ) 2 ( ) x n+3/4 x n+1/4 then du dx n+1/2 because 2 ( ξ n+3/4 ξ n+1/4 ) = ξn+1 ξ n. u n+1 u n 2 ( x n+3/4 x n+1/4 ). (6) This formulae uses the value u N = u, but not x N =. The two nite dierence approximations (5) and (6) have order of accuracy O(N 2 ). First Prev Next Last

8 Page 8 of 22 A non-standard nite dierence scheme on quasi-uniform grids A second order nite dierence scheme can be written as follows U n+1 U n a n+1/2 f ( x n+1/2, b n+1/2 U n+1 + c n+1/2 U n ) = 0, g (U 0, U N ) = 0, where the d dimensional vector U n is the numerical approximation to the solution u(x n ) at the points of the mesh, and a n+1/2 = 2 ( ) x n+3/4 x n+1/4 b n+1/2 = x n+1/2 x n x n+1 x n (7) c n+1/2 = x n+1 x n+1/2 x n+1 x n for n = 0, 1,..., N 1. First Prev Next Last

9 Page 9 of 22 We notice that but when b n+1/2 c n+1/2 1/2 for all n = 0, 1,..., N 2, n = N 1, then b N 1/2 = 0 and c N 1/2 = 1. On the contrary, we choose to set b N 1/2 = b N 3/2 and c N 1/2 = c N 3/2 in order to avoid a suddenly jump for the coecients of the nite dierence scheme. As it will be clear from the results reported in the next section this produces a much smaller error in the numerical solution of the system at x N. First Prev Next Last

10 The method (7) is a nonlinear system of d (N + 1) equations Page 10 of 22 in the d (N + 1) unknowns U = (U 0, U 1,..., U N ) T. For the solution of (7) we can apply the classical Newton's method along with the simple termination criterion 1 d N U nl TOL, d(n + 1) l=1 n=0 where U nl, for n = 0, 1,..., N and l = 1, 2,..., d, is the dierence between two successive iterate components and TOL is a xed tolerance. The results listed in the next sections were computed by setting TOL = 1E 6. First Prev Next Last

11 Page 11 of 22 The Falkner-Skan model The Falkner-Skan model of boundary layer theory is given by [ d 3 ( ) ] 2 u dx + u du 3 ud2 dx + P 1 = 0 2 dx u(0) = du du (0) = 0, dx dx ( ) = 1. It is a BVP dened on a semi-innite interval. We rewrite the (8) as a rst order system with du 1 dx = u 2 du 2 dx = u 3 du 3 dx = u 1u 3 P (1 u 2 2 ). (8) First Prev Next Last

12 Then, we have u = (u 1, u 2, u 3 ) T f(x, u) = ( u 2, u 3, u 1 u 3 P (1 u 2 2 ) ) T g (u(0), u( )) = (u 1 (0), u 2 (0), u 2 ( ) 1) T, with the boundary conditions u 1 (0) = u 2 (0) = 0, u 2 ( ) = 1. Page 12 of 22 For all values of N we used the initial iterate u 1 (x) = u 2 (x) = 1/2, u 3 (x) = First Prev Next Last

13 u 1 (x) u 2 (x) Page 13 of 22 u 3 (x) Figure 2 Numerical solution of Falkner-Skan model with P = 1 obtained with the map x = x(ξ) dened by x = c ln(1 ξ) with c = 5 for N = 80. First Prev Next Last

14 Table 1: Numerical approximation of d2 u dx 2(0) and d2 u dx 2( ). Page 14 of 22 N iter d 2 u dx 2(0) d 2 u dx 2( ) First Prev Next Last

15 Table 2: Comparison of d2 u dx 2(0) and free or truncated boundary (x ɛ and x respectively) for P = 0.5 and P = 1. Nasr et al. [1] Fazio [2] Asaithambi [3] This paper Chebyshev method Free BF Finite dierence Quasi-uniform P x d 2 u dx 2 (0) x ɛ d 2 u dx 2 (0) x d 2 u dx 2 (0) x N d 2 u dx 2 (0) Page 15 of [1] H. Nasr et al. Int. J. Computer Math., 33 : , [2] R. Fazio. Calcolo, 31 : , [3] A. Asaithambi. Appl. Math. Comput., 92 : , First Prev Next Last

16 Page 16 of 22 A pile in soil Let u(x) be the deection of a semi-innite pile embedded in soft soil at a distance x below the surface of the soil. The dierential equation for the movement of the pile is given by d 4 u dx = P ( ) e P 2 u, x [0, ), (9) d 2 u dx 2(0) = 0, d 3 u dx 3(0) = P 3, u( ) = du dx ( ) = 0, where P 1 and P 2 are positive material constants. We rewrite the (9) as a rst order system du 1 dx = u 2, du 3 dx = u 4, du 2 dx = u 3, du 4 dx = P ( ) 1 1 e P 2 u 1. First Prev Next Last

17 Then, we have u = (u 1, u 2, u 3, u 4 ) T f(x, u) = ( u 2, u 3, u 4, P 1 ( 1 e P 2 u 1 )) T g (u(0), u( )) = (u 3 (0), u 4 (0) P 3, u 1 ( ), u 2 ( )) T with the boundary conditions u 3 (0) = 0, u 4 (0) = P 3, u 1 ( ) = 0, u 2 ( ) = 0. Page 17 of 22 For all values of N we used the initial iterate and u 1 (x) = u 2 (x) = u 3 (x) = u 4 (x) = 1 P 1 = 1, P 2 = 1 2 and P 3 = 1 2. First Prev Next Last

18 u 1 (x) u 3 (x) u 4 (x) Page 18 of 22 u 2 (x) Figure 3 Numerical solution of pile model (9) obtained with the map x = x(ξ) dened by x = c ln(1 ξ) with c = 5 for N = 80. First Prev Next Last

19 Table 3: Numerical approximation of u(0) and du dx (0). Page 19 of 22 N iter u(0) du dx (0) First Prev Next Last

20 Table 4: Comparison of u(0), du dx (0) and free or truncated boundary (x ɛ and x respectively) for the pile problem. Lentini and Keller [1] Fazio [2] This paper Asymptotic BCs x = 10 Free BF x ɛ = Quasi-uniform x N = u(0) du dx (0) u(0) du dx (0) u(0) du dx (0) Page 20 of [1] M. Lentini and H. B. Keller. SIAM J. Numer. Anal., 17 : , [2] R. Fazio. Calcolo, 31 : , First Prev Next Last

21 Page 21 of 22 Concluding Remarks Let us discuss, at the end of this work, a possible way to extend the nite dierence schemes on quasi-uniform grids to the numerical solutions of problems dened on the whole real line. For these problems, all boundary conditions are imposed at ±. In such a case it is possible to use the tangential quasi-uniform grid ( nπ ) x n = c tan, 2N where c > 0 is a control parameter. In fact, if n = N, N + 1, 1, 0, 1,, N 1, N, then this grid covers the whole innite line, and in particular we have that x N =. First Prev Next Last