1. 9/20/20151 Numerical Solution of Partial Differential Equations Using Compactly Supported Radial Basis Functions C.S. Chen Department of Mathematics University of Southern Mississippi

2. 9/20/20152 PURPOSE OF THE LECTURE TO SHOW HOW COMPACTLY SUPPORTED RADIAL BASIS FUNCTIONS (CS-RBFs) CAN BE USED TO EXTEND THE APPLICABILITY OF BOUNDARY METHODS TO PROVIDE 'MESH FREE' METHODS FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS.

3. 9/20/20153

4. 4 MESH METHOD MESHLESS METHOD

5. 9/20/20155 The Method of Particular Solutions Consider the Poisson’s equation Whereis a bounded open nonempty domain with sufficiently regular boundary Letwheresatisfying but does not necessary satisfy the boundary condition in (2). (1) (2) v satisfies (3) (4) (5)

6. 9/20/20156 Domain integral Atkinson’s method (C.S. Chen, M.A. Golberg & Y.C. Hon, The MFS and quasi-Monte Carlo method for diffusion equations, Int. J. Num. Meth. Eng. 43,1421-1435, 1998) (Requires no meshing if = circle or sphere) Radial Basis Function Approximation of Others

7. 9/20/20157 Assume that and that we can obtain an analytical solutionto Then To approximate f bywe usually require fitting the given data set of pairwise distinct centres with the imposed conditions The Method of Particular Solutions (6)

8. 9/20/20158 The linear system is well-posed if the interpolation matrix is non-singular Once in (6) has been established, (7) (8) where and (9)

9. 9/20/20159 Compactly Supported RBFs References Z. Wu, Multivariate compactly supported positive definite radial functions, Adv. Comput. Math., 4, pp. 283-292, 1995. R. Schaback, Creating surfaces from scattered data using radial basis functions, in Mathematical Methods for Curves and Surface, eds. M. Dahlen, T. Lyche and L. Schumaker, Vanderbilt Univ. Press, Nashville, pp. 477-496, 1995 W. Wendland, Piecewise polynomial, positive definite and compactly supported RBFs of minumal degree, Adv. Comput. Math., 4, pp 389- 396, 1995.

10. 9/20/201510 C.S. Chen, C.A. Brebbia and H. Power, Dual receiprocity method using compactly supported radial basis functions, Communications in Numerical Methods in Engineering, 15, 1999, 137-150. C.S. Chen, M. Marcozzi and S. Choi, The method of fundamental solutions and compactly supported radial basis functions - a meshless approach to 3D problems, Boundary Element Methods XXI, eds. C.A. Brebbia, H. Power, WIT Press, Boston, Southampton, pp. 561-570, 1999. C.S. Chen, M.A. Golberg, R.S. Schaback, Recent developments of the dual reciprocity method using compactly supported radial basis functions, in: Transformation of Domain Effects to the Boundary, ed. Y.F. Rashed, WIT PRESS, pp183-225, 2003. M.A. Golberg, C.S. Chen, M. Ganesh, Particular solutions of the 3D modified Helmholtz equation using compactly supported radial basis functions, Engineering Analysis with Boundary Elements, 24, pp. 539- 547, 2000. References

11. 9/20/201511 Wendland’s CS-RBFs Define For d=1, For For For d=2, 3, (10)

12. 9/20/201512 Globally Supported RBFs φ=1+r

13. 9/20/201513 Compactly Supported RBFs

14. 9/20/201514 Compact support cut-off parameter (scaling factor)

15. 9/20/201515 Analytic Particular Solutions L= in 2D

16. 9/20/201516 Choose A= B = 0, we have

17. 9/20/201517 Note that From (*), we have

18. 9/20/201518

19. 9/20/201519 Example in 2D

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21. 9/20/201521

22. 9/20/201522 Example 2D 10,000 uniform grid points

23. 9/20/201523 Sparse matrix for uniform grid points.

24. 9/20/201524 CS-RBF used: MFS: Fictitious circle with r = 4. Interpolation points: 100x100 uniform grid points α RMSENZC PU (sec) 0.103.76E-4561,0131.65 0.201.55E-52,171,0573.36 0.258.43E-63,376,0255.32

25. 9/20/201525 Analytic Particular Solutions L= in 3D Recall Since we have (11)

26. 9/20/201526 Numerical Example in 3D Consider the following Poisson’s problem Physical Domain

27. 9/20/201527 The effect of various scaling factor α We choose to approximate the forcing term. To evaluate particular solutions, we choose 300 quasi-random points in a box [-1.5,1.5]x[-0.5,0.5]x[-0.5,0.5]. The numerical results are compute along the x-axis with y=z=0.

28. 9/20/201528 Modified Helmholtz Equation in 3D (16) (15) (17)

29. 9/20/201529 Hence, (15) is equivalent to The general solution of (18) is of the form q(r) can be obtained by the method of undetermined coefficients, or by symbolic ODE solver (MATHEMATICA or MAPLE). A,B,C and D in (19) are to be chosen so that in (16) is twice differentiable at r = 0 and (18) (19)

30. 9/20/201530 Theorem 1. Let w be a solution of (18) with w(0)=0. Then defined by (16) is twice continuously differentiable at 0 with (0)=w'(0), '(0)=0, ''(0)=[s 2 w'(0)+p(0)]/3. Furthermore, (r) satisfies (15) as lim r→0 + Choose D = 0. Consequently, we get (20) (21)

31. 9/20/201531 Hence, the particular solution Φ is given by Notice that for r > α and large wave number λ (22)

32. 9/20/201532 Example The general solution of is given by Hence, A,B,C can be obtain from (21).

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34. 9/20/201534 Numerical Results with Consider (23) (24)

35. 9/20/201535 We choose f and g such that the u(x,y,z)=e x+y+z /400 is the exact solution of (23)-(24). To evaluate particular solutions, we choose N=400 quasi- random points. We choose the CS-RBFs

36. 9/20/201536 Error estimates for various compact support cut-off parameter

37. 9/20/201537 Multilevel Interpolation Given a set of X scattered points, we decompose X into a nested sequence M subsets Algorithm:

38. 9/20/201538 The basic idea is to set α 1 relatively large with few interpolation points, and to let the α k decrease as k increases with more points. Reference: M.S. Floater, A. Iske, Multistep scattered data interpolation using compactly supported radial basis functions, J. Comp. and Appl. Math., 73, 65-78, 1996. 30 interpolation points 100 interpolation points

39. 9/20/201539 300 interpolation points

40. 9/20/201540 Numerical Results

41. 9/20/201541 α/1.5, 1.0, 0.6, 0.2/ n/30, 60, 200, 400/

42. 9/20/201542 For time-dependent problems, we consider two approaches to convert problems to Helmholtz equation LAPLACE TRANSFORM FINITE DIFFERENCES IN TIME ALSO POSSIBLE OPERATOR SPLITTING (RAMACHANDRAN AND BALANKRISHMAN)

43. 9/20/201543 (HEAT EQUATION) Consider the BVP where D is a bounded domain in 2D and 3D. Let (28) Then (25) (26) (27) satisfies (29) (30)

44. 9/20/201544 (WAVE EQUATION) Consider BVP Thensatisfies Similar approach can be applied to hyperbolic-heat equation and heat equations with memory (31) (32) (33) (34) (35)

45. 9/20/201545 (HEAT EQUATION) and (36) (37) (38) (39) (40) (41)

46. 9/20/201546 (J. Su & B. Tabarrok, A time-marching integral equation method for unsteady state problems, Comp. Meths. Appl. Mech. Eng. 142, 203-214, 1997) (WAVE EQUATION) CONVECTION-DIFFUSION EQUATION Similar approach works for non-linear equation (41) (42) (43) (44) (45) (46)