An Error Minimized Pseudospectral Penalty Direct Poisson Solver

Transcription

1 An Error Minimized Pseudospectral Penalt Direct Poisson Solver b Horng Tz-Leng Dept. of Applied Mathematics Feng Chia Universit Taichung Taiwan tlhorng123@gmail.com and Teng Chun-Hao Dept. of Applied Mathematics ational Chiao Tung Universit Hsin-Chu Taiwan chunhao.teng@gmail.com

2 Abstract This paper presents a direct Poisson solver based on an error minimized Chebshev pseudospectral penalt formulation for problems defined on rectangular domains. The current method is based on eigenvalueeigenvector matri diagonalization methods developed long time before and elaborated b Chen Su and Shizgal (2000) [1]. However this kind of fast Poisson solvers utilizing pseudospectral discretization is usuall restricted to periodic or Dirichlet boundar conditions onl for efficient implementation. Enforcement of eumann or Robin boundar conditions requests mess row operations that reduces the easiness of operation. The current method can easil accommodate all three tpes of boundar conditions: Dirichlet eumann and Robin boundar conditions. The reason for that is that it emplos penalt method. Besides the penalt parameters are determined analticall such that the discrete L2 error is minimized. 2D and 3D numerical eperiments are conducted and the results show that the penalt scheme computes numerical solutions with better accurac compared to the traditional approaches with boundar conditions enforced strongl. The current method can be etended to generall linear 2nd order elliptic-tpe partial differential boundar value problems with arbitrar coefficients with onl one restriction that those arbitrar coefficients must be able to be separable to coordinate-variable functions. Under this generalization this method can be surel applied to solve famous Helmholtz equation too. As to computing efficienc the asmptotic operation count for current method in 3D case is 2z ( + + z) and its counterpart for 2D case is 2( + ). Obviousl it is far superior to method epressing Laplace operator in tensor product. For an FFT-based method the asmptotic operation count is basicall 2z (log() + log() + log(z)) for 3D case and 2(log() + log()) for 2D case. Indeed the current method which relies on etensive matri-matri multiplications is inferior to FFT-based methods in theor. However this inferiorit also depends on hardware. For moderate and z it is not necessaril inferior in computers nowadas. Of course for large grid resolution FFTbased methods remain the the best choice if handling boundar conditions is not a problem. [1] H. CHE AD Y. SU AD B.D. SHIZGAL. A Direct Spectral Collocation Poisson Solver in Polar and Clinder Coordinates. J. Comput. Phs. 160 (2000)

3 Approimation b Chebshev polnomials 1 Suppose that u is approimated b a truncated series of Chebshev polnomials as u uˆ ktk k 0 where T denotes Chebshev polnomial and uˆ epansion coefficient. k 2 Choosing the Chebshev Gauss-Lobatto collocations points: j j cos j 01 3 We can form the following discrete transform and inverse transform 2 1 u u T j k k j k j k j c k j0 cj u u T k 0 2 j 0 2 k 0 where cj and ck. 1 otherwise 1 otherwise k

4 4 For pseudospectral method interpolating u at the collocation points j 01. j 5 Hence u can be also epressed as u Lk u k 0 where L is the Lagrange interpolating polnomial defined as k L k1 2 l ( 1) (1 ) T( ) k 2. l0 k l ck ( k) lk 2 du d u 6 Then the derivatives and can be approimated as 2 d d 2 2 du dlk d u d Lk j uk j and 2 juk 2 j d d d d j j dl d derivative matri. k0 k0 k where j is usuall referred to the Chebshev collocation k

5 Chebshev Collocation Derivative Matri 1 The entries of first-order Chebshev collocation derivative matri with respect to based on Gauss-Lobatto collocation points are given as below: D D 6 6 j 2 for i 0 or D j 12 1 where jj 2 ci 21 1 otherwise j i j ci 1 D i j and i j 01 2 ij c j i j j here j cos j 012 are Chebshev Gauss-Lobatto collocation points. Also there is a version for avoiding round-off error when is large. (W. Don and S. Solomonoff in SIAM J. Sci. Comp. Vol. 6 pp (1994)) 2 D D 2.

6 Chebshev Gauss-Lobatto collocation mesh

7 Eample: 1D Poisson equation (PE) with Dirichlet BC s 1 Consider an 1D Poisson equation with Dirichlet boundar conditions as follows u f 11 u 1 g and u 1 g. 2 Discretization: ( The first/last row/column of D can due to Dirichlet boundar conditions.) be stripped u 1 1 f D g D 10 g 1 u f 2 2 D g D 20 g 2 D ij u 2 f 2 D g 20 D g 2 u 1 f 1 D g 10 D g 1 where D D i j ij ij (3) Eponential order of convergence.

8 1 Consider a 2D Poisson problem with Dirichlet boundar conditions in a rectangular domain shown as follows u u f u g u g u g u g 2 Discretizing collocation points in the and directions for u Eample: 2D Poisson equation with Dirichlet BC s respectivel and keeping u in the shape of a rectangular matri: t t t ( +1) ( +1) D u D u D u ud f with u f R. 3 The first and last rows and columns of D and D can be stripped due to Dirichlet BCs then we can obtain D D i j D D i j ij ij ij ij g j D g j D g i D g i ( -1) ( -1) t D u ud f where u f R where f f D ij ij i0 i 1 2 1; j ( ) ( ) ( ) ( ) with i j0 j

9 References for Diagonalization Method (Dirichlet BC s) 1 D. B. Haidvogel Thomas Zang The accurate solution of Poisson's equation b epansion in Chebshev polnomials J. Comput. Phs. 30 (1979) (2D diagonalization under spectral method) (2) P. Haldenwang G. Labrosse and S. Abboudi Chebshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation J. Comput. Phs. 55 (1984) Hwar C. Ku Richard S. Hirsh and Thomas D. Talor A pseudospectral method for solution of the three-dimensional incompressible avier-stokes equations J. Comput. Phs. 10 (1987) (earliest 2D) (4) D. Funaro and D. Gottlieb A new method of imposing boundar conditions in pseudospectral approimations of hperbolic equations Mathematics of Computation 51:184 (1988) (penalt method analsis) 5 U. Ehrenstein R. Peret A Chebshev collocation method for the avier-stokes equations with application to double-diffusive convection International Journal for umerical Methods in Fluids 9 (1989) (2D Helmholtz) 6 Henr H. Yang Bernie Shizgal Chebshev pseudospectral multi-domain technique for viscous flow calculation Comput. Methods Appl. Mech. Engrg. 118 (1994) (2D Helmholtz eumann BC)

10 7 S. Zhao and M. J. Yedlin A new iterative Chebshev spectral method for solving the elliptic equation u f J. Comput. Phs. 113 (1994) (2D) (8) H. B. Chen K. andakumar W. H. Finla and H. C. Ku Three-dimensional viscous flow through a rotating channel: a pseudospectral matri method approach 23 (1996): (3D Helmholtz Dirichlet BC) 9 Y. Su Collocation spectral methods in the solution of Poisson equation MS thesis Universit of British Columbia Canada (deatiled review and 2D derivation) 10 Heli Chen Yuhong Su and B. D. Shizgal A direct spectral collocation Poisson solver in polar and clindrical coordinates J. Comput. Phs. 160 (2000) (2D/3D clindrical coordinate and smbolic derivation) 11 Roger Peret Spectral methods for incompressible viscous flow Springer-Verlag ew York (2D p 88) 12 S. guen C. Delcarte A spectral collocation method to solve Helmholtz problems with boundar conditions involving mied tangential and normal derivatives J. Comput. Phs. 200 (2004) (complicated Helmholtz solver single/multi-block)

11 Diagonalization via Eigenvector De where i ie D i Y Y 1 here and e represent the eigenvalue and eigenvector of D and Y e e. 1 e Similarl i t t D e e D X X j i j j 1 2 where. 1

12 D u ud f 1 DYY ux ux fx 1 YY ux ux fx Y ux Y ux Y fx where and 1 1 V V B V Y ux B Y fx t D ux ud X fx t D ux ux fx i.e. v v b i and j v ij i ij ij j ij i b ij j where v ij and b are the elements of V and B respectivel. ij 1 Once V is solved u can be recovered from u YVX. The diagonalization method so far onl works easil for Dirichlet BC's. For non-dirichlet BC's such as eumann and Robin BC's it requests mess row operations.

13 Eample: 1D Poisson equation with BC s involving derivative 1 Consider an 1D Poisson equation with Dirichlet/eumann boundar conditions as follows u f 11 u 1 g and u 1 g. 2 Stripping off first/last row/column of (3) Discretization: D no longer works. u g 0 u 1 f D 1 0 j D ˆ where Dˆ D ij ij u 1 f 1 I j u g 1 1 here I j stands for the last row of the 1 1 identit matri.

14 Eample: 2D Poisson equation with BC s involving derivative 1 Consider a 2D Poisson equation with Dirichlet/eumann boundar conditions as follows u u f u g u g u g u g 2 Following the idea of 1D case discretizing collocation points in and directions respectivel for u and keeping u in the form of a rectangular matri: D u ud fˆ u fˆ R ˆ ˆ t ( +1) ( +1) where Dˆ fˆ D 0 j D 0 j D Dˆ D ij ij I I j j ( +1) ( +1) g or g g g g or g g g fij g g g or g g g g or g ( +1) ( + 1) ( +1) ( +1) i here. j 1 2 1

15 ˆ ˆ t 3 However using 55 matri as an eample D u ud fˆ will become u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u uu u u u u u u uu g or g g g g g or g g f f f g g f f f g g f f f g g or g g g g g or g which is wrong at BC's. How to fi this? Resorting to Kronecker product I Dˆ Dˆ I is both time and memor consuming.

16 Diagonalization with Penalt Method 1 Consider the 2D Poisson equation with all kind of boundar conditions in a rectangular domain i.e. where Bu 1 g ( ) Bu 1 g ( ) B and and are specified constants. At least one of and B must be non-zero for a unique solution to eist. 2 At this time we can not strip first/last row/column of D and D as we did for the case of Dirichlet BCs. We follow the previous idea but modif the substitution b penalt method shown as belows.

17 3 Replace D and D b the following D and D respectivel D D D I0 j D j D D I j D 0 0 j ij D j j 0 j I 0 j D I j D j j D 0 j where and are the penalt weights which are usuall large numbers. ij D

18 t 4 Using 55 matri as an eample D uud f will become u Bu Bu u Bu u Bu u Bu u Bu Bu ubu u u u ubu ubu u u u ubu u Bu u u u u Bu u Bu Bu u Bu u Bu u Bu u Bu Bu f g g f g f g f g f g g f g f f f f g f g f f f f g f g f f f f g f g g f g f g f g f g g which well approimates u f with the specified BC's when 's are large enough.

19 Diagonalization via Eigenvector again D e where i ie D i Y Y here and e represent the eigenvalue and eigenvector of D and Y e e. 0 e 1 2 Similarl j i t t j j i D e e D X X 0 1 where.

20 D uud f 1 DYY ux ux fx 1 YY ux ux fx Y ux Y ux Y fx where and 1 1 V V B V Y ux B Y fx t D ux ud X fx t D ux ux fx i.e. v v b i 01 and j 01 v ij i ij j ij ij i b ij respectivel. j where v and b are the elements of V and B ij 1 Once is solved can be recovered from. V u u YVX ij

21 1 D D 3D Case D 0 j I j D D D I j D 0 0 j ij D j j D 0 j I j D D ij D I j D 0 0 j j j z z z Dzz 0 j I j Dz D z z z Dzz I z j Dz 0 0 j zz zz ij j j z z z z

22 (2) D Y Y Y D Y 1 il lm m im mi il lm m l0 i0 l0 D X X X D X t 1 t lj jn n ln nl lj jn n j0 l0 j0 D Z Z Z D Z. z z z t 1 t zz lk ko o lo ol zz lk ko o k0 l0 k0

23 (3) z t t il ljk ilk lj ijl zz lk ijk l0 l0 l0 D u u D u D F z z D u X Z u X Z il ljk jn ko ilk n ln ko k0 j0 l0 k0 l0 z z u Z X F X Z ijl o lo jn ijk jn ko j0 l0 k0 j0 z D z 1 Y il lmyml uljk X jnzko uilk XlnZko n k0 j0 l0 m0 k0 l0 z u X Z ijl jn lo o ijk jn ko j0 l0 k0 j0 z F X Z

24 z z 1 YimmYml uljk X jnzko uilk XlnZkon k0 j0 l0 k0 l0 z u X Z ijl jn lo o ijk jn ko j0 l0 k0 j0 z F X Z z z 1 1 myml uljk X jnzko Ymi uilk XlnZkon k0 j0 l0 i0 k0 l0 z z 1 1 Ymi uijl X jnzloo Ymi Fijk X jnzko i0 l0 j0 i0 k0 j0

25 z z 1 1 m Yml uljk X jnzko n Ymi uilk XlnZko i0 k0 j0 i0 k0 l0 z z 1 1 o Ymi uijl X jnzlo Ymi Fijk X jnzko i0 l0 j0 i0 k0 j0 Hence W B W m n o mno mno mno z z 1 1 where Wmno Ymi uilk XlnZko Bmno Ymi Fijk X jnzko. i0 k0 l0 i0 k0 j0 m B mno n o mno z 1 1 ijk im mno nj ok m0 o0 n0 u Y W X Z.

26 Error Minimization for Penalt Parameter 2 1 Taking 1D PE as eample u( ) f( ). 2 f( ) 16 sin 4 so eact solution u( ) sin(4 ). 3 BC's: (a) u( 1) 0 u(1) 4 (b) u( 1) 4 u(1) u(1) 4. Figure 1: BC (a) =16. Figure (2): BC (b) =16.

27 Error-Minimized Tau vs. Strongl Enforcement Strongl-enforced BC is equivalen to. Table 1: BC (a) Table 2: BC (b)

28 Error-Minimized Tau Can Be Derived Details see: Tz-Leng Horng and Chun-Hao Teng* 2012 "An error minimized pseudospectral penalt direct Poisson solver" Journal of Computational Phsics 231(6):

29 Error-Minimized Tau Can Be Etended to Higher-Dimension PE Taking 2D PE as an eample u u u f( ) with BC's Bu 1 g ( ) Bu 1 g ( ). There eists a unique set u( ) u( ) f ( ) f ( ) such that u ( ) u( ) u( ) and f( ) f ( ) f ( ) with u f subject to B u 1 g ( ) u f subject to B u 1 g ( ) So we can calculate error-minimized and in each direction alone according to previous formula.

30 umerical eperiments for 2D PE 1 u f( ). 2 2 f( ) 32 sin(4 )sin(4 ) so eact solution is u ( ) sin(4 )sin(4 ). u( 1 ) (3) BC's: (a) u( 1 ) 4 sin(4 ) u ( 1) 4sin(4 ).

31 BC's: (b) u( 1 ) 0 u( 1) 0. CT (Chebshev tau): D.B. Haidvogel T. Zang The accurate solution of Poisson s equation b epansion in Chebshev polnomials J. Comput. Phs. 30 (1979) CC (Chebshev collocation): U. Ehrenstein R. Peret A Chebshev collocation method for the avier Stokes equations with application to double-diffusive convection Int. J. umer. Methods fluids 9 (1989) CG (Chebshev Galerkin): J. Shen Efficient spectral-galerkin method II. Direct solvers of second and fourth order equations b using Chebshev polnomials SIAM J. Sci. Comput. 16 (1995)

32 u(1 ) BC's: (c) u1 0 4sin(4 ) u ( 1) u ( 1) 0 u ( 1) 4sin(4 ).

33 umerical eperiment for 3D PE 1 u f( z). 2 2 f( z) 48 sin(4 )sin(4 )sin(4 z) so eact solution is u ( ) sin(4 )sin(4 )sin(4 z). u(1 z) (3) BC's: u( 1 z) 0 4sin(4 )sin(4 z) u ( 1 z) u ( 1 z) 4sin(4 )sin(4 z) u ( 1 z) 4sin(4 )sin(4 z) u ( 1) u ( 1) u ( 1) 4 sin(4 )sin(4 ) 4sin(4 )sin(4 ). z z

34 1 Asmptotic operation account: Conclusion 2D: 2 3D: 2 compared with FFT-based method z z 2D: 2 log log 3D: 2 log log log. z z (2) This method can be easil etended to the following tpe of elliptic problem subjected to all kinds of BC's in a rectangular domain: where L a L L Lzu f z b c L a b c Lz a3 z b 2 3 z c3 z z z Sure it includes Helmholtz equation. (3) So far analtic formula for error-minimized tau can be onl derived for PE.. 4 For all BC's being eumann in PE which has infinitel man solutions and the difference of an two of them will be a constant the current method will give ou one of the solutions. 5 The basis function is not limited to Chebshev polnomials. It can be replaced b Legendre polnomials or an other basis function as far as there is an associated collocation derivative matri. It can be applied to finite difference too. (6) Sample matlab code is available at

35 Thank ou for our attention. An question?