Lanza, Alessandro

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Almae Matris Studiorum Campus

 Center for Subsurface Modeling 1 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     PDE -  Poisson Equation 222222)(yuxuufufu                  Dirichlet B.C.   Neumann B.C.        on    :    Mixed B.C. gnubaugnuguf=0  Laplace Equation R Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 2 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM                      ,,,,,,4321xgdxuxgcxuygybuygyauyxfyxuc b a d y x mesh:x a ih           h (b - a) / Niy c jk   k (d - c) / Mj    PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 3 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     1iju ijjijijijiji fhuuuuu21,1,,,1,1 4  We want u(x,y) at inner nodes, with h=k.   Centered differences: ),(2221,,1,2,1,,12222jijijijijijijiyxfhuuuhuuufyuxu  iju 1i ju 1i ju 1iju PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 4 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     ijuijjijijijiji fhuuuuu21,1,,,1,1 4  N=M=4 ...  4x4 mesh (grid): (N-1)2 unknowns uij     i,j=1,..,N-1  Unknowns: , , , N , , N ,N( u ,u ,u , ...,u ,u , ...,u )  1 1 2 1 3 1 1 1 1 2 1 1j=3 1 j=2 j=1 i=1 i=2 i=3 2 4 5 6 7 8 9 3 ijug13 g12 g11 g41 g42 g43 g23 g22 g21 g31 g32 g33 PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 5 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     Symmetric, diagonally dominant (not strictly), positive definite.  M-matrix! 10    , 1, 2,...,       and   0ija i j n i j A   One equation for each inner node ,,,,,,,,,u huuuuuuuu                                                 21 12 13 11 22 23 21 32 33 34 1 0 1 0 0 0 0 01 4 1 0 1 0 0 0 00 1 4 0 0 1 0 0 01 0 0 4 1 0 1 0 00 1 0 1 4 1 0 1 00 0 1 0 1 4 0 0 10 0 0 1 0 0 4 1 00 0 0 0 1 0 1 4 10 0 0 0 0 1 0 1 4,,,,,,,,,f g gh f gh f g gh f gh fh f gh f g gh f gh f g g                  1 122 123 121 222 223 221 322 323 311 313221 33122213 414243 23PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 6 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     ( 1) 1 ( 1) ( )( )k k kx D b Lx Ux    2, 1 1, , 1, , 14i j i j i j i j i j iju u u u u h f        How to solve the linear system (N-1)x(N-1)?  Iterative method: Gauss-Seidel: (step k) We do not need to build the matrix...solution row by row PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 7 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     ( 1) ( 1) ( ) ( 1) ( ) 2, 1, 1, , 1 , 1Gauss Seidel iteration1,2,...2,..,2,..,1( )4k k k k ki j i j i j i j i j ijfor kfor i Nfor j Nu u u u u h fendendend         ijjijijijiji fhuuuuu21,1,,,1,1 4  PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 8 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     The structure of the matrix depends on the nodes indexing Block Tridiagonal PDE -  Poisson Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 9 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     ijjijijijiji fhuuuuu21,1,,,1,1 4  u-1,j u0,j u0,j+1 u0,j-1 u1,j , j , j , j , j , j , j, j , j, j , j, j , j , j , j , ju u u u u h fu uux xuu u xxuu x u u u h fx                 21 1 0 0 1 0 1 01 11 121 0 0 1 0 1 04222 2 4Poisson Equation:  Neumann B.C. Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 10 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     , j , j , j , j , j , j, j , j, j , j, j , j , j , j , ju u u u u h fu uux xuu u xxuu x u u u h fx                 21 1 0 0 1 0 1 01 11 121 0 0 1 0 1 04222 2 4Poisson Equation:  Neumann B.C. Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 11 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     Multi-dimensional Heat Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna  Center for Subsurface Modeling 12 Metodi Numerici per l‘Ingegneria  F. Sgallari Dip. Matematica-CIRAM     Multi-dimensional Heat Equation Numerical methods – I    F. Sgallari Dept. Mathematics-CIRAM, Univ. Bologna 

INM_8_4

http://campus.unibo.it/93846/1/INM_8_4.pdf

INM_8_4

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