偏微分方程式の数値解法 - nkl.cc.u-tokyo.ac.jp

1 ( 500080) Partial Differential Equations PDE mesh, grid, particle 2 PDEPDE3 Finite-Difference Method, FDM 1 2 3 PDE4 1/2 Partial Differential Equations, PDE local governing equation boundary condition initial condition Integral Equations global Green function fundamental function PDE5 2/2 Partial Differential Equations, PDE local Integral Equations global conditional Variational Method functional PDE6 1 f(x) u (x)u(x) x.

2 1 x (x) x su(x)u(x+ x) + xf(x) s 0sin1sin1 sxf xx+ x 0,0 xu xdxuddxdudxdudxduxxxxxxxxx 22tansintansin xxdxduxuxxuxs 2221)( 022 xfdxud PDE8 x G(x, ) x= f( ) x dfxGxu 10, G(x, ) x influence function Green s function PDE9 functional 102102101021111dxdxdudxdxdudxds 102101022121dxxuxfdxdudxxuxfdxdxduuJ 10dxxuxf stationarize Euler PDE10 Finite-Difference Method, FDM 1 2 3 PDE11 1/5 3 (x1,x2,x3) u(x1,x2,x3)

3 2 G=0 2 Second order PDE linear non-linear c=0 homogeneous inhomogeneous 0,,,,,,,,,,,,321321312212232222212 xxxuxuxuxuxxuxxuxuxuxuG 0,,,,,,3213132131,2321 xxxcxuxxxbxxuxxxaiiijijiijPDE12 2/5 u aij aij=aji 3 A x(x1,x2,x3) A(x) 0 x parabolic x hyperbolic x elliptic ijjixxuxxu 22 333231232221131211aaaaaaaaaAPDE13 3/5 2 x,y A AC-B2 =0 <0 >0 CBBAA yxcuyuxuyuCyxuBxuA,222222 22 BACCACBBA IAPDE14 4/5 022 YuXu02222 YuXu02222 YuXuPDE15 5/5 2 0222222 Qzuyuxu 02222222 ukzuyuxuQzuyuxutuc 222222 222222222zuyuxuctuSzuyuxuctu PDE16 conduction convection radiation diffusion convection PDE17 u heat flux q Fourier c 22xuxuxtuc xuq C mass flux J Fick D 22xCDxCDxtCxCDJ Fourier q(x,t)

4 X+ x PDE18 1/2 1 u(x,t) T temperature xq(x,t)q(x+ x,t) xtxutxq ,, xxtxutxqxxtxuxtxuxtxxutxxq 2222,,,,,, Q(x,t) heat source PDE19 2/2 x q xxtxutxqtxxqq 22,,, xttxucxttxucq ,, ttxu , 22xutuc Qxutuc 22 PDE20 Initial/Boundary Conditions time-marching unsteady, transient time dependent steady 1 Dirichlet 2 Neumann 0 3 Robin 1 2 u PDE21 x T 2 TT x su(x)u(x+ x) + xf(x)

5 Sxx+ x xxfTTxtu sinsin22 xfxuTtu 2222 Tcxuctu222222 PDE22 1 2 3 PDE23 Discretization Taylor 24 PDE Finite Difference Method x x i-1 i i+1 25 PDE xxxxxxxxdxdx 0lim i i+1 x x i-1 i i+1 xdxdiii 12/1 x 0 i 211112/12/1222xxxxxdxddxddxdiiiiiiiiii 26 PDE Taylor 1/3 x x i-1 i i+1 iiiiixxxxxx 3332221!3!2 iiiiixxxxxx 3332221!3!2 27 PDE Taylor 2/3 x x i-1 i i+1 iiiiixxxxxx 3332221!3!2 iiiiixxxxxx 3332221!3!2 iiiiixxxxxx 332221!3!2 x iiiiixxxxxx 332221!

6 3!2 x 28 PDE Taylor 3/3 x x i-1 i i+1 iiiiixxxxxx 3332221!3!2 iiiiixxxxxx 3332221!3!2 iiiixxxx 33211!322 ( x)2 29 PDE x x i-1 i i+1 2/13332/12222/12/11!32/!22/2/ iiiiixxxxxx 2/13322/11!32/2 iiiixxxx ( x)2 2/13332/12222/12/1!32/!22/2/ iiiiixxxxxx x 30 PDEPDE 211112/12/1222xxxxxdxddxddxdiiiiiiiiii 222111)(,2)(,1)()1()()()()(xiAxiAxiANiiB FiAiAiARDLiRiDiL 022 BFdxd )1(0)(121)1(0)(212212211 NiiBFxxxNiiBFxiiiiii 31 Discretization Taylor Finite Element Method FEM weak form weak solution 32 PDE Handbook of Grid Generation 33 FEM-introPDE34 Finite-Difference Method, FDM 1 2 3 PDE35 1 Steady Convection-Diffusion Equation 1022 xxuxua 11,00 uu a 0 v xPexaBeABeAxu aaLPe Pe: Peclet L.

7 =1 [L2T-1] [L1T-1] PDE36 2D Navier-Stokes + Continuity 01122222222 yvxuyvxvypyvvxvutvyuxuxpyuvxuutu PDE37 Pe x=0 PexPePexPeeePexueexu 111 PDE38 2111122xuuuxuuaiiiii Rc: Reynolds xaRuRuuRciciic 024211 iiiqcqcu2211 02422 ccRqqRq1, q2: 024211 iciicqRqqR: : icciccRRccuRRqq 2222,12121q2 0 |Rc|<2 PDE39 Rc= x= , a= , v= x= , a= , v= DifferenceExactPDE40 Rc a= , v= , Rc= , Rc= 1/2 2111122xuuuxuuaiiiii 21112xuuuxuuaiiiii / 1storder upwinding iu1 iu1 iua>0xuuaxuaii 1iu1 iu1 iua<0xuuaxuaii 1 icicccRccuRqqRqRq 11,101)2(21212.

8 Rc 0 PDE42 2/2 211111222xuuuxaxuuaxuuaiiiiiii 21111222xuuuxaxuuaiiiii scheme PDE43 Rc= x= , a= , v= x= , a= , v= DifferenceUpwindingExactPDE44 1/2 implicit REAL*8(A-H,O-Z)real(kind=8), dimension(:), allocatable:: U1, U2real(kind=8), dimension(:,:), allocatable:: AMAT!C!C-- INIT. read (*,*) NE, VELO, DIFFDX= (NE)N = NE + 1 LENGTH= (U1(N), U2(N), AMAT(N,N))REc= VELO*DX/DIFFPECLET= VELO/DIFFCOEFc2= VELO/( *DX)COEFc1= VELO/( *DX)COEFd= DIFF/(DX*DX)!C!C-- Central Diff. do j= 1, Ndo i= 1, NAMAT(i,j)= i= 1, NU1(i)= : N: =NE+1 DT: tDX: xDIFF: vVELO: a 1022 xxuxua 11,00 uuPDE45 123456789012345678L= NE NE=8 x=L/NEN grid point N=NE+1 FortranC 1/2 implicit REAL*8(A-H,O-Z)real(kind=8), dimension(:), allocatable:: U1, U2real(kind=8), dimension(:,:), allocatable:: AMAT!

9 C!C-- INIT. read (*,*) NE, VELO, DIFFDX= (NE)N = NE + 1 LENGTH= (U1(N), U2(N), AMAT(N,N))REc= VELO*DX/DIFFPECLET= VELO/DIFFCOEFc2= VELO/( *DX)COEFc1= VELO/( *DX)COEFd= DIFF/(DX*DX)!C!C-- Central Diff. do j= 1, Ndo i= 1, NAMAT(i,j)= i= 1, NU1(i)= PECLET: =aL/v=a/v REc: =a x/v COEFc1: a/ xCOEFc2: a/(2* x)COEFd: v/ x2 PDE47 2/2 do i= 2, N-1 AMAT(i,i )= * COEFdAMAT(i,i-1)= -COEFc2 - COEFdAMAT(i,i+1)= COEFc2 - COEF denddo!C!C-- Boundary ConditionsU1(1)= (N)= (1,1 )= (N,N )= !C!C-- Gaussian Eliminationcall GAUSS (AMAT, U1, N, N)stopend xaRwhereuRuuRuxxauxuxxaxuuuxuuaciciiciii iiiii 0242022222111221221111 COEFc1: a/ xCOEFc2: a/(2* x)COEFd: v/ x2123456789i=2~N-1 PDE48 2/2 do i= 2, N-1 AMAT(i,i )= * COEFdAMAT(i,i-1)= -COEFc2 - COEFdAMAT(i,i+1)= COEFc2 - COEF denddo!

10 C!C-- Boundary ConditionsU1(1)= (N)= (1,1 )= (N,N )= !C!C-- Gaussian Eliminationcall GAUSS (AMAT, U1, N, N)stopend101 Nuu123456789 1022 xxuxua 11,00 uuPDE49 Finite-Difference Method, FDM 1 2 3 PDE50 2 1storder wave equation a 0),300(),0(11050:6050sin1000,300110,500: 00,3000 tutuxxxuxxxuaaxuatu PDE51 xuuatuii 211 explicit forward Euler ninininininininiuuxtauuxuuatuu11111122 implicit backward Euler ninininininininiuuxtauuxtaxuuatuu 1111111111222 niun- PDE52 FTCSF orward-Time/Center-Space xtacuucuuuuxtauuxuuatuuninininininininin ininini ,222111111111 Courant 1 x= , t= , c~ x= , t= , c= x= , t=.