$B#3!]#4!%JPHyJ,J}Dx<0$N$^$H$a$H1i=,(J

Elliptic $B!JBJ1_7?!K(J Equations

Laplace Equation:  $B"_(J²u/$B"_(Jx² + $B"_(J²u/$B"_(Jy² = 0

Poisson Equation:  $B"_(J²u/$B"_(Jx² + $B"_(J²u/$B"_(Jy²  + f (x, y) = 0

• $B@~7A$NB?85O"N)J}Dx<0$K4T85$5$l$k$N$G!"#4>O$N9TNs1i;;$G

$B!!(J

Hyperbolic $B!JAP6J7?!K(J Equations

Wave Equation:  $B"_(J²y/$B"_(Jt² - c² $B"_(J²y/$B"_(Jx² = 0

• Simple method
  
  u_jⁿ⁺¹ = u_jⁿ - cdt/2dx (u_j+1ⁿ - u_j-1ⁿ)
  y_jⁿ⁺¹ = y_jⁿ + cdt/2dx (y_j+1ⁿ - y_j-1ⁿ + 2u_jⁿ⁺¹)

• the Lax method
  
  u_jⁿ⁺¹ = (u_j+1ⁿ + u_j-1ⁿ )/2 - cdt/2dx (u_j+1ⁿ - u_j-1ⁿ)
  y_jⁿ⁺¹ = (y_j+1ⁿ + y_j-1ⁿ )/2  + cdt/2dx (y_j+1ⁿ - y_j-1ⁿ + 2u_jⁿ⁺¹)

$B!!(J

Parabolic $B!JJ|J*7?!K(J Equations

Diffusion Equation:  $B"_(Ju/$B"_(Jt - k $B"_(J²u/$B"_(Jx² = 0

• Simple method
  
  u_jⁿ⁺¹ = u_jⁿ - kdt/dx² (u_j+1ⁿ - 2u_jⁿ + u_j-1ⁿ)

• the Crank-Nicholson method
  
  u_jⁿ⁺¹ = u_jⁿ - kdt/2dx² (u_j+1ⁿ⁺¹ - 2u_jⁿ⁺¹ + u_j-1ⁿ⁺¹ + u_j+1ⁿ - 2u_jⁿ + u_j-1ⁿ)
  
  $B!J$3$l$O1"2rK!!#>C5nK!$GO"N)J}Dx<0$r2r$/$+!"H?I|K!$G2r$r5a$a$k!#!K(J

$B!!(J

        $B!T1i=,!U(J

1. $B=i4|29EYJ,I[(J  u(x,0) = 2x  [0 < x < 0.5]$B!"(J2(1-x)  [0.5 < x < 1]$B!"(J
   $B6-3&>r7o(J  u(0,t) = u(1,t) = 0 [for any t]$B!"$H$7$F;~9o(Jt=0.1$B$K$*$1$k29EYJ,I[$r7W;;$9$k!#(J
   dx=0.1$B$r8GDj$7!"(Jdt$B$rJQ2=$5$;$F(JSimple method$B$N0BDj@-$r5DO@$;$h!#(Ja=kdt/dx²$B$,$I$N$"$?$j$GIT0BDj2=$9$k$+!)(J
   $B!J(JDiffusion Equation$B$GG.3H;678?t(Jk=1$B$H$9$k!#!K(J
   
   $B%5%s%W%k%W%m%0%i%`$r;29M$K$7$F$b$h$$!#(J[fortran, C]
   f77 -O -o simple simple.f
   ./simple
   
   $B7k2L$O(Jgnuplot$B$r;H$C$F%0%i%U2=$;$h!#(J
   gnuplot
   .....
   gnuplot> plot 'fort.10' w lines, 'fort.11' w points     $B"+2r@O2r$H$NHf3S(J
   
   $B2r@O2r!'!!(Ju(x,t) = 8/p²$B-t(J_i=n^$B!g(J(1/n²)sin(np/2)sin(npx)exp(-n²p²t)  (data at t=0.1[fort.10])
   
   $B!!(J
2. $BF1$8LdBj$G(JGauss$B$N>C5nK!$r;H$C$?(JCrank-Nicholson method$B$r;n$;!#(J[fortran, C]
   $B!!(J
3. $B$5$i$K(JGauss-Seidel$BH?I|K!$r;H$C$?(JCrank-Nicholson method$B$r;n$;!#(J[fortran, C]
   $B!!(J
4. $B!J%"%I%P%s%9%H$J<+=,LdBj!KNW3&E@6aK5$K$"$k7O$r8=>]O@E*$K5-=R$9$kJ}Dx<0$H$7$F(JTDGL(Time Dependent Ginzburg-Landau)$B<0$,$"$k!#<'@-$J$ICa=xJQ?t(J(j)$B$,J]B8NL$G$J$$>l9g$K$O!"7O$N;~4VH/E8$O0J2<$NHsJ]B87?(JTDGL$B<0$G5-=R$G$-$k!#$3$NJ|J*7?$NJPHyJ,J}Dx<0$r(JSimple method$B$G(J$B?tCM%7%_%e%l!<%7%g%s$9$k%W%m%0%i%`$r:n$j!"(Jb$B$N@5Ii$G7O$N%@%$%J%_%/%9$,$I$&JQ2=$9$k$+$r5DO@$;$h!#(J
   
   GL Free Energy:  F=$B"i(Jdxdy ( j⁴/4 + bj²/2 + c($B"`(Jj)<sup>2</sup>)
   TDGL$BJ}Dx<0!JHsJ]B8!K(J:  $B"_(Jj/$B"_(Jt = -L dF/dj = -L(j<sup>3</sup> + bj - c $B"`(J²j) = 0
   
   $B4JC1$N$?$a!"(Jb=$B!^(J0.5$B!"(Jc = 0.1$B!"(JL = 1$B!"(Jdx = 1$B!"(Jdt = 0.1$B$H$7!"(J$B6u4V$N$B%7%9%F%`%5%$%:$O(J128x128$B$G<~4|E*6-3&>r7o!"=i4|>r7o$O(Jj=$B!^(J0.025$B$NHO0O$G%i%s%@%`$KM?$($h!#%@%$%J%_%/%9$N5DO@$K$O!"Nc$($P3&LL@Q!J#2Z$;$h!#(J
   
   Simple method$B$G=q$$$?%5%s%W%k%W%m%0%i%`$O$3$3(J [fortran, C]
   f77 -O -o tdgl_simple tdgl_simple.f
   ./tdgl_simple
   
   gnuplot
   .....
   gnuplot> set contour
   gnuplot> set data style lines
   gnuplot> splot 'fort.10'$B!!!!!!!!!!!!!!"+Ca=xJQ?t(J(j)$B$N6u4VJ,I[(J
   gnuplot> set logscale xy
   gnuplot> plot 'fort.11' u 1:2$B!!!!!!!!(J $B"+;~4V(J-$B3&LL%(%M%k%.!<$NN>BP?t%W%m%C%H(J
   gnuplot> plot 'fort.11' u 1:4$B!!!!!!(J $B!!"+;~4V(J-$B3&LLD9$NN>BP?t%W%m%C%H(J
   $B!!(J
5. $B!J%"%I%P%s%9%H$J<+=,LdBj!K5$1UAjE>0\$J$ICa=xJQ?t(J(j)$B$,J]B8NL$G$"$k>l9g$K$O!"7O$N;~4VH/E8$O(J$B0J2<$NJ]B87?(JTDGL$B<0$G5-=R$G$-$k!#(JSimple method$B$G(J$B?tCM%7%_%e%l!<%7%g%s$9$k(J$B#2$B%W%m%0%i%`$r:n$j!"7O$N%@%$%J%_%/%9$rHsJ]B8$N>l9g$HHf3S$;$h!#(J
   
   TDGL$BJ}Dx<0!JJ]B8!K(J:  $B"_(Jj/$B"_(Jt = L $B"`(J<sup>2</sup>dF/dj = L$B"`(J²(j³ + bj - c $B"`(J²j) = 0
   
   Simple method$B$G=q$$$?%5%s%W%k%W%m%0%i%`$O$3$3(J [fortran]
   $B!!(J
6. $B!J$5$i$K%"%I%P%s%9%H$J<+=,LdBj!K(J$B>e$N#4!"#5$K$D$$$F(JCrank-Nicholson$BK!$X$N2~NI$r;n$_$h!#(J

$B!!(J