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Universita di Roma ”La Sapienza”
Programmi utilizzati durante il corso
ed esercizi
Dott. Francesco Battista
a.a. 2016-2017
Laboratorio di Calcolo di Aerodinamica
Laurea in Ingegneria Aerospaziale
Chapter 1
Prefazione
Questo libretto e stato creato per raccogliere alcuni dei programmi che vengono mostrati durante
il Laboratorio di Calcolo di Aerodinamica. Esso e in continuo aggiornamento quindi si consiglia
di controllare di tanto in tanto sul sito se vi e una versione aggiornata. La maggior parte dei
codici mostrati sono senza errori, alcuni, non indicati, presentano errori il cui scopo e mostrare
esplicitamente la corretta sintassi. Si consiglia vivamente percio di utilizzare questo libretto come
linea guida per la scrittura dei programmi e non come una dispensa tradizionale: la semplice lettura
dei codici non e di alcuna utilia.
I passi da seguire sono i seguenti:
• leggete la traccia dell’esercizio
• provate a risolvere l’esercizio scrivendo il programma in autonomia ovvero senza vederne la
soluzione sul libretto
• compilate il programma: il successo di questo passo vi permette di passare al passo successivo
in quanto il vostro programma non presenta errori sintattici, altrimenti cercate di correggere
l’errore. Ricordate in caso di errore il compilatore vi indica posizione e tipo di errore quindi
leggete con attenzione i messaggi del compilatore.
• eseguite il programma. A questo punto l’unica domanda che dovete porvi e: “il programma
esegue le istruzioni richieste dall’esercizio?” Se sı, non vi sono errori, altrimenti bisogna
rivedere il programma e le istruzioni in esso contenute.
• confrontate il vostro programma con quello riportato in questo libretto, ricordate “Non esiste
un unico modo per eseguire ognuno degli esercizi in questo libretto!!” quindi non necessaria-
mente i vostri programmi devono essere uguali a quelli riportati nel libretto.
Alcuni codici per il corretto funzionamento necessitano di un file di input che in questo libretto
non e riportato. Infine si ricorda ancora una volta che questo libretto e solo un’aiuto allo studio
che raccoglie tutti i programmi che sono stati discussi in modo approfondito durante le lezioni.
Buon Lavoro
2
Chapter 2
I primi programmi
Questo primo capitolo e dedicato ai primi programmi visti a lezione. Sono da ritenersi utili per
avere degli esempi della sintassi di base necessaria per poter scrivere programmi piu complicati.
2.1
Scrivere a schermo il saluto “Ciao Mondo”: definizione delle sezioni (esecutiva e conclusiva) fonda-
mentali ed imprescindibili di un programma.
1 ! File: hello.f90
2 ! The first program in Fortran 90
3 ! This code print a message on the screen
4 PROGRAM hello
5
6 ! Executive section
7 WRITE (*,*) ’Hello world!’
8
9 ! End section
10 STOP
11 END PROGRAM hello
2.2
Scrivere un numero, scelto dall’utente, a schermo: definizione della sezione dichiarativa.
1 ! File: assegnazione.f90
2 ! This program reads and prints a number on the screen
3 PROGRAM assegnazione
4
5 ! declaration section
6 IMPLICIT NONE
7 INTEGER :: i
8
9 ! execution section
10 WRITE (*,*) ’write and integer on the screen ’
11 READ (*,*) i
12
13 WRITE (*,*) ’You write ’,i
14
15 ! end section
3
16 STOP
17 END PROGRAM assegnazione
2.3
Scrivere due numeri scelti dall’utente a schermo ed effettuare calcoli aritmetici con essi scrivendo
a schermo il risultato: assegnazione del valore ad una variabile.
2.3.1
Operazione eseguita n2 = 10n1
1 ! File: assegnazione2.f90
2 PROGRAM assegnazione2
3
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2
7
8 ! Sezione esecutiva
9 WRITE (*,*) ’Inserisci due interi (separati da spazio) e premi INVIO’
10 READ (*,*) num1 , num2
11
12 WRITE (*,*) ’Hai scritto:’,num1 , num2
13
14 num2 = num1 * 10
15
16 WRITE (*,*) ’Le nuove variabili sono:’,num1 , num2
17
18 ! Sezione conclusiva
19 STOP
20 END PROGRAM assegnazione2
2.3.2
Operazione eseguita n2 = 10 + n1
1 ! File: assegnazione3.f90
2 PROGRAM assegnazione3
3
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2
7
8 ! Sezione esecutiva
9 WRITE (*,*) ’Inserisci due interi (separati da spazio) e premi INVIO’
10 READ (*,*) num1 , num2
11
12 WRITE (*,*) ’Hai scritto:’,num1 , num2
13
14 num2 = num1 + 10
15
16 WRITE (*,*) ’Le nuove variabili sono:’,num1 , num2
17
18 ! Sezione conclusiva
4
19 STOP
20 END PROGRAM assegnazione3
2.3.3
Operazione eseguita n1 = 10 + n1
1 ! File: assegnazione4.f90
2 PROGRAM assegnazione4
3
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2
7
8 ! Sezione esecutiva
9 WRITE (*,*) ’Inserisci due interi (separati da spazio) e premi INVIO’
10 READ (*,*) num1 , num2
11
12 WRITE (*,*) ’Hai scritto:’,num1 , num2
13
14 num1 = num1 + 10
15
16 WRITE (*,*) ’Le nuove variabili sono:’,num1 , num2
17
18 ! Sezione conclusiva
19 STOP
20 END PROGRAM assegnazione4
2.3.4
Operazione eseguita n2 + 10 = n1
1 ! File: assegnazione5.f90
2 PROGRAM assegnazione5
3
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2
7
8 ! Sezione esecutiva
9 WRITE (*,*) ’Inserisci due interi (separati da spazio) e premi INVIO’
10 READ (*,*) num1 , num2
11
12 WRITE (*,*) ’Hai scritto:’,num1 , num2
13
14 num2 + 10 = num1
15
16 WRITE (*,*) ’Le nuove variabili sono:’,num1 , num2
17
18 ! Sezione conclusiva
19 STOP
20 END PROGRAM assegnazione5
2.3.5
Operazione eseguita n2 = n1 + n2
5
1 ! File: assegnazione6.f90
2 PROGRAM assegnazione6
3
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2
7
8 ! Sezione esecutiva
9 WRITE (*,*) ’Inserisci due interi (separati da spazio) e premi INVIO’
10 READ (*,*) num1 , num2
11
12 WRITE (*,*) ’Hai scritto:’,num1 , num2
13
14 num2 = num1 + num2
15
16 WRITE (*,*) ’Le nuove variabili sono:’,num1 , num2
17
18 ! Sezione conclusiva
19 STOP
20 END PROGRAM assegnazione6
2.4
Dato il raggio (numero reale) scrivere l’area ed il perimetro del cerchio: definizione di varibili ed
utilizzo delle stesse.
1 ! File: cerchio.f90
2 !Questo programma legge un reale dallo schermo
3 !e calcola l’area e la circonferenza del cerchio
4 !di cui il reale e’ il raggio
5 PROGRAM cerchio
6
7 ! Sezione dichiarativa
8 IMPLICIT NONE
9 REAL :: radius
10 REAL , PARAMETER :: pi =3.141592
11
12 ! Sezione esecutiva
13 WRITE (*,*) ’Qual il raggio del cerchio?’
14 READ (*,*) radius
15
16 WRITE (*,*) ’Il perimetro del cerchio e’’:’ ,2. * pi * radius
17 WRITE (*,*) ’L’’area del cerchio e’’:’, pi * radius **2
18
19 ! Sezione conclusiva
20 STOP
21 END PROGRAM cerchio
2.5
Dati due interi calcolarne il massimo.
6
2.5.1
Utilizzo della funzione max(a,b).
1 ! File: massimo1.f90
2 ! Calcolo del massimo dati due numeri
3 PROGRAM massimo
4
5 ! Sezione dichiarativa
6 IMPLICIT NONE
7 INTEGER :: num1 , num2 , massimo
8
9 ! Sezione esecutiva
10 WRITE (*,*) ’Inserisci due interi (separati da spazio)’
11 READ (*,*) num1 , num2
12
13 massimo = max(num1 ,num2)
14
15 WRITE (*,*) ’Il massimo e’’:’,massimo
16
17 ! Sezione conclusiva
18 STOP
19 END PROGRAM massimo
2.5.2
Utilizzo della sintassi if-then-else.
1 ! File: massimo.f90
2 ! Calcolo del massimo dati due numeri
3 PROGRAM massimo
4
5 ! Sezione dichiarativa
6 IMPLICIT NONE
7 INTEGER :: num1 , num2 , massimo
8
9 ! Sezione esecutiva
10 WRITE (*,*) ’Inserisci due interi (separati da spazio)’
11 READ (*,*) num1 , num2
12
13 IF (num1.GT.num2) then
14 massimo = num1
15 ELSE
16 massimo = num2
17 ENDIF
18
19 WRITE (*,*) ’Il massimo e’’:’,massimo
20
21 ! Sezione conclusiva
22 STOP
23 END PROGRAM massimo
2.6
Dato un intero scelto dall’utente dire se e pari o dispari: utilizzo della struttura if-then-else e della
funzione mod(a,b).
7
1 ! file even_odd.f90
2 ! This code reads one integer and say if it is even or odd
3 PROGRAM even_odd
4 !sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER i1
7 CHARACTER (4) num
8 !sezione esecutiva
9 WRITE (*,*) ’Insert one integer , then press ENTER’
10 READ (*,*) i1
11 if (mod(i1 ,2).eq.0) then
12 num=’even’
13 else
14 num=’odd’
15 endif
16 write (*,*) ’the inserted number is ’,num
17 !sezione conclusiva
18 STOP
19 END
2.7
Data una sequenza di 3 numeri (interi) calcolarne la somma e scriverla a schermo: utilizzo dell’istruzione
ciclica do-loop
1 !file: somma.f90
2 !Show the cicle statements use
3 PROGRAM somma
4 !Declaration section
5 IMPLICIT NONE
6 INTEGER :: amount , num1 , summ , i
7 !Execution section
8 amount =3
9 WRITE (*,*)’Insert ’, amount ,’integer and press each time ENTER:’
10 summ=0
11 DO i=1,amount
12 READ (*,*) num1
13 summ = summ + num1
14 ENDDO
15 WRITE (*,*) ’The sum reads:’, summ
16 !End section
17 STOP
18 END
2.8
Dato un intero, scelto dall’utente, calcolare il fattoriale del numero stesso: utilizzo dell’istruzione
ciclica do-loop
1 !file: fattoriale.f90
2 !This program reads a number and compute the factorial
3 PROGRAM fattoriale
4 !sezione dichiarativa
5 IMPLICIT NONE
8
6 INTEGER :: n, m
7 INTEGER :: fact
8 !sezione esecutiva
9 WRITE (*,*) ’Write the number on the screen and press ENTER’
10 READ (*,*) n
11 fact=n
12 DO m=1,n-1
13 fact=fact*(n-m)
14 ENDDO
15 if (n.eq.0) fact=1
16 WRITE (*,*) ’The factorial of’, n,’is:’,fact
17 !sezione esecutiva
18 STOP
19 END PROGRAM fattoriale
2.9
Data una sequenza di interi (di 0 e 1) scelta dall’utente dire da quanti zeri e formata la sequenza
di zeri, consecutivi, piu lunga: utilizzo dell’istruzione ciclica indefinita do-loop
1 ! file: sequenza.f90
2 ! This program is an example of the undefined DO -LOOP
3 PROGRAM sequenza
4 !declaration section
5 IMPLICIT NONE
6 INTEGER :: bit , cont , maxleng
7 cont = 0
8 maxleng = 0
9 !execution section
10 WRITE (*,*) ’Insert a 0,1,2 sequence (press ENTER each time)’
11 DO
12 WRITE (*,*) ’Insert a number (no 0 or 1 to end):’
13 READ (*,*) bit
14 IF (bit .ne. 1 .and. bit .ne. 0) EXIT
15 IF (bit .eq. 0) THEN
16 cont = cont + 1
17 IF (cont .gt. maxleng) maxleng = maxleng + 1
18 ELSE
19 cont = 0
20 ENDIF
21 ENDDO
22 WRITE (*,*) ’The most length sequence of only 0 counts ’&
23 ,maxleng ,’ zeros ’
24 !end section
25 STOP
26 END
2.10
Dati due numeri interi scelti dall’utente calcolare il massimo comune denominatore: utilizzo delle
istruzioni if-then-else, do-loop e della funzione mod(a,b)
1 !file: mcd.f90
2 !This program prints the greater common divisor of two integers
9
3 PROGRAM mcd
4 !sezione dichiatativa
5 IMPLICIT NONE
6 INTEGER :: a, b, m
7 !sezione esecutiva
8 WRITE (*,*) ’Insert two number separated by a space:’
9 READ (*,*) a, b
10
11 m=a
12 IF (b.lt.a) m=b
13 DO
14 IF (mod(b,m).eq.0. and.mod(a,m).eq.0) EXIT
15 m = m-1
16 ENDDO
17 WRITE (*,*) ’The greater common divisor is’,m
18 !sezione conclusiva
19 STOP
20 END
2.11
Dati due numeri interi scelti dall’utente calcolare il massimo:
2.11.1
Utilizzo della sotto-unita subroutine
1 ! File: massimo.f90
2 ! Calcolo del massimo dati due numeri
3 PROGRAM massimo
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2 , mas
7 ! Sezione esecutiva
8 WRITE (*,*) ’Inserisci due interi (separati da spazio)’
9 READ (*,*) num1 , num2
10 call calc(num1 ,num2 ,mas)
11 WRITE (*,*) ’Il massimo e’’:’,mas
12 ! Sezione conclusiva
13 STOP
14 END PROGRAM massimo
15 !____________________________________________________________
16 !____________________________________________________________
17 SUBROUTINE calc(x,y,z)
18 IMPLICIT NONE
19 INTEGER :: x,y,z
20 IF (x.GT.y) then
21 z = x
22 ELSE
23 z = y
24 ENDIF
25 RETURN
26 END SUBROUTINE calc
10
2.11.2
Utilizzo della sotto-unita function
1 ! File: massimo.f90
2 ! Calcolo del massimo dati due numeri
3 PROGRAM massimo
4 ! Sezione dichiarativa
5 IMPLICIT NONE
6 INTEGER :: num1 , num2 , mas , calc
7 ! Sezione esecutiva
8 WRITE (*,*) ’Inserisci due interi (separati da spazio)’
9 READ (*,*) num1 , num2
10 mas = calc(num1 ,num2)
11 WRITE (*,*) ’Il massimo e’’:’,mas
12 ! Sezione conclusiva
13 STOP
14 END PROGRAM massimo
15 !____________________________________________________________
16 !____________________________________________________________
17 FUNCTION calc(x,y)
18 IMPLICIT NONE
19 INTEGER :: x,y,calc
20 IF (x.GT.y) then
21 calc = x
22 ELSE
23 calc = y
24 ENDIF
25 RETURN
26 END FUNCTION calc
2.12
Creare una stringa (nome di un file) concatenando tre stringhe:
• la versione ‘a’
• il tempo ‘10’
• l’estensione ‘.dat’.
La stringa finale sara ‘a00010.dat’
1 ! File: name_file.f90
2 ! Gestire e creare una stringa corrispondende
3 ! al nome di un file
4 PROGRAM name_file
5 IMPlICIT NONE
6 CHARACTER (30):: name
7 CHARACTER (1):: ver
8 CHARACTER (6):: time
9 CHARACTER (4):: ext
10 INTEGER :: it
11
12 it= 10
13
14 ext=’.dat’
11
15 ver=’a’
16 WRITE(time ,’(i6.6)’) it
17 name = ver//time//ext
18
19 WRITE (*,*) ’Apro il file ’,name
20
21 STOP
22 END PROGRAM name_file
2.13
Leggere un numero da un file chiamato ‘input.dat’ e scrivere il numero a schermo
1 ! File: name_file.f90
2 ! Gestire e creare una stringa corrispondende
3 ! al nome di un file
4 PROGRAM read_file
5 IMPlICIT NONE
6 CHARACTER (30):: name_file
7 CHARACTER (7):: ver
8 CHARACTER (4):: ext
9 REAL :: pi
10
11 ext=’.dat’
12 ver=’input’
13 name_file = trim(ver)//ext
14
15 WRITE (*,*) ’Apro il file ’,name_file
16 OPEN(UNIT=1,FILE=name_file ,STATUS=’old’,ACTION=’read’)
17 READ (1,*) pi
18 CLOSE (1)
19
20 WRITE (*,*) ’Il contenuto del file ’,name_file ,’ e’’ ’, pi
21 WRITE (*,10) name_file , pi
22
23 10 FORMAT(’Il contenuto del file ’,A9,’ e’’’, F7.4)
24 STOP
25 END PROGRAM read_file
12
Chapter 3
Soluzione di ODE
3.1
Calcolare e la soluzione esatta della seguente equazione differenziale ordinaria:
dy
dx= −3y (3.1)
y(0) = y0 = 1 (3.2)
nel punto x=10, tramite l’uso di una subroutine.
1 PROGRAM main
2 IMPLICIT NONE
3 REAL :: y,x,alpha ,y0
4
5 x=1.0
6 alpha =3.0
7 y0=1.0
8 CALL exact_sol(y,x,y0,alpha)
9
10 WRITE (*,*) ’La soluzione al tempo t=’,x,’ e’’ uguale a y=’,y
11
12 STOP
13 END PROGRAM main
14 !_____________________________________________________________
15
16 SUBROUTINE exact_sol(y,x,y0 ,a)
17 IMPLICIT NONE
18 REAL ,INTENT(IN):: y0,x,a
19 REAL ,INTENT(OUT):: y
20
21 y = y0*exp(-x*a)
22
23 RETURN
24 END SUBROUTINE exact_sol
13
3.2
Calcolare e la soluzione esatta della seguente equazione differenziale ordinaria:
dy
dx= −3y (3.3)
y(0) = y0 = 1 (3.4)
dal punto x=0 al punto x=10, tramite l’uso di una subroutine.
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nx=10
4 INTEGER :: i
5 REAL :: y,x,alpha ,y0
6 REAL :: Dx ,x_max ,x_min
7
8 x_max =2.
9 x_min =0.
10 Dx=(x_max -x_min)/(Nx)
11 alpha =3.0
12 y0=1.
13
14 do i=0,Nx
15 x=x_min+Dx*i
16 CALL exact_sol(y,x,y0,alpha)
17
18 WRITE (*,*) x,y
19 enddo
20
21 WRITE (*,*) ’Fine della simulazione ’
22
23 STOP
24 END PROGRAM main
25 !_____________________________________________________________
26
27 SUBROUTINE exact_sol(y,x,y0 ,a)
28 IMPLICIT NONE
29 REAL ,INTENT(IN):: y0,x,a
30 REAL ,INTENT(OUT):: y
31
32 y = y0*exp(-x*a)
33
34 RETURN
35 END SUBROUTINE exact_sol
3.3
Calcolare e la soluzione esatta e la soluzione approssimata con il metodo di eulero della seguente
equazione differenziale ordinaria:
dy
dx= −3y (3.5)
y(0) = y0 = 1 (3.6)
dal punto x=0 al punto x=10. Scrivere infine la soluzione su un file.
14
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nx=10
4 INTEGER :: i
5 REAL :: y,x,alpha ,y0
6 REAL :: Dx ,y_old ,y_new
7 REAL :: rhs ,x_max ,x_min
8 REAL :: error , err_max
9
10 error = 0.0
11 err_max = 0.0
12
13 call input(x_min ,x_max ,alpha ,y0)
14 Dx=(x_max -x_min)/float(Nx)
15 y_old=y0
16
17 call write_sol(x_min ,y0,y0)
18
19 do i=1,Nx
20 x=x_min+Dx*i
21 CALL exact_sol(y,x,y0,alpha)
22
23 y_new = y_old + Dx*rhs(y_old ,alpha)
24
25 call write_sol(x,y,y_new)
26
27 y_old = y_new
28
29 error = abs(y_new -y)
30 err_max = max(err_max ,error)
31 enddo
32
33 WRITE (*,*) ’Dx , errore ’,Dx ,error
34 WRITE (*,*) ’Fine della simulazione ’
35
36 STOP
37 END PROGRAM main
38 !_____________________________________________________________
39 !_____________________________________________________________
40 SUBROUTINE exact_sol(y,x,y0 ,a)
41 IMPLICIT NONE
42 REAL ,INTENT(IN):: y0,x,a
43 REAL ,INTENT(OUT):: y
44
45 y = y0*exp(-x*a)
46
47 RETURN
48 END SUBROUTINE exact_sol
49 !_____________________________________________________________
50 SUBROUTINE input(x_min ,x_max ,alpha ,y0)
51 IMPLICIT NONE
52 REAL ,INTENT(OUT):: y0,alpha ,x_min ,x_max
53
54 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
55 READ (1,*) x_min
56 READ (1,*) x_max
15
57 READ (1,*) y0
58 READ (1,*) alpha
59 CLOSE (1)
60
61 RETURN
62 END SUBROUTINE input
63 !_____________________________________________________________
64 FUNCTION rhs(y,a)
65 IMPLICIT NONE
66 REAL ,INTENT(IN) :: a,y
67 REAL :: rhs
68
69 rhs=-a*y
70
71 RETURN
72 END FUNCTION rhs
73 !_____________________________________________________________
74 SUBROUTINE write_sol(x,ye ,ya)
75 IMPLICIT NONE
76 REAL ,INTENT(IN)::x,ye,ya
77
78 OPEN(UNIT=1,FILE=’output.dat’,STATUS=’unknown ’,ACTION=’write ’,ACCESS=’append ’)
79 WRITE (1,*) x,ye ,ya
80 CLOSE (1)
81
82 RETURN
83 END SUBROUTINE write_sol
84 !_____________________________________________________________
3.4
Risolvere il problema della sezione 3.3 con il metodo di Heun
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nx=10
4 INTEGER :: i
5 REAL :: y,x,alpha ,y0
6 REAL :: Dx ,y_old ,y_new ,y_ast
7 REAL :: rhs ,x_max ,x_min
8 REAL :: error ,err_max
9
10 error = 0.
11 err_max = 0.
12
13 call input(x_min ,x_max ,alpha ,y0)
14 Dx=(x_max -x_min)/float(Nx)
15 y_old=y0
16
17 call write_sol(x_min ,y0,y0)
18
19 do i=1,Nx
20 x=x_min+Dx*i
21 CALL exact_sol(y,x,y0,alpha)
22
23 y_ast = y_old + Dx*rhs(y_old ,alpha)
16
24 y_new = y_old + Dx *.5*( rhs(y_old ,alpha)+rhs(y_ast ,alpha))
25
26
27 error = abs(y_new -y)
28 err_max=max(err_max ,error)
29
30 call write_sol(x,y,y_new)
31
32 y_old = y_new
33 enddo
34
35 WRITE (*,*) ’ERRORE MAX’,Dx ,err_max
36 WRITE (*,*) ’Fine della simulazione ’
37
38 STOP
39 END PROGRAM main
40 !_____________________________________________________________
41 !_____________________________________________________________
42 SUBROUTINE exact_sol(y,x,y0 ,a)
43 IMPLICIT NONE
44 REAL ,INTENT(IN):: y0,x,a
45 REAL ,INTENT(OUT):: y
46
47 y = y0*exp(-x*a)
48
49 RETURN
50 END SUBROUTINE exact_sol
51 !_____________________________________________________________
52 SUBROUTINE input(x_min ,x_max ,alpha ,y0)
53 IMPLICIT NONE
54 REAL ,INTENT(OUT):: y0,alpha ,x_min ,x_max
55
56 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
57 READ (1,*) x_min
58 READ (1,*) x_max
59 READ (1,*) y0
60 READ (1,*) alpha
61 CLOSE (1)
62
63 RETURN
64 END SUBROUTINE input
65 !_____________________________________________________________
66 FUNCTION rhs(y,a)
67 IMPLICIT NONE
68 REAL ,INTENT(IN) :: a,y
69 REAL :: rhs
70
71 rhs=-a*y
72
73 RETURN
74 END FUNCTION rhs
75 !_____________________________________________________________
76 SUBROUTINE write_sol(x,ye ,ya)
77 IMPLICIT NONE
78 REAL ,INTENT(IN)::x,ye,ya
79
17
80 OPEN(UNIT=1,FILE=’output.dat’,STATUS=’unknown ’,ACTION=’write ’,POSITION=’append ’)
81 WRITE (1,*) x,ye ,ya
82 CLOSE (1)
83
84 RETURN
85 END SUBROUTINE write_sol
86 !____________________________________________________________
3.5
Risolvere il problema della sezione 3.3 con il metodo di Runge-Kutta al IV ordine
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nx =5000
4 INTEGER ,PARAMETER :: nrk=4
5 INTEGER :: i,l
6 REAL :: y,x,alpha ,y0
7 REAL :: Dx ,y_old ,y_new ,y_ast
8 REAL :: rhs ,x_max ,x_min
9 REAL :: error ,err_max
10 REAL ,DIMENSION (1: nrk) :: crk1 ,crk2
11 REAL ,DIMENSION (0: nrk) :: k
12 REAL :: rhsrk
13
14 crk1= (/0., .5, .5, 1./)
15 crk2= (/1., 2., 2., 1./)
16
17 error = 0.
18 err_max = 0.
19
20 call input(x_min ,x_max ,alpha ,y0)
21 Dx=(x_max -x_min)/float(Nx)
22 y_old=y0
23
24 call write_sol(x_min ,y0,y0)
25
26 do i=1,Nx
27 x=x_min+Dx*i
28 CALL exact_sol(y,x,y0,alpha)
29
30 rhsrk =0.
31 y_new = y_old
32 k(0)=0.
33 do l=1,nrk
34 y_old = y_new + crk1(l) * Dx * k(l-1)
35 k(l) = rhs(y_old ,alpha)
36 rhsrk = rhsrk + k(l)*crk2(l)
37 enddo
38
39 y_new = y_new + Dx/6.* rhsrk
40
41
42 error = abs(y_new -y)
43 err_max=max(err_max ,error)
44
18
45 call write_sol(x,y,y_new)
46
47 y_old = y_new
48 enddo
49
50 WRITE (*,*) ’ERRORE MAX’,Dx ,err_max
51 WRITE (*,*) ’Fine della simulazione ’
52
53 STOP
54 END PROGRAM main
55 !_____________________________________________________________
56 !_____________________________________________________________
57 SUBROUTINE exact_sol(y,x,y0 ,a)
58 IMPLICIT NONE
59 REAL ,INTENT(IN):: y0,x,a
60 REAL ,INTENT(OUT):: y
61
62 y = y0*exp(-x*a)
63
64 RETURN
65 END SUBROUTINE exact_sol
66 !_____________________________________________________________
67 SUBROUTINE input(x_min ,x_max ,alpha ,y0)
68 IMPLICIT NONE
69 REAL ,INTENT(OUT):: y0,alpha ,x_min ,x_max
70
71 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
72 READ (1,*) x_min
73 READ (1,*) x_max
74 READ (1,*) y0
75 READ (1,*) alpha
76 CLOSE (1)
77
78 RETURN
79 END SUBROUTINE input
80 !_____________________________________________________________
81 FUNCTION rhs(y,a)
82 IMPLICIT NONE
83 REAL ,INTENT(IN) :: a,y
84 REAL :: rhs
85
86 rhs=-a*y
87
88 RETURN
89 END FUNCTION rhs
90 !_____________________________________________________________
91 SUBROUTINE write_sol(x,ye ,ya)
92 IMPLICIT NONE
93 REAL ,INTENT(IN)::x,ye,ya
94
95 OPEN(UNIT=1,FILE=’output.dat’,STATUS=’unknown ’,ACTION=’write ’,POSITION=’append ’)
96 WRITE (1,*) x,ye ,ya
97 CLOSE (1)
98
99 RETURN
100 END SUBROUTINE write_sol
19
101 !____________________________________________________________
3.6
Risolvere il problema di una particella inerziale sottoposta a gravita ed a resistenza di Stokes:
dvpdt
= −vpτp
+ g (3.7)
vp(0) = 0 (3.8)
Il sistema precedente ha la seguente soluzione esatta:
vp(t) =(
1− e−t/τp)g τp (3.9)
Si usi un metodo di integrazione accurato al primo ordine.
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nt =100000
4 INTEGER :: n
5 REAL :: v,t,tau ,v0,g
6 REAL :: Dt ,v_old ,v_new
7 REAL :: rhs ,t_max ,t_min
8 REAL :: error ,err_max
9
10 error = 0.
11 err_max = 0.
12
13 CALL input(t_min ,t_max ,tau ,g,v0)
14 Dt=(t_max -t_min)/float(Nt)
15 v_old=v0
16
17 CALL write_sol(t_min ,v0,v0,error)
18
19 DO n=1,Nt
20 t=t_min+Dt*n
21 CALL exact_sol(v,t,v0,tau ,g)
22
23 v_new = v_old + rhs(Dt ,v_old ,tau ,g)
24
25
26 error = abs(v_new -v)
27 err_max=max(err_max ,error)
28
29 CALL write_sol(t,v,v_new ,error)
30
31 v_old = v_new
32 ENDDO
33
34 WRITE (*,*) ’ERRORE MAX’,Dt ,err_max
35 WRITE (*,*) ’Fine della simulazione ’
36
37 STOP
38 END PROGRAM main
39 !_____________________________________________________________
20
40 !_____________________________________________________________
41 SUBROUTINE exact_sol(v,t,v0 ,tau ,g)
42 IMPLICIT NONE
43 REAL ,INTENT(IN):: v0,t,tau ,g
44 REAL ,INTENT(OUT):: v
45
46 v = (v0 -g*tau)*exp(-t/tau)+g*tau
47
48 RETURN
49 END SUBROUTINE exact_sol
50 !_____________________________________________________________
51 SUBROUTINE input(t_min ,t_max ,tau ,g,v0)
52 IMPLICIT NONE
53 REAL ,INTENT(OUT):: v0,tau ,g,t_min ,t_max
54
55 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
56 READ (1,*) t_min
57 READ (1,*) t_max
58 READ (1,*) v0
59 READ (1,*) tau
60 READ (1,*) g
61 CLOSE (1)
62 RETURN
63 END SUBROUTINE input
64 !_____________________________________________________________
65 FUNCTION rhs(dt,v,tau ,g)
66 IMPLICIT NONE
67 REAL ,INTENT(IN) :: dt,tau ,g,v
68 REAL :: rhs
69
70 rhs = (-v/tau + g) * dt
71
72 RETURN
73 END FUNCTION rhs
74 !_____________________________________________________________
75 SUBROUTINE write_sol(x,ye ,ya ,err)
76 IMPLICIT NONE
77 REAL ,INTENT(IN)::x,ye,ya,err
78
79 OPEN(UNIT=1,FILE=’output.dat’,STATUS=’unknown ’,ACTION=’write ’,POSITION=’append ’)
80 WRITE (1,*) x,ye ,ya ,err
81 CLOSE (1)
82
83 RETURN
84 END SUBROUTINE write_sol
85 !_____________________________________________________________
3.7
Al problema della sezione precedente, § 3.6 si aggiunga l’equazione per la posizione della particella:
dxpdt
= vp (3.10)
xp(0) = 0 (3.11)
21
La soluzione esatta per questa nuova equazione :
xp(t) = τp g[t+ τp
(e−t/τp − 1
)](3.12)
1 PROGRAM main
2 IMPLICIT NONE
3 INTEGER ,PARAMETER :: Nt =1000
4 INTEGER :: n
5 REAL :: x,v,t,tau ,x0,v0,g
6 REAL :: Dt ,v_old ,v_new
7 REAL :: x_old ,x_new
8 REAL :: rhs_x ,rhs_v ,t_max ,t_min
9 REAL :: error ,err_max
10
11 error = 0.
12 err_max = 0.
13
14 CALL input(t_min ,t_max ,tau ,g,v0,x0)
15 Dt=(t_max -t_min)/float(Nt)
16 v_old=v0
17
18 CALL write_sol(t_min ,v0,v0,x0,x0)
19
20 DO n=1,Nt
21 t=t_min+Dt*n
22 CALL exact_sol(v,x,t,v0,tau ,g)
23
24 v_new = v_old + rhs_v(Dt ,v_old ,tau ,g)
25 x_new = x_old + rhs_x(Dt ,v_old)
26
27 CALL write_sol(t,v,v_new ,x,x_new)
28
29 v_old = v_new
30 x_old = x_new
31 ENDDO
32
33 WRITE (*,*) ’ERRORE MAX’,Dt ,err_max
34 WRITE (*,*) ’Fine della simulazione ’
35
36 STOP
37 END PROGRAM main
38 !_____________________________________________________________
39 !_____________________________________________________________
40 SUBROUTINE exact_sol(v,x,t,v0 ,tau ,g)
41 IMPLICIT NONE
42 REAL ,INTENT(IN):: v0,t,tau ,g
43 REAL ,INTENT(OUT):: v,x
44
45 x = tau*g*(t+tau*(exp(-t/tau) -1.))
46 v = (1.-exp(-t/tau))*g*tau
47
48 RETURN
49 END SUBROUTINE exact_sol
50 !_____________________________________________________________
51 SUBROUTINE input(t_min ,t_max ,tau ,g,v0 ,x0)
52 IMPLICIT NONE
22
53 REAL ,INTENT(OUT):: x0,v0,tau ,g,t_min ,t_max
54
55 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
56 READ (1,*) t_min
57 READ (1,*) t_max
58 READ (1,*) x0
59 READ (1,*) v0
60 READ (1,*) tau
61 READ (1,*) g
62 CLOSE (1)
63 RETURN
64 END SUBROUTINE input
65 !_____________________________________________________________
66 FUNCTION rhs_v(dt,v,tau ,g)
67 IMPLICIT NONE
68 REAL ,INTENT(IN) :: dt,tau ,g,v
69 REAL :: rhs_v
70
71 rhs_v = (-v/tau + g) * dt
72
73 RETURN
74 END FUNCTION rhs_v
75 !_____________________________________________________________
76 FUNCTION rhs_x(dt,v)
77 IMPLICIT NONE
78 REAL ,INTENT(IN) :: dt,v
79 REAL :: rhs_x
80
81 rhs_x = v * Dt
82
83 RETURN
84 END FUNCTION rhs_x
85 !_____________________________________________________________
86 SUBROUTINE write_sol(x,ye ,ya ,xe ,xa)
87 IMPLICIT NONE
88 REAL ,INTENT(IN)::x,ye,ya,xe,xa
89
90 OPEN(UNIT=1,FILE=’output.dat’,STATUS=’unknown ’,ACTION=’write ’,POSITION=’append ’)
91 WRITE (1,*) x,ye ,ya ,xe ,xa
92 CLOSE (1)
93
94 RETURN
95 END SUBROUTINE write_sol
96 !_____________________________________________________________
23
Chapter 4
Soluzione di PDE
4.1
Scrivere un programma per il calcolo della soluzione dell’equazione di convezione lineare se ne
confronti il risultato approssimato con quello analitico. Il sistema differenziale da risolvere :
∂u
∂t+ a
∂u
∂x= 0 [0, 1]× [0, 10] (4.1)
u(0, t) = 0 (4.2)
u(x, 0) = u0(x) = max [0, (x− 0.25)(0.5− x)] (4.3)
Scrivere il risultato su diversi file per diversi istanti temporali ed integrare il sistema con un metodo
esplicito accurato al primo ordine nel tempo e nello spazio.
1 !____. file: eulero_upwind.f90
2 !____. This program applies the eulero integration to the
3 !____. linear convection partial differential equation
4 PROGRAM main
5 !____.__________________________________________________________________
6 IMPLICIT NONE
7 INTEGER :: it , j
8 INTEGER :: itmin ,itmax ,itout
9 INTEGER ,PARAMETER :: Nx=100,Ng=1
10 !____.__________________________________________________________________
11 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old ,v_exact ,x
12 REAL:: delta_t ,delta_x ,a_conv ,rhs ,rhs_uw ,rhs_uw2
13 REAL:: error ,error_max ,time ,v0
14 !____.__________________________________________________________________
15 ! metric definition
16 CALL griglia(Nx,Ng,x,delta_x)
17 !____.__________________________________________________________________
18 ! some initializations
19 DO j=1-ng,nx+ng
20 v_old(j)=v0(x(j))
21 ENDDO
22 error =0.
23 error_max = -999999.9
24 !____.__________________________________________________________________
25 ! read on file input.dat
26 CALL input(itmin ,itmax ,itout ,a_conv ,delta_t)
27 !____.__________________________________________________________________
24
28 ! compute maximun number of iterations and open write unit
29 !____.__________________________________________________________________
30 write (*,*) ’cfl’,delta_t*a_conv/delta_x
31 CALL write_disk(x,v_old ,v_old ,0,Nx,Ng)
32 DO it= itmin , itmax
33 DO j=1,nx
34 rhs = rhs_uw(v_old(j),v_old(j-1),a_conv ,delta_x)
35 v_new(j) = v_old(j) + delta_t * rhs
36 ENDDO
37 time = it*delta_t
38 !____. boundary conditions
39 DO j=1-ng ,0
40 v_new(j)=0.
41 ENDDO
42 !____. update the solution
43 DO j=1-ng ,Nx+ng
44 v_old(j)= v_new(j)
45 ENDDO
46 !____. compute the error
47 CALL exact_sol(Nx,Ng,x,time ,a_conv ,v_exact)
48
49 error =0.
50 DO j=1,Nx
51 error = error + abs(v_new(j)-v_exact(j))
52 ENDDO
53 error_max=max(error_max ,error/float(Nx))
54 !____. write on disk
55 if (mod(it ,itout).eq.0) CALL write_disk(x,v_new ,v_exact ,it,Nx,Ng)
56 ENDDO
57
58 WRITE (*,*) ’dt , error=’,delta_t ,error_max
59 WRITE (*,*) ’solution stored in file velocity.dat’
60 WRITE (*,*) ’Simulation ended!’
61 STOP
62 END
63 !____.__________________________________________________________________
64 !____.__________________________________________________________________
65 !____.__________________________________________________________________
66 !____.__________________________________________________________________
67 FUNCTION rhs_uw(vj,vjm ,a,Dx)
68 IMPLICIT NONE
69 REAL:: rhs_uw ,vj,vjm ,a,Dx
70
71 rhs_uw=-a*(vj-vjm)/Dx
72
73 RETURN
74 END
75 !____.__________________________________________________________________
76 !____.__________________________________________________________________
77 FUNCTION v0(x)
78 IMPLICIT NONE
79 REAL:: v0 ,x,fx
80
81 fx=(.25-x)*(x-.5)
82
83 v0=max(0.,fx)
25
84
85 RETURN
86 END
87 !____.__________________________________________________________________
88 !____.__________________________________________________________________
89 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,a_conv ,v_exact)
90
91 IMPLICIT NONE
92 INTEGER :: Nx ,Ng ,j
93 REAL ,DIMENSION (1-ng:nx+ng):: x, v_exact
94 REAL:: v0 , appo , time , a_conv
95
96 DO j=1-ng,Nx+ng
97 appo=x(j)-a_conv*time
98 v_exact(j)=v0(appo)
99 ENDDO
100
101 RETURN
102 END
103 !____.__________________________________________________________________
104 !____.__________________________________________________________________
105 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
106
107 IMPLICIT NONE
108 INTEGER :: it ,nx ,ng ,j
109 REAL ,DIMENSION (1-ng:Nx+ng):: x,v,u
110 CHARACTER (32):: opfile
111
112 opfile=’bcl12345678.dat’
113 WRITE(opfile (4:11) ,1000) it
114 !!!!! WRITE(opfile (4:11) ,’(i8.8) ’) it
115 opfile = ’./DATA/’// opfile
116 1000 FORMAT(i8.8)
117
118 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
119 DO j=1-Ng,Nx
120 WRITE (13,*) x(j),v(j),u(j)
121 ENDDO
122 CLOSE (13)
123
124 RETURN
125 END
126 !____.__________________________________________________________________
127 !____.__________________________________________________________________
128 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
129 IMPLICIT NONE
130 INTEGER Nx ,Ng ,j
131 REAL ,DIMENSION (1-ng:Nx+ng):: x
132 REAL L,delta_x
133 L=1.
134 delta_x=L/float(Nx+1)
135
136 DO j=1-ng,Nx+ng
137 x(j)=j*delta_x
138 ENDDO
139
26
140 RETURN
141 END
142 !____.__________________________________________________________________
143 !____.__________________________________________________________________
144 SUBROUTINE input(itmin ,itmax ,itout ,a_conv ,delta_t)
145 IMPLICIT NONE
146 INTEGER :: itmin ,itmax ,itout
147 REAL :: a_conv ,delta_t
148 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
149 READ (1,*) itmin
150 READ (1,*) itmax
151 READ (1,*) itout
152 READ (1,*) a_conv
153 READ (1,*) delta_t
154 CLOSE (1)
155 RETURN
156 END SUBROUTINE input
4.2
Integrare il sistema della sezione precedente, § 4.1, con un metodo esplicito accurato al primo ordine
nel tempo e al secondo ordine nello spazio.
1 !____. file: eulero_upwind2.f90
2 !____. This program applies the eulero integration to the
3 !____. linear convection partial differential equation
4 PROGRAM main
5 !____.__________________________________________________________________
6 IMPLICIT NONE
7 INTEGER :: it , j
8 INTEGER :: itmin ,itmax ,itout
9 INTEGER ,PARAMETER :: Nx=100,Ng=2
10 !____.__________________________________________________________________
11 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old ,v_exact ,x
12 REAL:: delta_t ,delta_x ,a_conv ,rhs ,rhs_uw ,rhs_uw2
13 REAL:: error ,error_max ,time ,v0
14 !____.__________________________________________________________________
15 ! metric definition
16 CALL griglia(Nx,Ng,x,delta_x)
17 !____.__________________________________________________________________
18 ! some initializations
19 DO j=1-ng,nx+ng
20 v_old(j)=v0(x(j))
21 ENDDO
22 error =0.
23 error_max = -999999.9
24 !____.__________________________________________________________________
25 ! read on file eulero_stokes.dat
26 CALL input(itmin ,itmax ,itout ,a_conv ,delta_t)
27 !____.__________________________________________________________________
28 ! compute maximun number of iterations and open write unit
29 !____.__________________________________________________________________
30 write (*,*) ’cfl’,delta_t*a_conv/delta_x
31 CALL write_disk(x,v_old ,v_old ,0,Nx,Ng)
32 DO it= itmin , itmax
27
33 DO j=1,nx
34 rhs = rhs_uw2(v_old(j),v_old(j-1),v_old(j-2),a_conv ,delta_x)
35 v_new(j) = v_old(j) + delta_t * rhs
36 ENDDO
37 time = it*delta_t
38 !____. boundary conditions
39 DO j=1-ng ,0
40 v_new(j)=0.
41 ENDDO
42 !____. update the solution
43 DO j=1-ng ,Nx+ng
44 v_old(j)= v_new(j)
45 ENDDO
46 !____. compute the error
47 CALL exact_sol(Nx,Ng,x,time ,a_conv ,v_exact)
48
49 error =0.
50 DO j=1,Nx
51 error = error + abs(v_new(j)-v_exact(j))
52 ENDDO
53 error_max=max(error_max ,error/float(Nx))
54 !____. write on disk
55 if (mod(it ,itout).eq.0) CALL write_disk(x,v_new ,v_exact ,it,Nx,Ng)
56 ENDDO
57
58 WRITE (*,*) ’dt , error=’,delta_t ,error_max
59 WRITE (*,*) ’solution stored in file velocity.dat’
60 WRITE (*,*) ’Simulation ended!’
61 STOP
62 END
63 !____.__________________________________________________________________
64 !____.__________________________________________________________________
65 !____.__________________________________________________________________
66 !____.__________________________________________________________________
67 FUNCTION rhs_uw2(vj,vjm ,vjmm ,a,Dx)
68 IMPLICIT NONE
69 REAL:: rhs_uw2 ,vj,vjm ,vjmm ,a,Dx
70
71 rhs_uw2=-a*(3.*vj -4.* vjm+vjmm)/(2.*Dx)
72
73 RETURN
74 END
75 !____.__________________________________________________________________
76 !____.__________________________________________________________________
77 FUNCTION v0(x)
78 IMPLICIT NONE
79 REAL:: v0 ,x,fx
80
81 fx=(.25-x)*(x-.5)
82
83 v0=max(0.,fx)
84
85 RETURN
86 END
87 !____.__________________________________________________________________
88 !____.__________________________________________________________________
28
89 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,a_conv ,v_exact)
90
91 IMPLICIT NONE
92 INTEGER :: Nx ,Ng ,j
93 REAL ,DIMENSION (1-ng:nx+ng):: x,v_exact
94 REAL:: v0 , appo , a_conv , time
95
96 DO j=1-ng,Nx+ng
97 appo=x(j)-a_conv*time
98 v_exact(j)=v0(appo)
99 ENDDO
100
101 RETURN
102 END
103 !____.__________________________________________________________________
104 !____.__________________________________________________________________
105 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
106
107 IMPLICIT NONE
108 INTEGER it ,nx ,ng ,j
109 REAL ,DIMENSION (1-ng:nx+ng):: x,v,u
110 CHARACTER (32) opfile
111
112 opfile=’acl12345678.dat’
113 WRITE(opfile (4:11) ,1000) it
114 opfile = ’./DATA/’// opfile
115 1000 FORMAT(i8.8)
116
117 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
118 DO j=1-Ng,Nx
119 WRITE (13,*) x(j),v(j),u(j)
120 ENDDO
121 CLOSE (13)
122
123 RETURN
124 END
125 !____.__________________________________________________________________
126 !____.__________________________________________________________________
127 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
128 IMPLICIT NONE
129 INTEGER :: Nx ,Ng ,j
130 REAL ,DIMENSION (1-ng:nx+ng):: x
131 REAL:: L,delta_x
132 L=1.
133 delta_x=L/float(Nx)
134
135 DO j=1-ng,Nx+ng
136 x(j)=j*delta_x
137 ENDDO
138
139 RETURN
140 END
141 !____.__________________________________________________________________
142 !____.__________________________________________________________________
143 SUBROUTINE input(itmin ,itmax ,itout ,a_conv ,delta_t)
144 IMPLICIT NONE
29
145 INTEGER :: itmin ,itmax ,itout
146 REAL :: a_conv ,delta_t
147 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
148 READ (1,*) itmin
149 READ (1,*) itmax
150 READ (1,*) itout
151 READ (1,*) a_conv
152 READ (1,*) delta_t
153 CLOSE (1)
154 RETURN
155 END SUBROUTINE input
4.3
Scrivere un programma per il calcolo della soluzione dell’equazione del calore se ne confronti il
risultato approssimato con quello analitico. Il sistema differenziale da risolvere :
∂u
∂t= ν
∂2u
∂x2[0, 1]× [0, 10] (4.4)
u(0, t) = 100 (4.5)
u(1, t) = 100 (4.6)
u(x, 0) = u0(x) = u(0, t) + [v(1, t)− v(0, t)] x−∞∑m=1
400
(2m− 1)πsin((2m− 1)πx) (4.7)
La cui soluzione esatta :
u(x, t) = u(0, t) + [v(1, t)− v(0, t)] x−∞∑m=1
400
(2m− 1)πsin((2m− 1)πx) exp(−ν (2m− 1)2π2 t)
Scrivere il risultato su diversi file per diversi istanti temporali ed integrare il sistema con un
metodo esplicito accurato al primo ordine nel tempo e al secondo ordine nello spazio.
1 !____. file: FTCS.f90
2 !____. This program applies the eulero explicit time integration
3 !____. and a second order spatial discretization to the
4 !____. Fourier partial differential equation
5 PROGRAM main
6 !____.__________________________________________________________________
7 IMPLICIT NONE
8 INTEGER :: it , j
9 INTEGER :: itmin ,itmax ,itout
10 INTEGER ,PARAMETER :: Nx=20,Ng=1
11 !____.__________________________________________________________________
12 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old ,v_exact ,x
13 REAL:: delta_t ,delta_x ,nu,rhs_cent ,rhs ,va,vb
14 REAL:: error ,error_max ,time ,v0
15 !____.__________________________________________________________________
16 ! metric definition
17 CALL griglia(Nx,Ng,x,delta_x)
18 !____.__________________________________________________________________
19 ! read on file eulero_stokes.dat
20 CALL input(itmin ,itmax ,itout ,nu,delta_t ,va,vb)
21 !____.__________________________________________________________________
30
22 ! some initializations
23 call exact_sol(Nx,Ng,x,0.,nu,va,vb,v_exact)
24
25 DO j=1,nx
26 v_old(j)=0.!v_exact(j)
27 ENDDO
28 v_old (0)=1.
29 v_old(Nx+1) =0.5
30
31 error =0.
32 error_max = -999999.9
33 !____.__________________________________________________________________
34 ! compute maximun number of iterations and open write unit
35 !____.__________________________________________________________________
36 write (*,*) ’cfl’,delta_t*nu/delta_x **2
37 CALL write_disk(x,v_old ,v_old ,0,Nx,Ng)
38 DO it= itmin , itmax
39 DO j=1,nx
40 rhs = rhs_cent(v_old(j+1),v_old(j),v_old(j-1),nu ,delta_x)
41 v_new(j) = v_old(j) + delta_t * rhs
42 ENDDO
43 time = it*delta_t
44 !____. boundary conditions
45 DO j=1-ng ,0
46 v_new( j) = 1.
47 v_new(Nx+1+j) = 0.5
48 ENDDO
49 !____. update the solution
50 DO j=1-ng ,Nx+ng
51 v_old(j)= v_new(j)
52 ENDDO
53 !____. compute the error
54 CALL exact_sol(Nx,Ng,x,time ,nu,va,vb,v_exact)
55
56 error =0.
57 DO j=1,Nx
58 error = error + abs(v_new(j)-v_exact(j))
59 ENDDO
60 error_max=max(error_max ,error/float(Nx))
61 !____. write on disk
62 if (mod(it ,itout).eq.0) CALL write_disk(x,v_new ,v_exact ,it,Nx,Ng)
63 ENDDO
64
65 WRITE (*,*) ’dt , error=’,delta_t ,error_max
66 WRITE (*,*) ’Simulation ended!’
67 STOP
68 END
69 !____.__________________________________________________________________
70 !____.__________________________________________________________________
71 !____.__________________________________________________________________
72 !____.__________________________________________________________________
73 FUNCTION rhs_cent(vjp ,vj,vjm ,a,Dx)
74 IMPLICIT NONE
75 REAL:: rhs_cent ,vjp ,vj,vjm ,a,Dx
76
77 rhs_cent=a*(vjp -2.*vj+vjm)/Dx**2
31
78
79 RETURN
80 END
81 !____.__________________________________________________________________
82 !____.__________________________________________________________________
83 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,nu ,va ,vb ,v_exact)
84
85 IMPLICIT NONE
86 INTEGER :: Nx ,Ng ,j, MW , m
87 REAL ,DIMENSION (1-ng:nx+ng):: x,v_exact
88 REAL:: appo , va, vb, vs, appo2 ,time ,nu
89 REAL ,PARAMETER :: pi=acos (-1.)
90
91 MW = 1000
92
93 DO j=1-ng,Nx+ng
94 vs = va + (vb - va) * x(j)
95 appo2 = 0.
96 DO m = 1, MW
97 appo = (2 * m -1) * pi
98 appo2 = appo2 + 400./ appo * sin(appo*x(j)) &
99 * exp(-nu*appo **2* time)
100 ENDDO
101
102 v_exact(j) = - appo2 + vs
103 ENDDO
104
105 RETURN
106 END
107 !____.__________________________________________________________________
108 !____.__________________________________________________________________
109 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
110
111 IMPLICIT NONE
112 INTEGER :: it ,nx ,ng ,j
113 REAL ,DIMENSION (1-ng:nx+ng):: x,v,u
114 CHARACTER (32):: opfile
115
116 opfile=’acl12345678.dat’
117 WRITE(opfile (4:11) ,1000) it
118 opfile = ’./DATA/’// opfile
119 1000 FORMAT(i8.8)
120
121 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
122 DO j=1-Ng,Nx+ng
123 WRITE (13,*) x(j),v(j),u(j)
124 ENDDO
125 CLOSE (13)
126
127 RETURN
128 END
129 !____.__________________________________________________________________
130 !____.__________________________________________________________________
131 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
132 IMPLICIT NONE
133 INTEGER :: Nx ,Ng ,j
32
134 REAL ,DIMENSION (1-ng:nx+ng):: x
135 REAL:: L,delta_x
136 L=1.
137 delta_x=L/float(Nx+1)
138
139 DO j=1-ng,Nx+ng
140 x(j)=j*delta_x
141 ENDDO
142
143 RETURN
144 END
145 !____.__________________________________________________________________
146 !____.__________________________________________________________________
147 SUBROUTINE input(itmin ,itmax ,itout ,nu ,delta_t ,va ,vb)
148 IMPLICIT NONE
149 INTEGER :: itmin ,itmax ,itout
150 REAL :: nu,delta_t ,va,vb
151 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
152 READ (1,*) itmin
153 READ (1,*) itmax
154 READ (1,*) itout
155 READ (1,*)nu
156 READ (1,*) delta_t
157 READ (1,*)va
158 READ (1,*)vb
159 CLOSE (1)
160 RETURN
161 END SUBROUTINE input
4.4
Scrivere un programma per il calcolo della soluzione dell’equazione del calore, sistema differenziale
presentato nella $ 4.3, con un metodo implicito applicando il metodo di Jacobi per la soluzione del
sistema lineare,
1 !____. file: jacobi_BTCS.f90
2 !____. This program uses the Jacobi method to integrate
3 !____. the heat equation with an implicit method.
4 PROGRAM main
5 !____.__________________________________________________________________
6 IMPLICIT NONE
7 INTEGER :: it , j
8 INTEGER :: itmin ,itmax ,itout
9 INTEGER ,PARAMETER :: Nx=20,Ng=1
10 !____.__________________________________________________________________
11 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old ,v_exact ,x
12 REAL:: delta_t ,delta_x ,nu,rhs_cent ,rhs ,va,vb
13 REAL:: error ,error_max ,time ,v0
14 !____.__________________________________________________________________
15 ! metric definition
16 CALL griglia(Nx,Ng,x,delta_x)
17 !____.__________________________________________________________________
18 ! read on file eulero_stokes.dat
19 CALL input(itmin ,itmax ,itout ,nu,delta_t ,va,vb)
20 !____.__________________________________________________________________
33
21 ! some initializations
22 call exact_sol(Nx,Ng,x,0.,nu,va,vb,v_exact)
23 DO j=1-ng,nx+ng
24 v_old(j)=v_exact(j)
25 ENDDO
26 error =0.
27 error_max = -999999.9
28 !____.__________________________________________________________________
29 write (*,*) ’cfl’,delta_t*nu/delta_x **2
30 CALL write_disk(x,v_old ,v_old ,0,Nx,Ng)
31 DO it= itmin , itmax
32
33 call jacobi(Nx,Ng,v_old ,v_new ,delta_x ,delta_t ,nu)
34 time = it*delta_t
35
36 !____. boundary conditions
37 DO j=1-ng ,0
38 v_new( j) = va
39 v_new(Nx+1+j) = vb
40 ENDDO
41
42 !____. update the solution
43 DO j=1-ng ,Nx+ng
44 v_old(j)= v_new(j)
45 ENDDO
46
47 !____. compute the error
48 CALL exact_sol(Nx,Ng,x,time ,nu,va,vb,v_exact)
49
50 error =0.
51 DO j=1,Nx
52 error = error + abs(v_new(j)-v_exact(j))
53 ENDDO
54 error_max=max(error_max ,error/float(Nx))
55 !____. write on disk
56 if (mod(it ,itout).eq.0) CALL write_disk(x,v_new ,v_exact ,it,Nx,Ng)
57 ENDDO
58
59 WRITE (*,*) ’dt , error=’,delta_t ,error_max
60 WRITE (*,*) ’Simulation ended!’
61 STOP
62 END
63 !____.__________________________________________________________________
64 !____.__________________________________________________________________
65 !____.__________________________________________________________________
66 !____.__________________________________________________________________
67 SUBROUTINE jacobi(nx ,ng ,v_old ,v_new ,delta_x ,Dt ,nu)
68 !____.
69 IMPLICIT NONE
70 INTEGER :: nx ,ng ,i,j,m,itmax
71 REAL ,DIMENSION (1-ng:nx+ng),INTENT(OUT):: v_new
72 REAL ,DIMENSION (1-ng:nx+ng),INTENT(IN):: v_old
73 REAL ,DIMENSION (1-ng:nx+ng):: v_it
74 REAL:: tol ,error ,appo ,delta_x , cfl , dt, nu
75
76 tol =1.E-10
34
77 m=0
78 cfl = Dt*nu/delta_x **2
79
80
81 DO j=1-ng,nx+ng
82 v_it(j)=v_old(j)
83 ENDDO
84
85 DO
86 error =0.
87 DO j=1,nx
88 v_new(j)=(v_old(j)+cfl*(v_it(j-1)+v_it(j+1)))/(1.+ cfl *2.)
89 error = error + abs(v_new(j)-v_it(j))
90 ENDDO
91 if (error/float(Nx).lt.tol) then
92 exit
93 else
94 DO j=1,nx
95 v_it(j)=v_new(j)
96 ENDDO
97 endif
98 m = m + 1
99 ENDDO
100 write (*,*) ’it=’,m,’error=’,error
101 !____.
102 RETURN
103 END SUBROUTINE jacobi
104 !____.__________________________________________________________________
105 !____.__________________________________________________________________
106 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,nu ,va ,vb ,v_exact)
107
108 IMPLICIT NONE
109 INTEGER :: Nx ,Ng ,j, MW , m
110 REAL ,DIMENSION (1-ng:nx+ng):: x,v_exact
111 REAL:: appo , va, vb, vs, appo2 ,time ,nu
112 REAL ,PARAMETER :: pi=acos (-1.)
113
114 MW = 10
115
116 DO j=1-ng,Nx+ng
117 vs = va + (vb - va) * x(j)
118 appo2 = 0.
119 DO m = 1, MW
120 appo = (2 * m -1) * pi
121 appo2 = appo2 + 4./ appo * sin(appo*x(j)) &
122 * exp(-nu*appo **2* time)
123 ENDDO
124
125 v_exact(j) = - appo2 + vs
126 ENDDO
127
128 RETURN
129 END SUBROUTINE exact_sol
130 !____.__________________________________________________________________
131 !____.__________________________________________________________________
132 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
35
133
134 IMPLICIT NONE
135 INTEGER :: it ,nx ,ng ,j
136 REAL ,DIMENSION (1-ng:nx+ng):: x,v,u
137 CHARACTER (32):: opfile
138
139 opfile=’acl12345678.dat’
140 WRITE(opfile (4:11) ,1000) it
141 opfile = ’./DATA/’// opfile
142 1000 FORMAT(i8.8)
143
144 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
145 DO j=1-Ng,Nx+ng
146 WRITE (13,*) x(j),v(j),u(j)
147 ENDDO
148 CLOSE (13)
149
150 RETURN
151 END SUBROUTINE write_disk
152 !____.__________________________________________________________________
153 !____.__________________________________________________________________
154 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
155 IMPLICIT NONE
156 INTEGER :: Nx ,Ng ,j
157 REAL ,DIMENSION (1-ng:nx+ng):: x
158 REAL:: L, delta_x
159 L=1.
160 delta_x=L/float(Nx+1)
161
162 DO j=1-ng,Nx+ng
163 x(j)=j*delta_x
164 ENDDO
165
166 RETURN
167 END SUBROUTINE griglia
168 !____.__________________________________________________________________
169 !____.__________________________________________________________________
170 SUBROUTINE input(itmin ,itmax ,itout ,nu ,delta_t ,va ,vb)
171 IMPLICIT NONE
172 INTEGER :: itmin ,itmax ,itout
173 REAL :: nu,delta_t ,va,vb
174 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
175 READ (1,*) itmin
176 READ (1,*) itmax
177 READ (1,*) itout
178 READ (1,*)nu
179 READ (1,*) delta_t
180 READ (1,*)va
181 READ (1,*)vb
182 CLOSE (1)
183 RETURN
184 END SUBROUTINE input
36
4.5
Scrivere un programma per il calcolo della soluzione dell’equazione del calore, sistema differenziale
presentato nella $ 4.3, con un metodo implicito applicando il metodo di Gauss-Seidel per la soluzione
del sistema lineare,
1 !____. file: gauss_seidel_BTCS.f90
2 !____. This program uses the Gauss -Seidel method to integrate
3 !____. the heat equation with an implicit method.
4 PROGRAM main
5 !____.__________________________________________________________________
6 IMPLICIT NONE
7 INTEGER :: it , j
8 INTEGER :: itmin ,itmax ,itout
9 INTEGER ,PARAMETER :: Nx=20,Ng=1
10 !____.__________________________________________________________________
11 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old ,v_exact ,x
12 REAL:: delta_t ,delta_x ,nu,rhs_cent ,rhs ,va,vb
13 REAL:: error ,error_max ,time ,v0
14 !____.__________________________________________________________________
15 ! metric definition
16 CALL griglia(Nx,Ng,x,delta_x)
17 !____.__________________________________________________________________
18 ! read on file eulero_stokes.dat
19 CALL input(itmin ,itmax ,itout ,nu,delta_t ,va,vb)
20 !____.__________________________________________________________________
21 ! some initializations
22 call exact_sol(Nx,Ng,x,0.,nu,va,vb,v_exact)
23 DO j=1-ng,nx+ng
24 v_old(j)=v_exact(j)
25 ENDDO
26 error =0.
27 error_max = -999999.9
28 !____.__________________________________________________________________
29 write (*,*) ’cfl’,delta_t*nu/delta_x **2
30 CALL write_disk(x,v_old ,v_old ,0,Nx,Ng)
31 DO it= itmin , itmax
32
33 call gauss_seidel(Nx,Ng,v_old ,v_new ,delta_x ,delta_t ,nu)
34 time = it*delta_t
35
36 !____. boundary conditions
37 DO j=1-ng ,0
38 v_new( j) = va
39 v_new(Nx+1+j) = vb
40 ENDDO
41
42 !____. update the solution
43 DO j=1-ng ,Nx+ng
44 v_old(j)= v_new(j)
45 ENDDO
46
47 !____. compute the error
48 CALL exact_sol(Nx,Ng,x,time ,nu,va,vb,v_exact)
49
50 error =0.
51 DO j=1,Nx
37
52 error = error + abs(v_new(j)-v_exact(j))
53 ENDDO
54 error_max=max(error_max ,error/float(Nx))
55 !____. write on disk
56 if (mod(it ,itout).eq.0) CALL write_disk(x,v_new ,v_exact ,it,Nx,Ng)
57 ENDDO
58
59 WRITE (*,*) ’dt , error=’,delta_t ,error_max
60 WRITE (*,*) ’Simulation ended!’
61 STOP
62 END
63 !____.__________________________________________________________________
64 !____.__________________________________________________________________
65 !____.__________________________________________________________________
66 !____.__________________________________________________________________
67 SUBROUTINE gauss_seidel(nx ,ng ,v_old ,v_new ,delta_x ,Dt ,nu)
68 !____.
69 IMPLICIT NONE
70 INTEGER nx ,ng ,i,j,m,itmax
71 REAL ,DIMENSION (1-ng:nx+ng),INTENT(OUT):: v_new
72 REAL ,DIMENSION (1-ng:nx+ng),INTENT(IN):: v_old
73 REAL ,DIMENSION (1-ng:nx+ng):: v_it
74 REAL tol ,error ,appo ,delta_x , cfl , dt, nu
75
76 tol =1.E-10
77 m=0
78 cfl = Dt*nu/delta_x **2
79
80
81 DO j=1-ng,nx+ng
82 v_it(j)=v_old(j)
83 ENDDO
84
85 DO
86 error =0.
87 DO j=1,nx
88 v_new(j)=(v_old(j)+cfl*( v_new(j-1)+v_it(j+1)))/(1.+ cfl *2.)
89 error = error + abs(v_new(j)-v_it(j))
90 ENDDO
91 if (error/float(Nx).lt.tol) then
92 exit
93 else
94 DO j=1,nx
95 v_it(j)=v_new(j)
96 ENDDO
97 endif
98 m = m + 1
99 ENDDO
100 write (*,*) ’it=’,m,’error=’,error
101 !____.
102 RETURN
103 END SUBROUTINE gauss_seidel
104 !____.__________________________________________________________________
105 !____.__________________________________________________________________
106 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,nu ,va ,vb ,v_exact)
107
38
108 IMPLICIT NONE
109 INTEGER :: Nx ,Ng ,j, MW , m
110 REAL ,DIMENSION (1-ng:nx+ng):: x,v_exact
111 REAL:: appo , va, vb, vs, appo2 , time , nu
112 REAL ,PARAMETER :: pi=acos (-1.)
113
114 MW = 10
115
116 DO j=1-ng,Nx+ng
117 vs = va + (vb - va) * x(j)
118 appo2 = 0.
119 DO m = 1, MW
120 appo = (2 * m -1) * pi
121 appo2 = appo2 + 4./ appo * sin(appo*x(j)) &
122 * exp(-nu*appo **2* time)
123 ENDDO
124
125 v_exact(j) = - appo2 + vs
126 ENDDO
127
128 RETURN
129 END SUBROUTINE exact_sol
130 !____.__________________________________________________________________
131 !____.__________________________________________________________________
132 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
133
134 IMPLICIT NONE
135 INTEGER :: it ,nx ,ng ,j
136 REAL ,DIMENSION (1-ng:nx+ng):: x,v,u
137 CHARACTER (32) opfile
138
139 opfile=’acl12345678.dat’
140 WRITE(opfile (4:11) ,1000) it
141 opfile = ’./DATA/’// opfile
142 1000 FORMAT(i8.8)
143
144 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
145 DO j=1-Ng,Nx+ng
146 WRITE (13,*) x(j),v(j),u(j)
147 ENDDO
148 CLOSE (13)
149
150 RETURN
151 END SUBROUTINE write_disk
152 !____.__________________________________________________________________
153 !____.__________________________________________________________________
154 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
155 IMPLICIT NONE
156 INTEGER :: Nx ,Ng ,j
157 REAL ,DIMENSION (1-ng:nx+ng):: x
158 REAL:: L, delta_x
159 L=1.
160 delta_x=L/float(Nx+1)
161
162 DO j=1-ng,Nx+ng
163 x(j)=j*delta_x
39
164 ENDDO
165
166 RETURN
167 END SUBROUTINE griglia
168 !____.__________________________________________________________________
169 !____.__________________________________________________________________
170 SUBROUTINE input(itmin ,itmax ,itout ,nu ,delta_t ,va ,vb)
171 IMPLICIT NONE
172 INTEGER :: itmin ,itmax ,itout
173 REAL :: nu,delta_t ,va,vb
174 OPEN(UNIT=1,FILE=’input.dat’,STATUS=’old’,ACTION=’read’)
175 READ (1,*) itmin
176 READ (1,*) itmax
177 READ (1,*) itout
178 READ (1,*)nu
179 READ (1,*) delta_t
180 READ (1,*)va
181 READ (1,*)vb
182 CLOSE (1)
183 RETURN
184 END SUBROUTINE input
4.6
Risolvere il seguente problema di Poisson attraverso il metodo di Jacobi/Gauss-Seidel:
d2u
dx2= −1 x ∈ [−1.1]u(−1, t) = u(1, t) = 0 (4.8)
La soluzione esatta :
u(x, t) =1
2(1− x2)
Si considerino i metodi di risoluzione del sistema algebrico lineare convergenti quando due iterate
successive presentano una differenza pari al pi a 10−10. Scrivere il numero di iterate necessarie ad
ogni metodo per la convergenza.
1 !____. file: poisson.f90
2 !____. This program applies the Jacobi/Gauss -Seidel
3 !____. iterative method to integrate
4 !____. an ordinary differential equation of second order
5 PROGRAM poisson
6 !____.__________________________________________________________________
7 IMPLICIT NONE
8 INTEGER :: i,j
9 INTEGER :: ntime ,nout
10 INTEGER ,PARAMETER :: Nx=100,Ng=1
11 !____.__________________________________________________________________
12 REAL ,DIMENSION (1-ng:Nx+ng):: v_new , v_old , v_exact , x
13 REAL:: delta_t ,delta_x ,rhs ,rhs_cent
14 REAL:: time ,cfl
15 REAL:: error ,error_max
16 !____.__________________________________________________________________
17 ! metric definition
18 call griglia(Nx,Ng,x,delta_x)
19 !____.__________________________________________________________________
40
20 ! some initializations
21 call exact_sol(Nx,Ng,x,0., v_exact)
22
23 DO i=1-Ng,Nx+Ng
24 v_old(i)=0.
25 ENDDO
26 error =0.
27 error_max = -999999.9
28 !____.__________________________________________________________________
29
30 !call jacobi(nx ,ng ,v_old ,v_new ,delta_x)
31 call gauss_seidel(nx,ng,v_old ,v_new ,delta_x)
32
33 error =0.
34 DO j=1,Nx
35 error = error + abs(v_new(j)-v_exact(j))
36 ENDDO
37 error_max=max(error_max ,error/float(Nx))
38
39 !____. write on disk
40 call write_disk(x,v_new ,v_exact ,0,Nx,Ng)
41
42 WRITE (*,*) ’dt , error=’,delta_t ,error_max
43 WRITE (*,*) ’solution stored in file velocity.dat’
44 WRITE (*,*) ’the end falks!’
45 STOP
46 !____.__________________________________________________________________
47 END
48 !____.__________________________________________________________________
49 !____.__________________________________________________________________
50 !____.__________________________________________________________________
51 !____.__________________________________________________________________
52 SUBROUTINE jacobi(nx ,ng ,v_old ,v_new ,delta_x)
53 !____.
54 IMPLICIT NONE
55 INTEGER :: nx ,ng ,i,j,m,itmax
56 REAL ,DIMENSION (1-Ng:Nx+Ng):: v_new ,v_old
57 REAL:: tol ,error ,appo ,delta_x
58
59 tol =1.E-10
60 m=0
61
62 DO
63 error =0.
64 DO j=1,nx
65 v_new(j)=(-delta_x **2+ v_old(j-1)+v_old(j+1))/2.
66 error = error + abs(v_new(j)-v_old(j))
67 ENDDO
68 do j=1-ng ,0
69 v_new(j)=0.
70 v_new(-j+nx+1)=0.
71 enddo
72 if (error/float(Nx).lt.tol) then
73 exit
74 else
75 DO j=1,nx
41
76 v_old(j)=v_new(j)
77 ENDDO
78 endif
79 m = m + 1
80 ENDDO
81 write (*,*) ’it=’,m,’error=’,error
82 !____.
83 RETURN
84 END SUBROUTINE jacobi
85 !____.__________________________________________________________________
86 !____.__________________________________________________________________
87 SUBROUTINE gauss_seidel(nx ,ng ,v_old ,v_new ,delta_x)
88 !____.
89 IMPLICIT NONE
90 INTEGER :: nx ,ng ,i,j,m,itmax
91 REAL ,DIMENSION (1-ng:nx+ng):: v_new , v_old
92 REAL:: tol ,error ,appo ,delta_x
93
94 tol =1.E-10
95 itmax =40
96 m=0
97
98
99 write (*,*) ’sono qui’
100 DO
101 error =0.
102 DO j=1,nx
103 v_new(j)=(-delta_x **2+ v_new(j-1)+v_old(j+1))/2.
104 error = error + abs(v_new(j)-v_old(j))
105 ENDDO
106 if (error.lt.tol) then
107 exit
108 else
109 DO j=1,nx
110 v_old(j)=v_new(j)
111 ENDDO
112 endif
113 m = m + 1
114 ENDDO
115 write (*,*) ’it=’,m,’error=’,error
116 !____.
117 RETURN
118 END SUBROUTINE gauss_seidel
119 !____.__________________________________________________________________
120 !____.__________________________________________________________________
121 SUBROUTINE exact_sol(Nx ,Ng ,x,time ,v_exact)
122
123 IMPLICIT NONE
124 INTEGER :: Nx ,Ny ,Ng ,i,j,WN ,m,n
125 REAL ,DIMENSION (1-ng:nx+ng)::x,v_exact
126 REAL:: appo , appo1 , appo2 , pi, time
127
128 WN=30.
129 pi = acos (-1.)
130
131 DO i=1-Ng,Nx+Ng
42
132 v_exact(i)= 0.5*(x(i)**2 -1.)
133 ENDDO
134
135 RETURN
136 ENDSUBROUTINE exact_sol
137 !____.__________________________________________________________________
138 !____.__________________________________________________________________
139 SUBROUTINE write_disk(x,v,u,it ,nx ,ng)
140
141 IMPLICIT NONE
142 INTEGER :: it ,nx ,ny ,ng ,i,j
143 REAL ,DIMENSION (1-ng:Nx+ng):: x,v,u
144 CHARACTER (32):: opfile
145
146 opfile=’acl12345678.dat’
147 WRITE(opfile (4:11) ,1000) it
148 1000 FORMAT(i8.8)
149
150 OPEN(UNIT=13,FILE=opfile ,STATUS=’unknown ’,ACTION=’write ’)
151
152 DO i=1-Ng,Nx+ng
153 WRITE (13,*) x(i),v(i),u(i)
154 ENDDO
155
156 CLOSE (13)
157
158 RETURN
159 END SUBROUTINE write_disk
160 !____.__________________________________________________________________
161 !____.__________________________________________________________________
162 SUBROUTINE griglia(Nx ,Ng ,x,delta_x)
163 IMPLICIT NONE
164 INTEGER :: Nx ,j,Ng
165 REAL ,DIMENSION (1-ng:nx+ng):: x
166 REAL:: L, delta_x
167 L=2.
168 delta_x=L/float(Nx+1)
169
170
171 DO j=1-ng,Nx+ng
172 x(j)=-1.+j*delta_x
173 ENDDO
174
175 RETURN
176 ENDSUBROUTINE griglia
43
