### A sample of toy FDM code

Solving equation

Time integration is full implicit, and during each time step Newton’s method is employed.

PROGRAM MAIN !-------------------------------------------------- !Implicit time evolution of equation !u_t-u''=f(x,u) !f(x,u)=-2+r*(sin(u)-sin(x^2)),r=2 !-------------------------------------------------- ! t=time points ! dt=time step ! h=neighboring nodes distance ! u=unknown ! x=node location ! n=No. of unknowns !USE SUNPERF IMPLICIT NONE INTEGER(4) :: n REAL(4) :: elapse,time(2),etime REAL(8) :: t=0,dt=0.1 REAL(8) :: h REAL(8),ALLOCATABLE :: u(:),x(:) INTEGER(4) :: count,i n=50 h=1/(DBLE(n)+1.) ALLOCATE (x(n),u(n)) x=(/(i*h,i=1,n)/) u=(/((i*h)**0.5,i=1,n)/) OPEN(UNIT=100,FILE='data',STATUS='UNKNOWN',ACTION='& &WRITE') DO i=1,n WRITE(100,*) t,x(i),u(i) END DO DO WHILE (t<1.0) CALL newton(n,h,dt,u,x) t=t+dt DO i=1,n WRITE(100,*) t,x(i),u(i) END DO END DO CLOSE(UNIT=100) elapse=etime(time) WRITE(*,*) 'elapsed:',elapse END PROGRAM MAIN !-------------------------------------------------- SUBROUTINE newton(n,h,dt,u,x) IMPLICIT NONE INTEGER(4),INTENT(IN) :: n REAL(8),INTENT(IN) :: x(n),h,dt REAL(8),INTENT(INOUT) :: u(n) REAL(8) :: r=2,c(n),error,b(n,1),b0(n,1),a(2,n) REAL(8) :: alpha=1.0,beta=-1.0 INTEGER(4) :: info,i,iter c=u*h*h/dt iter=0 error=1 DO WHILE (error>1.e-13) iter=iter+1 a(1,:)=-1 a(2,:)=2+h*h/dt b(:,1)=(-2+r*(SIN(u)-SIN(x*x)))*h*h+c b(n,1)=b(n,1)+1 CALL DSBMV('u',n,1,alpha,a,2,u,1,beta,b,1) b=-b a(2,:)=a(2,:)-r*COS(u)*h*h CALL DGTSV(n,1,a(1,2:),a(2,:),a(1,2:),b,n,info) error=MAXVAL(ABS(b)) u=u+b(:,1) END DO WRITE(*,*) 'INFO',info,'Number of iterations:',iter END SUBROUTINE newton !--------------------------------------------------

Output would be like:

% f95 -dalign implicit_newton.f95 -xlic_lib=sunperf % a.out INFO 0 Number of iterations: 4 INFO 0 Number of iterations: 4 INFO 0 Number of iterations: 4 INFO 0 Number of iterations: 4 INFO 0 Number of iterations: 4 INFO 0 Number of iterations: 3 INFO 0 Number of iterations: 3 INFO 0 Number of iterations: 3 INFO 0 Number of iterations: 3 INFO 0 Number of iterations: 3 elapsed: 0.061407

Of course the exact solution is . Here is time evolution plot of the numerical solution.