As part of my senior thesis work, I’ve been quickly getting up to speed with both programming in Fortran and various numerical methods for solving different types of equations. This, of course, will all come before I learn the relevant topics in my numerical analysis (MTH 451) class, but I don’t mind too much. This thesis is more important than that math course anyway, plus learning it beforehand means I can focus on my other courses at the end of the semester without stressing too much. Sounds good to me!

Anyway, I wrote three programs over the last few hours to solve the ODE u’ = u [ exact solution: u = exp(t) ] just to make sure I was programming the methods right. After all, I may need to use those methods in my final project, so getting them right now is a big step. In all three programs (that differ only in the “solve” function), the step size needs to be changed in two separate places and the function would need to be changed for different problems, both of which would require a re-compile, but for these programs that isn’t too bad. You’d need to do the same for a change to some sort of input file anyway, and I didn’t want to get around that restriction by having user-inputted values. Check them out!

PROGRAM Forward_Euler IMPLICIT None REAL, PARAMETER :: tstep = 0.2 REAL, PARAMETER :: tend = 3.0 PRINT '(A)', "Forward Euler Method" CALL solve(tend) CONTAINS FUNCTION f(u, t) REAL :: u, t, f f = u END FUNCTION f FUNCTION advance(u, t) RESULT(unew) REAL :: unew, u, t unew = u + tstep * f(u, t) END FUNCTION advance SUBROUTINE solve(tend) REAL :: tend, t = 0.0, u = 1.0 INTEGER :: k = 0 DO IF ( t > tend ) EXIT PRINT '(i2, f5.1, f7.3, f7.3, f7.3)', k, t, u, EXP(t), ABS(EXP(t)-u) u = advance(u, t) t = t + tstep k = k + 1 END DO END SUBROUTINE solve END PROGRAM Forward_Euler

PROGRAM RungeKutta_2 IMPLICIT None REAL, PARAMETER :: tstep = 0.2 REAL, PARAMETER :: tend = 3.0 PRINT '(A)', "2nd Order Runge-Kutta Method" CALL solve(tend) CONTAINS FUNCTION f(u, t) REAL :: u, t, f f = u END FUNCTION FUNCTION rungekutta2(u, t) RESULT(unew) REAL :: u, t, unew, ustep, tnew ustep = u + (tstep / 2.0) * f(u, t) tnew = t + tstep / 2.0 unew = u + tstep * f(ustep, tnew) END FUNCTION rungekutta2 SUBROUTINE solve(tend) REAL :: tend, t = 0.0, u = 1.0 REAL, PARAMETER :: tstep = 0.2 INTEGER :: k = 0.0 DO IF ( t > tend ) EXIT PRINT '(i2, f5.1, f7.3, f7.3, f7.3)', k, t, u, EXP(t), ABS(EXP(t)-u) u = rungekutta2(u, t) t = t + tstep k = k + 1 END DO END SUBROUTINE solve END PROGRAM RungeKutta_2

PROGRAM RungeKutta_4 IMPLICIT None REAL, PARAMETER :: tstep = 0.2 REAL, PARAMETER :: tend = 3.0 PRINT '(A)', "4th Order Runge-Kutta Method" CALL solve(tend) CONTAINS FUNCTION f(u, t) REAL :: u, t, f f = u END FUNCTION FUNCTION rungekutta4(u, t) RESULT(unew) REAL :: u, t, unew, k1, k2, k3, k4, tnew k1 = tstep * f(u, t) tnew = t + tstep / 2.0 k2 = tstep * f(u+0.5*k1, tnew) k3 = tstep * f(u+0.5*k2, tnew) tnew = t + tstep k4 = tstep * f(u+k3, t+tstep) unew = u + (1.0/6.0) * (k1 + 2*k2 + 2*k3 + k4) END FUNCTION SUBROUTINE solve(tend) REAL :: tend, t = 0.0, u = 1.0 REAL, PARAMETER :: tstep = 0.2 INTEGER :: k = 0 DO IF ( t > tend ) EXIT PRINT '(i2, f5.1, f7.3, f7.3, f7.3)', k, t, u, EXP(t), ABS(EXP(t)-u) u = rungekutta4(u, t) t = t + tstep k = k + 1 END DO END SUBROUTINE solve END PROGRAM RungeKutta_4

Again, the only difference is in the “solve” method. Eventually these will all be wrapped up in a module, along with the other algorithms I’ll need for the final project, but for now three separate files is nice. I could put them in a single file and have the program call each one in turn, or only run a user-selected method, but that’s relatively simple compared to actually getting the coding down and to work.

I am very glad that it works.