This program solves a semi-discretized form of the following system of three PDEs (each similar to a Burgers equation):
u(i) = -(u(1)+u(2)+u(3)) u(i) + eta(i) u(i) (i=1,2,3),
t x xx
on the interval -1 .le. x .le. 1, and with time t .ge. 0.
The diffusion coefficients are eta(*) = .1, .02, .01.
The boundary conditions are u(i) = 0 at x = -1 and x = 1 for all i.
The initial profile for each u(i) is a square wave:
u(i) = 0 on 1/2 .lt. abs(x) .le. 1
u(i) = amp(i)/2 on abs(x) = 1/2
u(i) = amp(i) on 0 .le. abs(x) .lt. 1/2
where the amplitudes are amp(*) = .2, .3, .5.
A simplified Galerkin treatment of the spatial variable x is used, with piecewise linear basis functions on a uniform mesh of 100 intervals. The result is a system of ODEs in the discrete values u(i,k) approximating u(i) (i=1,2,3) at the interior points (k = 1,...,99).
The ODEs are:
(u'(i,k-1) + 4 u'(i,k) + u'(i,k+1))/6 =
-(1/6dx) (c(k-1)dul(i) + 2c(k)(dul(i)+dur(i)) + c(k+1)dur(i))
+ (eta(i)/dx**2) (dur(i) - dul(i)) (i=1,2,3, k=1,...,99),
c(j) = u(1,j)+u(2,j)+u(3,j), dx = .02 = the interval size,
dul(i) = u(i,k) - u(i,k-1), dur(i) = u(i,k+1) - u(i,k).
Terms involving boundary values (subscripts 0 or 100) are dropped from the equations for k = 1 and k = 99 above.