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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | # Category: Differential Equations
# Name: Characteristics and shocks for the traffic flow equation
#
# We are looking at characteristics (looking for shocks)
# for the equation u_t + g(u) u_x = 0, with initial condition
# u(x,0) = phi(x)
# this is for flux u*(2-u)
#function g(u) = 2-2*u;
# Quadratic flow for flux u^2 / 2
function g(u) = u;
# ramp down
function phi(x) = (
if x < 0 then 1
else if x < 1 then 1-x
else 0
);
# rampup
#function phi(x) = (
# if x < 0 then 0
# else if x < 1 then x
# else 1
#);
# we'll have x going from
xstart = -3;
xend = 3;
xstep = 0.1;
# end at this t, start at 0
tend = 1.5;
LinePlotWindow = [xstart,xend,-0.1,tend+0.1];
LinePlotDrawLegends = false;
LinePlotClear();
PlotWindowPresent(); # Make sure the window is raised
# color in [red,green,blue]
# Color according to phi
function the_color(x0) = [phi(x0), #red
0.2, # green
1.0-phi(x0)]; #blue
# Color according to x0
#function the_color(x0) = [(x0-xstart)/(xend-xstart), #red
# 0.2, # green
# 1.0-(x0-xstart)/(xend-xstart)]; #blue
# Draw characteristics
for x0=xstart to xend by xstep do (
LinePlotDrawLine([x0,0,x0+g(phi(x0))*tend,tend], "color", the_color (x0))
)
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