From https://en.wikipedia.org/wiki/Heat_equation
"In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics".
The idea is to see how Rebol or Red can solve this problem. Let’s imagine a coin heated at its center at T0.0 and calculate the temperature distribution toward the edges of the coin. We get a nice Gaussian curve.
Now, let’s look at how the phenomenon changes over time (from 0.0 to 0.10). We still see nice Gaussian curves whose mean decreases over time. We can also model the cooling of the metal part.
Due to vector datatype (https://github.com/Oldes/Rebol3) these equations are very easy to solve.
#!/usr/local/bin/r3
REBOL [
]
b2d: import 'blend2d ;--use blend2d (draw module)
cv2: import 'opencv ;--use OpenCV for visualisation
;--Global Physical Parameters
k: 0.5 ;--Diffusion coefficient (Thermal diffusivity coefficient of material)
l: 1.0 ;--Domain size
itime: 0.1 ;--Integration time
;--Global Numerical Parameters
nx: 100 ;--Number of grid points
nt: 1000 ;--Number of time steps
dx: l / (nx - 1) ;--Grid step (Spatial step size: 0.0101010101010101)
dt: itime / nt ;--Grid step (Time step size :0.0001)
;--Based on Oldes's code
linSpace: func [
"Generates n linearly spaced numbers from a to b (inclusive)."
a [number!] "Start"
b [number!] "End"
n [integer!] "Number of samples"
/no-end "Don't include end value in the result"
/local vec div step i ;--local values
][
vec: make vector! [f64! :n]
div: either no-end [n][n - 1]
step: (b - a) / div
repeat i n [vec/:i: a + (step * (i - 1))]
vec
]
;--Initialisation (using global variables)
init: does [
x: linSpace 0.0 1.0 nx ;--linear spacing between 0.0 and 1.0
t: (x * 2.0 * pi) ;--t is a vector
repeat i nx [t/:i: sin t/:i] ;--in radians
;--use blend2d commands in plot block
plot: compose [
font %/System/Library/Fonts/Geneva.ttf
fill black
text 0x10 10 "+1.0"
text 0x204 10 "0"
text 0x390 10 "-1.0"
pen black
line 5x200 410x200
]
]
;--Heat diffusion equation (using global variables)
diffusion: does [
vect: make vector! [f64! :nx]
posy: 0
n: 0
while [n <= nt] [
j: 2
while [j < nx] [
vect/:j: dt * k * (t/(j - 1) - (2.0 * t/:j) + t/(j + 1)) / (dx ** 2)
++ j
]
repeat j nx [t/:j: t/:j + vect/:j]
;--Plot every 100 time steps
if zero? n % 100 [
acolor: random white
posy: posy + 15
either n = 0 [tt: 0.0] [tt: round/to n % (n * dt) 0.001]
tt: ajoin ["time: " tt]
pos: as-pair 340 posy
append plot reduce ['fill (acolor) 'text (pos) 12 (tt) 'pen (acolor) 'line]
xx: 2
foreach val t [
append plot as-pair xx * 4 200 - (val * 195)
++ xx
]
]
++ n
]
]
;************************ Main Program ************************
init
diffusion
img: b2d/draw 410x400 :plot
img: cv2/resize img 150%
cv2/imshow/name :img "Heat Diffusion"
print "Any key to close"
cv2/waitKey 0
