Heat diffusion equation

Schrick-Noah_Homework-5.jl

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+using Pkg
+Pkg.add("BenchmarkTools")
+Pkg.add("LinearAlgebra")
+Pkg.add("Plots")
+
+using LinearAlgebra
+using BenchmarkTools
+# R has a tictoc library or system time.
+
+function solve_QHO(n, xL, xR)
+    h = (xR-xL)/(n-2);  # step size
+    x = range(xL, xR, step=h); # grid of x values
+
+    # with for loops
+    H = zeros(Float64, n-1, n-1); # intialize
+    for i in range(1,n-1, step=1);
+      @inbounds H[i,i] = 1/h^2 + .5*x[i]^2;  # diagonal of H
+    end
+
+    for i in range(1,n-2)
+      @inbounds H[i+1,i] = -1/(2*h^2); # upper diag, one row above
+      @inbounds H[i,i+1] = -1/(2*h^2); # lower diag
+    end
+
+    vals, vecs = eigen(H);
+    
+    return x, vals, vecs
+end
+
+x, vals, vecs = solve_QHO(1000, -5, 5);
+vals[1]
+
+# data for timing comparison
+ttable = m = Array{Float64}(undef, 70, 4)
+itr = 1;
+for n in range(500,3500, step=500);
+  for x in range(1,10);
+    r = @belapsed solve_QHO($n, -$x, $x)
+    ttable[itr,1] = -x
+    ttable[itr,2] = x
+    ttable[itr,3] = n
+    ttable[itr,4] = r
+    itr+=1
+  end
+end
+
+r = @belapsed solve_QHO(1000, -5,5)
+@btime solve_QHO(2000, -5,5)
+
+
+#Pkg.add("Plots")
+using Plots;
+plot(x, [vecs[:,1], vecs[:,2], vecs[:,3]],
+    title="wave functions",
+    label=["E0", "E1", "E2"], linewidth=3, gridlinewidth=3)
+
+# some julia linear linear algebra
+A = [ [2 3 4]; [3 2 4]; [5 -3 8] ];
+bvec = [6 8 1]';
+y = A \ bvec;  # use this in your function to solve the BVP
+A * y # should match b = [6 8 1]
+# * is matrix multiplication
+y.*bvec # makes operator element wise
+
+Ainv = A^(-1)
+Ainv = inv(A)
+Ainv * bvec
+
+function Laplace_time(;alpha, L, T, h, u_tblr)
+  # ; lets you ref arguments by name
+  # Laplace_time(alpha=20, L=25, T=)
+  # u_tblr = [100, 0, 0, 0] # boundary conditions
+  dt = h^2/(4*alpha+1);  # ensures numerical stability
+  n = floor(Int64, L/h); # coerce to integer, spatial loops
+  m = floor(Int64, T/dt); # number of time steps
+  u = zeros(n,n,m) # intialize solution 3d array
+
+  function impose_bc(u_tblr, u, n, k)
+      u_top    = u_tblr[1];   u[1,:,k] .= u_top; # top bc at time k
+      u_bottom = u_tblr[2];   u[n,:,k] .= u_bottom; # top bc at time k
+      u_left   = u_tblr[3];   u[:,1,k] .= u_left; # top bc at time k
+      u_right  = u_tblr[4];   u[:,n,k] .= u_right; # top bc at time k
+      return u
+  end
+  u = impose_bc(u_tblr, u, n, 1)  # intial u at time k=1
+  for k in range(2,m-1)         # time
+    for i in range(2,n-1)     # x 
+        for j in range(2,n-1) # y
+          u[i,j,k+1] = (1-(4*alpha*dt)/(h^2))*u[i,j,k] + ((alpha*dt)/(h^2))*(u[i+1,j,k]+u[i-1,j,k]+u[i,j+1,k]+u[i,j-1,k]) # method of lines
+        end
+    end
+    u = impose_bc(u_tblr, u, n, k) # re-impose the bc
+  end
+  return u
+end
+
+my_u = Laplace_time(alpha=20, L=25, T=20, h=1, u_tblr=[100, 0, 100, 0]);
+my_u_b = Laplace_time(alpha=20, L=50, T=40, h=2, u_tblr=[100, 0, 100, 0]);
+my_u_c = Laplace_time(alpha=20, L=100, T=60, h=5, u_tblr=[100, 0, 100, 0]);
+
+
+my_dims = size(my_u)
+using Plots;
+heatmap(1:my_dims[1], 1:my_dims[2], my_u[:,:,1])
+
+@gif for k = 1:100
+  heatmap(1:my_dims[1], 1:my_dims[2], my_u[:,:,k],
+      xlabel = join(["time idx = ", k])
+  )
+end