function [w1, w2, w3] = vvheun3Dneumann(varargin)
% crankNicolson: uses Crank-Nicolson algorithm to approximate the solution
% to the parabolic PDE:
% u_{t}(x,t) - alpha^2 u_{xx}(x,t) = 0, 0<x<L, 0<t<T
% subject to the boundary conditions
% u(0,t) = u(L,t) = 0, 0<t<T
% and the initial conditions
% u(x,0) = f(x), 0<=x<=L
%
% arguments:
% L (vector 1by3) - upper bound of spatial (x) variable
% (Default L = [1,1,1])
% T (scalar) - upper bound of time (t) variable
% (Default T = 1)
% alpha (scalar) - square root of coefficient of u_{xx} term
% (Default alpha = 1)
% m (vector 1by3) - number of discrete spatial intervals
% (Default m = [100,100,100])
% n (scalar) - number of discrete time intervals
% (Default n = 100)
%
% w (m+1 x m+1 x m+1 x n) - approximation to u(x,t) at discrete space/time positions
%
% author: Troy J. Winkstern
% email: tjw8191@rit.edu
% date: 30 Jan 2011
% parse input arguments
[L,T,alpha,m,n] = parseInputs(varargin{:});
L_x = L(1);
L_y = L(2);
L_z = L(3);
m_x = m(1);
m_y = m(2);
m_z = m(3);
% initialize h, k, lambda, and w
h_x = L_x/m_x;
h_y = L_y/m_y;
h_z = L_z/m_z;
k = T./n;
w1 = zeros(m_y+1,m_x+1,m_z+1,n);
w2 = zeros(m_y+1,m_x+1,m_z+1,n);
w3 = zeros(m_y+1,m_x+1,m_z+1,n);
% initialize w1 (to sin(pix1/L1)*sin(pix2/L2)*sin(pix3/L3))
for i=1:m_x+1
for j=1:m_y+1
for k=1:m_z+1
w1(j,i,k,1) = cos(pi.*h_x.*(i-1)./L_x).*cos(pi.*h_y.*(j-1)./L_y)...
.*cos(pi.*h_z.*(k-1)./L_z);
end
end
end
% initialize w2 (to 4*sin(pix1/L1)*sin(pix2/L2)*sin(pix3/L3))
for i=1:m_x+1
for j=1:m_y+1
for k=1:m_z+1
w1(j,i,k,1) = cos(pi.*h_x.*(i-1)./L_x).*cos(pi.*h_y.*(j-1)./L_y)...
.*cos(pi.*h_z.*(k-1)./L_z);
end
end
end
% initialize w3 (to sin(pix1/L1)*sin(pix2/L2)*sin(pix3/L3))
for i=1:m_x+1
for j=1:m_y+1
for k=1:m_z+1
w1(j,i,k,1) = cos(pi.*h_x.*(i-1)./L_x).*cos(pi.*h_y.*(j-1)./L_y)...
.*cos(pi.*h_z.*(k-1)./L_z);
end
end
end
g1 = zeros(m_y+1,m_x+1,m_z+1);
p1 = zeros(m_y+1,m_x+1,m_z+1);
o1 = zeros(m_y+1,m_x+1,m_z+1);
g2 = zeros(m_y+1,m_x+1,m_z+1);
p2 = zeros(m_y+1,m_x+1,m_z+1);
o2 = zeros(m_y+1,m_x+1,m_z+1);
g3 = zeros(m_y+1,m_x+1,m_z+1);
p3 = zeros(m_y+1,m_x+1,m_z+1);
o3 = zeros(m_y+1,m_x+1,m_z+1);
for t=1:(n-1)
%Go back and check that these coefficients are correct.
f1 = (alpha^2)*laplacian(w1(:,:,:,t),h_y,h_x,h_z);
g1 = w1(:,:,:,t) + ((2*k)/3)*f1;
p1 = (alpha^2)*laplacian(g1,h_y,h_x,h_z);
o1 = p1 + ((2*k)/3)*(alpha^2)*laplacian(p1,h_y,h_x,h_z);
f2 = (alpha^2)*laplacian(w2(:,:,:,t),h_y,h_x,h_z);
g2 = w2(:,:,:,t) + ((2*k)/3)*f2;
p2 = (alpha^2)*laplacian(g2,h_y,h_x,h_z);
o2 = p2 + ((2*k)/3)*(alpha^2)*laplacian(p2,h_y,h_x,h_z);
f3 = (alpha^2)*laplacian(w3(:,:,:,t),h_y,h_x,h_z);
g3 = w3(:,:,:,t) + ((2*k)/3)*f3;
p3 = (alpha^2)*laplacian(g3,h_y,h_x,h_z);
o3 = p3 + ((2*k)/3)*(alpha^2)*laplacian(p3,h_y,h_x,h_z);
w1(:,:,:,t+1) = w1(:,:,:,t) + (k/4)*f1 + ((3*k)/4)*o1;
w2(:,:,:,t+1) = w2(:,:,:,t) + (k/4)*f2 + ((3*k)/4)*o2;
w3(:,:,:,t+1) = w3(:,:,:,t) + (k/4)*f3 + ((3*k)/4)*o3;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% subfunction parseInputs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [L,T,alpha,m,n] = parseInputs(varargin)
% check number of input arguments
nargs = length(varargin);
error(nargchk(0,5,nargs));
% get/set n
if nargs<5
n = [];
else
n = varargin{5};
end
if isempty(n)
n = 100;
end
% check n has valid value
if (numel(n)>1) || (n<1) || ~isequal(round(n),n)
error('n must be a positive integer.');
end
% get/set m
if nargs<4
m = [];
else
m = varargin{4};
end
if isempty(m)
m = [100,100,100];
end
% check m has valid value
if (numel(m)~=3) || any(m<1) || any(~isequal(round(m),m))
error('m must be a vector of positive integers.');
end
% get/set alpha
if nargs<3
alpha = [];
else
alpha = varargin{3};
end
if isempty(alpha)
alpha = 1;
end
% check alpha has valid value
if numel(alpha)>1
error('alpha must be a real scalar.');
end
% get/set T
if nargs<2
T = [];
else
T = varargin{2};
end
if isempty(T)
T = 1;
end
% check T has valid value
if (numel(T)>1) || (T<=0)
error('T must be a positive real scalar.');
end
% get/set L
if nargs<1
L = [];
else
L = varargin{1};
end
if isempty(L)
L = [1,1,1];
end
% check L has valid value
if (numel(L)~=3) || any(L<=0)
error('L must be a positive vector with 3 elements.');
end
% warning if L not integer
if any(~isequal(round(L),L))
warning('L should be an vector of integers so that the inital condition takes on the value of 0 for t = 0 and t = T.');
end
%another subfunction
function y = laplacian(x,h_y,h_x,h_z)
[m,n,p] = size(x);
y = zeros(m,n,p);
%compute the 8 corners
y(1,1,1) = (2*x(1,2,1)-2*x(1,1,1))/(h_x.^2)+(2*x(2,1,1)-2*x(1,1,1))/(h_y.^2)...
+(2*x(1,1,2)-2*x(1,1,1))/(h_z.^2);
y(m,1,1) = (2*x(m,2,1)-2*x(m,1,1))/(h_x.^2)+(2*x(m-1,1,1)-2*x(m,1,1))/(h_y.^2)...
+(2*x(m,1,2)-2*x(m,1,1))/(h_z.^2);
y(1,n,1) = (2*x(1,n-1,1)-2*x(1,n,1))/(h_x.^2)+(2*x(2,n,1)-2*x(1,n,1))/(h_y.^2)...
+(2*x(1,n,2)-2*x(1,n,1))/(h_z.^2);
y(1,1,p) = (2*x(1,2,p)-2*x(1,1,p))/(h_x.^2)+(2*x(2,1,p)-2*x(1,1,p))/(h_y.^2)...
+(2*x(1,1,p-1)-2*x(1,1,p))/(h_z.^2);
y(m,n,1) = (2*x(m,n-1,1)-2*x(m,n,1))/(h_x.^2)+(2*x(m-1,n,1)-2*x(m,n,1))/(h_y.^2)...
+(2*x(m,n,2)-2*x(m,n,1))/(h_z.^2);
y(1,n,p) = (2*x(1,n-1,p)-2*x(1,n,p))/(h_x.^2)+(2*x(2,n,p)-2*x(1,n,p))/(h_y.^2)...
+(2*x(1,n,p-1)-2*x(1,n,p))/(h_z.^2);
y(m,1,p) = (2*x(m,2,p)-2*x(m,1,p))/(h_x.^2)+(2*x(m-1,1,p)-2*x(m,1,p))/(h_y.^2)...
+(2*x(m,1,p-1)-2*x(m,1,p))/(h_z.^2);
y(m,n,p) = (2*x(m,n-1,p)-2*x(m,n,p))/(h_x.^2)+(2*x(m-1,n,p)-2*x(m,n,p))/(h_y.^2)...
+(2*x(m,n,p-1)-2*x(m,n,p))/(h_z.^2);
%compute the 12 edges
for i = 2:n-1
y(1,i,1) = (x(1,i-1,1)+x(1,i+1,1)-2*x(1,i,1))/(h_x.^2) + ...
(2*x(2,i,1)-2*x(1,i,1))/(h_y.^2) + (2*x(1,i,2)-2*x(1,i,1))/(h_z.^2);
end
for i = 2:n-1
y(m,i,1) = (x(m,i-1,1)+x(m,i+1,1)-2*x(m,i,1))/(h_x.^2) + ...
(2*x(m-1,i+1,1)-2*x(m,i,1))/(h_y.^2) + (2*x(m,i,2)-2*x(m,i,1))/(h_z.^2);
end
for i = 2:n-1
y(1,i,p) = (x(1,i-1,p)+x(1,i+1,p)-2*x(1,i,p))/(h_x.^2) + ...
(2*x(2,i+1,p)-2*x(1,i,p))/(h_y.^2) + (2*x(1,i,p-1)-2*x(1,i,p))/(h_z.^2);
end
for i = 2:n-1
y(m,i,p) = (x(m,i-1,p)+x(m,i+1,p)-2*x(m,i,p))/(h_x.^2) + ...
(2*x(m-1,i+1,p)-2*x(m,i,p))/(h_y.^2) + (2*x(m,i,p-1)-2*x(m,i,p))/(h_z.^2);
end
for j = 2:m-1
y(j,1,1) = (x(j-1,1,1)+x(j+1,1,1)-2*x(j,1,1))/(h_y.^2) + ...
(2*x(j,2,1)-2*x(j,1,1))/(h_x.^2) + (2*x(j,1,2)-2*x(j,1,1))/(h_z.^2);
end
for j = 2:m-1
y(j,n,1) = (x(j-1,n,1)+x(j+1,n,1)-2*x(j,n,1))/(h_y.^2) + ...
(2*x(j,n-1,1)-2*x(j,n,1))/(h_x.^2) + (2*x(j,n,2)-2*x(j,n,1))/(h_z.^2);
end
for j = 2:m-1
y(j,1,p) = (x(j-1,1,p)+x(j+1,1,p)-2*x(j,1,p))/(h_y.^2) + ...
(2*x(j,2,p)-2*x(j,1,p))/(h_x.^2) + (2*x(j,1,p-1)-2*x(j,1,p))/(h_z.^2);
end
for j = 2:m-1
y(j,n,p) = (x(j-1,n,p)+x(j+1,n,p)-2*x(j,n,p))/(h_y.^2) + ...
(2*x(j,n-1,p)-2*x(j,n,p))/(h_x.^2) + (2*x(j,n,p-1)-2*x(j,n,p))/(h_z.^2);
end
for k = 2:p-1
y(1,1,k) = (x(m,1,k-1)+x(m,1,k+1)-2*x(m,1,k))/(h_z.^2) + ...
(2*x(m,2,k)-2*x(m,1,k))/(h_x.^2) + (2*x(m-1,1,k)-2*x(m,1,k))/(h_y.^2);
end
for k = 2:p-1
y(m,1,k) = (x(1,1,k-1)+x(1,1,k+1)-2*x(1,1,k))/(h_z.^2) + ...
(2*x(1,2,k)-2*x(1,1,k))/(h_x.^2) + (2*x(2,1,k)-2*x(1,1,k))/(h_y.^2);
end
for k = 2:p-1
y(1,n,k) = (x(1,n,k-1)+x(1,n,k+1)-2*x(1,n,k))/(h_z.^2) + ...
(2*x(1,n-1,k)-2*x(1,n,k))/(h_x.^2) + (2*x(2,n,k)-2*x(1,n,k))/(h_y.^2);
end
for k = 2:p-1
y(m,n,k) = (x(m,n,k-1)+x(m,n,k+1)-2*x(m,n,k))/(h_z.^2) + ...
(2*x(m,n-1,k)-2*x(m,n,k))/(h_x.^2) + (2*x(m-1,n,k)-2*x(m,n,k))/(h_y.^2);
end
%compute the inside
for i = 2:n-1
for j = 2:m-1
for k = 2:p-1
y(j,i,k) = (x(j-1,i,k)+x(j+1,i,k)-2*x(j,i,k))/(h_y.^2) + ...
(x(j,i-1,k)+x(j,i+1,k)-2*x(j,i,k))/(h_x.^2) + ...
(x(j,i,k-1)+x(j,i,k+1)-2*x(j,i,k))/(h_z.^2);
end
end
end