solution of one problem from CFD Vol.I by K A.Hoffmann , with 4 method and result (Implicit crank nicolson - du fort-frankel - ftcs)

3. clc
clear all
close all
%%
% Book : CFD Vol.I by K A.Hoffmann
% Chapter : 3
% Number of example : 3.5 applications
% Author: Parham Sagharichi ha
%
parhamsagharchi@gmail.com
% inputs:
%
%
% outputs:
%
%%
%-------------------S------T------A------R------T-------------------------%
deltay = 0:1:40;
deltat = 0:0.001:2;
a = length(deltay);
b = length(deltat);
T = zeros(a,b);
T(a,:) = 50;
Tftcs = T;
Tfrankel = T;
Tcrank = T;
Timplicit = T;
% Forward time/central space (FTCS) method :
kappa = (0.000217*0.001)/(0.001^2);
for n1 = 2:b;
for m1 = 2:a-1;
Tftcs(m1,n1) = Tftcs(m1,n1-1) +
kappa*(Tftcs(m1+1,n1-1)-2*Tftcs(m1,n1-1) + Tftcs(m1-1,n1-
1));
end
end
% The DuFort-Frankel method :
Tfrankel(:,2)= Tftcs(:,2);
for n2 = 3:b;

4. for m2 = 2:a-1;
Tfrankel(m2,n2) = (((1-
2*kappa)/(1+2*kappa))*Tfrankel(m2,n2-
2))+(2*kappa/(1+2*kappa))*(Tfrankel(m2+1,n2-1)+Tfrankel(m2-
1,n2-1));
end
end
% The Crank-Nicolson method :
A1 = sparse(a,a);
A2 = sparse(a,a);
for i = 2:a-1;
A1(i,i-1) = -kappa/2;
A1(i,i) = 1+kappa;
A1(i,i+1) = -kappa/2;
A2(i,i-1) = kappa/2;
A2(i,i) = 1-kappa;
A2(i,i+1) = kappa/2;
end
A1(1,1) = 1;
A1(a,a) = 1;
A2(1,1) = 1;
A2(a,a) = 1;
figure(1)
spy(A1)
figure(2)
spy(A2)
for i = 1:b-1;
B1 = Tcrank(:,i);
Tcrank(:,i+1) = A1(A2*B1);
end
% Implicit method :
A3 = sparse(a,a);
for i = 2:a-1
A3(i,i-1) = -kappa;
A3(i,i) = (1+2*kappa);
A3(i,i+1) = -kappa;
end

5. A3(1,1) = 1;
A3(a,a) = 1;
figure(3)
spy(A3)
for i = 1:b-1;
B3 = Timplicit(:,i);
Timplicit(:,i+1) = A3B3;
end
% Timplicit(:,i+1) = A3B OR Timplicit(:,i+1) =
inv(A3)*B;
% an = a0 + (n-1)d --> a0 =0 --> n = an/d +1
n1 = 0.5/0.001 + 1;
n2 = 1/0.001 +1;
n3 = 1.5/0.001 +1;
deltay1 = deltay/1000;
% @ 2s
figure(4)
plot(Timplicit(:,end),deltay1,'-
ro',Tcrank(:,end),deltay1,'-
.b',Tftcs(:,end),deltay1,'+g',Tfrankel(:,end),deltay1,'*y')
ylabel(' delta y (m) ')
xlabel (' Velocity (m/s^2) ')
title(' @ t = 2s ')
legend('Implicit','Crank-Nicolson','FTCS','DuFort-
Frankel')
grid on
% @ 1.5s
figure(5)
plot(Timplicit(:,n3),deltay1,'-
ro',Tcrank(:,n3),deltay1,'-
.b',Tftcs(:,n3),deltay1,'+g',Tfrankel(:,n3),deltay1,'*y')
ylabel(' delta y (m) ')
xlabel (' Velocity (m/s^2) ')
title(' @ t = 1.5s ')
legend('Implicit','Crank-Nicolson','FTCS','DuFort-
Frankel')
grid on

6. % @ 1s
figure(6)
plot(Timplicit(:,n2),deltay1,'-
ro',Tcrank(:,n2),deltay1,'-
.b',Tftcs(:,n2),deltay1,'+g',Tfrankel(:,n2),deltay1,'*y')
ylabel(' delta y (m) ')
xlabel (' Velocity (m/s^2) ')
title(' @ t = 1s ')
legend('Implicit','Crank-Nicolson','FTCS','DuFort-
Frankel')
grid on
% @ 0.5s
figure(7)
plot(Timplicit(:,n1),deltay1,'-
ro',Tcrank(:,n1),deltay1,'-
.b',Tftcs(:,n1),deltay1,'+g',Tfrankel(:,n1),deltay1,'*y')
ylabel(' delta y (m) ')
xlabel (' Velocity (m/s^2) ')
title(' @ t = 0.5s ')
legend('Implicit','Crank-Nicolson','FTCS','DuFort-
Frankel')
grid on
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