Answer to Question #257561 in Mechanical Engineering for Garima

Question

discretize the 1 order and 2 order wave equation using lax-wendroff scheme.Explain the dispersion and dissipation errors encountered while solving the above equations

Answer 1

------------------- MATLAB CODE ----------------------

clear

close all

clc

dx = 1;

x = 0:dx:40;

% x will be the row direction

% t will be column direction

% BACKWARD DIFFERENCE IN TIME AND FORWARD DIFFERENCE IN SPACE

for c = [1,0.6,0.3]

dt = c*dx;

t = 0:dt:18;

u_a = zeros(length(x),length(t));

u_a(:,1) = sin(2*pi*x/40);

A = zeros(length(x)-2);

for i = 1:length(x)-2

if i == 1

A(1,1:2) = [1 c/2];

elseif i == length(x)-2

A(end,end-1:end) = [-c/2 1];

else

A(i,i-1:i+1) = [-c/2 1 c/2];

end

end

for i = 2:length(t)

b = u_a(2:end-1,i-1); % here we can add boundary values, in our case (0)

u_a(2:end-1,i) = A\b;

end

figure;contourf(u_a)

title(['Backward(in t),Forward(in x) c = ' num2str(c)])

xlabel('t')

ylabel('x')

end

% LAX SCHEME

for c = [1,0.6,0.3]

dt = c*dx;

t = 0:dt:18;

u_b = zeros(length(x),length(t));

u_b(:,1) = sin(2*pi*x/40);

for it = 2:length(t)

for ix = 2:length(x)-1

u_b(ix,it) = 0.5*(u_b(ix+1,it-1)+u_b(ix-1,it-1))...

- 0.5*c*(u_b(ix+1,it-1)-u_b(ix-1,it-1));

end

end

figure;contourf(u_b)

title(['Lax scheme, c = ' num2str(c)])

xlabel('t')

ylabel('x')

end

% LAX - WENDROFF SCHEME

for c = [1,0.6,0.3]

dt = c*dx;

t = 0:dt:18;

u_c = zeros(length(x),length(t));

u_c(:,1) = sin(2*pi*x/40);

for it = 2:length(t)

for ix = 2:length(x)-1

u_c(ix,it) = u_c(ix,it-1) - 0.5*c*(u_c(ix+1,it-1)-u_c(ix-1,it-1))...

+ 0.5*(c^2)*(u_c(ix+1,it-1)-2*u_c(ix,it-1)+u_c(ix-1,it-1));

end

end

figure;contourf(u_c)

title(['Lax - Wendroff scheme, c = ' num2str(c)])

xlabel('t')

ylabel('x')

end

-------------------------- CODE ENDS ------------------------