The 22_6xx-dg-for-pdes from userjjb

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Can't understand the formulation of ExactRHS in Module 2

I've watched your youtube tutorial on DG and cloned this git repo. Thank you for the excellent explanation.

What I'm concerned is your demo MATLAB script in module 2.

In the demo script, I can't understant why you formulate ExactRHS such way:

ExactRHS(1, :) = ...
    ( tau * deltax .* cos(tau * xNode(1:end-1)) ...
    + sin(tau * xNode(1:end-1)) ...
    - sin(tau * xNode(2:end)) ) ...
        ./ ( 4 * pi^2 * (deltax) );
ExactRHS(2, :) = ...
    -( tau * deltax .* cos(tau * xNode(2:end)) ...
    + sin(tau * xNode(1:end-1)) ...
    - sin(tau * xNode(2:end)) ) ...
        ./ ( 4 * pi^2 * (deltax) );

I know that the key idea behind the above is that $\int u_h \phi = \int u_0 \phi$. According to my understanding, I would like to formulate as follows,

f1 = @(x) - sin(2*pi*x)/(4*pi^2) - (cos(2*pi*x)/2 - (x*cos(2*pi*x))/2)/pi;
f1_int = zeros(K, 1);
for i = 1 : K
    x_init = xNode(i);
    x_end = xNode(i+1);
    f1_int(i) = f1(x_end) - f1(x_init);
end
f2 = @(x) (sin(2*pi*x) - 2*pi*x*cos(2*pi*x))/(4*pi^2);
f2_int = zeros(K, 1);
for i = 1 : K
    x_init = xNode(i);
    x_end = xNode(i+1);
    f2_int(i) = f2(x_end) - f2(x_init);
end

They are the integral of $\int u_0 \phi$ and the definite integral is over every single element. However, the above formulations give different result.

The plot of your formulation is

While my formulation output as follows,

Could you point out why I'm wrong? Thanks!

There maybe some errors in Module 3B

I directly execute the script SC1DLeg.m and get the following error

>> SC1DLeg
Error using toeplitz (line 23)
Inputs must be numeric.

Error in SC1DLeg (line 86)
SelfStencil = 2*(toeplitz(mod(0:N,2),0:N<0))-ones(N+1);

I suppose the input parameters for toeplitz maybe be wrong. Please check.

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