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So not quite sure what you're missing, but here's how you go about doing this sort of thing. So first, I am assuming this is 1D due to your description. Second, I'm assuming you know the relationship …

So the main issue with this is that you are trying to take uniqueness in your approximation, which is based on solving a difference equation, to infer something about your ill-posed PDE problem, like …

I am somewhat confused by this question. Why are you evaluating the fluxes at a midpoint of the time step instead of evaluating them at $t = t_{n}$ or $t = t_{n+1}? From what I have seen, fluxes are …

In the past semester, I was busy writing some CFD code for a simulation I needed for an engineering project. I stumbled upon a few methods that can handle complex geometry and still maintain decent si …

So something must be up with your code. I know this because when I just wrote a C++ code to solve this problem with my simulation framework, I got the following result using Explicit Euler: I would …

You shouldn't have any problem coding in a conditional boundary condition into the odefun, like the one you defined in your problem statement. Just use the last timestep's value of $u_b$ to do the co …

So your first big issue is one you bring up in point (4), where you say you get the error 'u(0, t[m]) = 0 "can't assign to function call"'. You're trying to store data by assigning data to a function. …

As an example, you should be able to build up a system of equations to solve for the time derivatives if you use something like the following scheme: \begin{align} A \frac{\partial ^2}{\partial x^2}\ …

There are some unknowns in what you are doing but for simplicity, suppose we want to find $u(t)$ as discrete times $t_1, t_2, \cdots, t_n$. Let $\textbf{F} = [F(t_1), F(t_2), \cdots, F(t_n)]^T$ and $\ …