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When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary conditions directly...

Since the initial values prescribe the values that the boundary has to take at $T = 0$ and we impose that these values must not change through no flux, wouldn't this be equivalent to simply prescribing the initial values at all times $T$ at the boundary (i.e. Dirichlet boundary, constant w.r.t. time)?

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It depends on the equation you are solving. For example, consider the following heat equation with a source term:

$$ \frac{dT}{dt} - \nabla^2 T = S$$

If $S \neq 0$, then you have a source term that changes the value of the temperature as time evolves. Consequently, even if your initial condition is $T(\textbf{x},t)=0$, your temperature will increase because of the source term. Then, having a Dirichlet or a Neumann boundary condition will lead to completely different result. If you have a Dirichlet (say $T=0$) then that boundary will remain at 0. If you have a Neumann, the gradient at the boundary will remain 0, thus the temperature will increase due to the source term. Consequently, you would get completely different temperature profile in the end.

There are many other examples you could use to explain the same phenomena. These are very different type of BCs, I would invite you to solve a 1D transient heat transfer problem and try it with both BCs and you will see that the resulting temperature profiles will be significantly different!

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Firstly, in more than one dimensional case, one prescribes with homogeneous Neumann boundary condition the zero value of the derivative only in one direction, e.g. the temperature can change along the boundary. Secondly, the time derivative can be nonzero there, and, moreover, even in 1D case, the flux can be nonzero in an arbitrary small neighborhood of the boundary point. The simplest function illustrating it is T(x,t)=t. The best way to see the difference between the mentioned two types of boundary conditions is to run related simulation.

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