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I think the difference between the first and second plots is explained by noting the error outlined above.

Regarding the conservation of the physical quantity. I believe the cause is that, as pointed out here, the advection equation in the form I have written it doesn't allow propagation in the reverse direction so the wave just passes through even with zero-flux boundary conditions. My initial intuition regarding conservation only applied when advection term is zero (this is solution in plot number 2 where the area is conserved).

Even with Neumann zero-flux boundary conditions $\frac{\partial \phi}{\partial x} = 0$ the mass can still leave the system, this is because the correct boundary conditions in this case are Robin boundary conditions in which the total flux is specified $j = D\frac{\partial \phi}{\partial x} + \boldsymbol{v}\phi = 0$. Moreover the Neunmann condition specifies that mass cannot leave the domain via diffusion, it says nothing about advection. In essence what we have hear are closed boundary conditions to diffusion and open boundary conditions to advection. For more information see the answer here, Implementation of gradient zero boundary conditon in advection-diffusion equation.

I think the difference between the first and second plots is explained by noting the error outlined above.

Regarding the conservation of the physical quantity. I believe the cause is that, as pointed out here, the advection equation in the form I have written it doesn't allow propagation in the reverse direction so the wave just passes through even with zero-flux boundary conditions. My initial intuition regarding conservation only applied when advection term is zero (this is solution in plot number 2 where the area is conserved).

Even with Neumann zero-flux boundary conditions $\frac{\partial \phi}{\partial x} = 0$ the mass can still leave the system, this is because the correct boundary conditions in this case are Robin boundary conditions in which the total flux is specified $j = D\frac{\partial \phi}{\partial x} + \boldsymbol{v}\phi = 0$. Moreover the Neunmann condition specifies that mass cannot leave the domain via diffusion, it says nothing about advection. In essence what we have hear are closed boundary conditions to diffusion and open boundary conditions to advection. For more information see the answer here, Implementation of gradient zero boundary conditon in advection-diffusion equation.

I think the difference between the first and second plots is explained by noting the error outlined above.

Regarding the conservation of the physical quantity. I believe the cause is that, as pointed out here, the advection equation in the form I have written it doesn't allow propagation in the reverse direction so the wave just passes through even with zero-flux boundary conditions. My initial intuition regarding conservation only applied when advection term is zero (this is solution in plot number 2 where the area is conserved).

Even with Neumann zero-flux boundary conditions $\frac{\partial \phi}{\partial x} = 0$ the mass can still leave the system, this is because the correct boundary conditions in this case are Robin boundary conditions in which the total flux is specified $j = D\frac{\partial \phi}{\partial x} + \boldsymbol{v}\phi = 0$

I think the difference between the first and second plots is explained by noting the error outlined above.

Regarding the conservation of the physical quantity. I believe the cause is that, as pointed out here, the advection equation in the form I have written it doesn't allow propagation in the reverse direction so the wave just passes through even with zero-flux boundary conditions. My initial intuition regarding conservation only applied when advection term is zero (this is solution in plot number 2 where the area is conserved).

Even with Neumann zero-flux boundary conditions $\frac{\partial \phi}{\partial x} = 0$ the mass can still leave the system, this is because the correct boundary conditions in this case are Robin boundary conditions in which the total flux is specified $j = D\frac{\partial \phi}{\partial x} + \boldsymbol{v}\phi = 0$. Moreover the Neunmann condition specifies that mass cannot leave the domain via diffusion, it says nothing about advection. In essence what we have hear are closed boundary conditions to diffusion and open boundary conditions to advection. For more information see the answer here, Implementation of gradient zero boundary conditon in advection-diffusion equation.