Trying to solve the following pde: $ \partial_{t}y + c\partial_{c}y = 0$ (for simplicity $ c=1$ ).

For the initial data I am using a Gaussian. The problem surges when I am trying to implement the outflow boundary condition as it was suggested to me, namely $ \frac{\partial{y}}{\partial{t}} =0$ at $ x =0$ .

So far my code is pretty simple:

` v = 1 ; L = 2; With[{y = y[t, x]}, eq = D[y, t] + vD[y, x] == 0; ic = y == Exp[-x^2] /. t -> 0; bc = {D[y, x] == 0 /. x -> 0 }]; mol[n_Integer, o_: "Pseudospectral"] := {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MaxPoints" -> n, "MinPoints" -> n, "DifferenceOrder" -> o}}; sol = NDSolveValue[{eq, ic, bc}, y, {t, 0, 1}, {x, 0, L}, Method -> mol[100, 4]]; {t0, tend} = sol["Domain"][[1]]; Manipulate[ Plot[sol[t, x], {x, 0, L}, PlotRange -> {-10, 10}], {t, 0, tend}]; `

It stems from answers to questions previously asked here, and I intend later to test finite difference methods (BTCS/FTCS) as done in Schemes for nonlinear advection equation.

However I am not being able to evolve the equation do to confusion when trying to implement the BC, I get the following:

` NDSolveValue: Boundary condition $ y^{(0,1)}[t,0]$ should have derivatives of order lower than the differential order of the partial differential equation. `

This is expected as I am not sure what would be the best way to impose BC on the problem.

If anyone has any suggestions they would be welcolmed.

Thanks.