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I have read some materials about ADI - PR method with the aim to understand how to put boundary conditions in my 2D scheme which solves the Time-Dependent Schrodinger Equation.

All the theory I read starts with the assumption that the equation to be solved has Dirichlet boundary conditions of the type $u(0, y, t) = g(y, t)$, if the operator acting on $u^{(n+1)*}$ is only differencing the x-coordinate. Then they derive boundary conditions for the intermediate timestep function, and they arrive at an expression similar to $U^*_{0, m} = ... = coeff_1 * g_{m}^{n} + coeff_2 * g_{m}^{n+1}$, where $g_m^{n+1} = g(mh, t_n + k)$, where $h$ is the cell length in both $x$ and $y$ directions and $k$ is the timestep duration. The $U^*_{0, m}$ approximates the solution to the first equation from the set of 2 from the ADI - PR method. Thus they assume one has $g$ in closed form and can query it at any timestep $t$ and at any $y$.

My problem requires the implementation of Transparent Boundary Conditions which read, for a 1D problem, $u^n_{0} = \alpha u^{n}_1$, where $\alpha = u^{n-1}_1 / u^{n-1}_2$. Thus the boundary condition is dependent on the previous iterations. I do not have that $g$ from mathematics literature, and I cannot query it at timestep $t_n + k$.

How can I go about implementing these transparent boundary conditions in my 2D ADI - PR solver?

Thank you!

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