Due to a lack of fruitful answers after 3 days, this question has been reformulated in a more direct way compared to its first version.
Consider the telegraph equation problem expressed as a system of two coupled PDEs in voltage v and current i:
eq1 = D[v[t, x], x] + D[i[t, x], t] == 0;
eq2 = D[i[t, x], x] + D[v[t, x], t] == 0;
sys = {eq1, eq2};
The initial conditions are a bell shape centered at x=0.5 for the voltage, and zero current i:
shape = D[0.125 Erf[(x - 0.5)/0.125], x];
ics ={v[0, x] == shape, i[0, x]==0};
We now want to find numerically v and i with NDSolve and the Finite Element method, between xMin and xMax, for t between 0 and 2, and for perfectly reflective or perfectly non-reflective boundary conditions.
xMin = 0; xMax = 1;
region = Line[{{xMin}, {xMax}}];
Case 1: perfect reflection on open ends => i=0
bcs1 = {i[t, xMin] == 0, i[t, xMax] == 0};
sol1 = NDSolve[{sys, ics, bcs1} // Flatten, {v, i}, {t,0,2}, {x} \[Element] region][[1]];
which we plot (current in cyan) using
myInterpPlot[interpolatingFunc_, t_, opts___] :=Block[
{grid = ((#[[2]] &) /@ PropertyValue[interpolatingFunc, Grid][[1]]) //Sort,
points},
points = {#, interpolatingFunc[t, #]} & /@ grid;
ListPlot[{points, points}, Joined -> {True, False}, opts,
PlotStyle -> {{Thin, Red}, {Medium, Blue}}, Frame -> True]]
myPlot [sol_,t_]:=Show[
myInterpPlot[sol[[1, 2]], t, PlotRange -> {-1.5, 1.5}],
myInterpPlot[sol[[2, 2]], t, PlotRange -> {-1.5, 1.5},
PlotStyle -> {{Thin, Cyan}, {Medium, Cyan}}],
Plot[shape, {x, xMin, xMax}, PlotStyle -> {Dashed, Blue}]]
Manipulate[myPlot[sol1,t],{t,0,2}]
Case 2: perfect reflection on shorted ends => v=0
bcs2 = {v[t, xMin] == 0, v[t, xMax] == 0};
sol2 = NDSolve[{sys, ics, bcs2} // Flatten, {v, i}, {t,0,2}, {x} \[Element] region][[1]];
Manipulate[myPlot[sol2,t],{t,0,2}]
Case 3 that I don't know how to encode: perfect non-reflective conditions. Physically, this case corresponds to the two equations {eq1,eq2} applying also on the boudaries, since any wave should propagate as if there were no boundaries. How to encode this case keeping the problem in v and i at first order? Can it be done with bcs3=... ? Should one use Neumann values in eq1 and eq2, but what would they mean for first order equations in space?
Case 4: No specified boundary conditions. What does NDSolve do in this case? What are the default boundary conditions used?
sol = NDSolve[{sys, ics} // Flatten, {v, i}, {t,0,2}, {x} \[Element] region][[1]];
Manipulate[myPlot[sol,t],{t,0,2}]
Thank you for your help. Any reference to the doc is welcome. Note that I know how to encode the non reflective case with a second order eq in v and Neumann values. So this is not the answer to the question.