April 24th, 2020, 5:52 pm
Maybe the answer is that your problem is always ill-posed.
Suppose there was a time-independent solution [$]u(x,y)[$], which is then a solution to a 2D hyperbolic problem. After 45 degree coordinate rotation to [$](\xi,\eta)[$], the characteristics leaving the origin are [$]\xi = \pm \eta[$]. Then, rotating back, aren't the characteristics in the original coordinates just the coordinate axes?
Now, in general, to make any hyperbolic problem well-posed, you have to specify data along curves which are non-characteristic.
But, if the horizontal and vertical lines are exactly the characteristics here, then you are asking for illegal boundary conditions.
This argument suggests the time-independent problem is ill-posed. It makes one suspicious about the time-dependent problem.