# Thread: Unconditionally stable scheme for solving first order system of PDEs

1. ## Unconditionally stable scheme for solving first order system of PDEs

Hello,

I would like to numerically solve the following system of PDEs using a stable computational scheme. The system involves three equations and three unknowns ( $q_x$, $q_y$, and $p$). I've tried to solve this system using an explicit scheme, but I found that the solution could become unstable. An implicit scheme may work, but then how would I deal with the three unknowns in a computationally efficient fashion?

Could someone suggest a stable scheme to numerically solve this system?

$
$A\frac{{\partial p}}{{\partial t}} + Bp = \frac{{\partial q_x }}{{\partial x}} + \frac{{\partial q_y }}{{\partial y}}$
$

$
$\frac{{\partial q_x }}{{\partial t}} = \frac{{\partial p}}{{\partial x}}$
$

$
$\frac{{\partial q_y }}{{\partial t}} = \frac{{\partial p}}{{\partial y}}$
$

2. Perhaps one way of proceeding is to set this up as a block tridiagonal system of equations, and then solve for all of the unknowns using an efficient algorithm.