The 3D open problem on the existence and uniqueness of the solution to the incompressible nonhomogeneous Euler equation has received a lot of attention in the last few years. It is a known fact that this problem does not admit ‘weak’ solutions. Theoretically, existence  and uniqueness for the homogeneous incompressible case receives attention in [1-3] and [5] with varying results. However, there is an agreement that the solution to the problem is unique for a finite interval of time, and thereafter loses regularity.

There are three unknown solutions to the problem: mass density (r), pressure (p) and velocity (v). In this short numerical simulation, we approximate the solution, to the nonhomogeneous incompressible case, over a finite time interval, in  using forward finite-differences. For this purpose, we develop a numerical scheme which is tested on a numerical example, and assuming the velocity is the same in all the three Cartesian directions.

The numerical scheme development is similar to the case in [4], where we assume  For the nonhomogeneous Euler equations, uniqueness for the solution is confirmed in [6] and [7], for a finite time interval in the space  This has led to this paper where we use numerical simulation to show that the space  for the study of solutions to the nonhomogeneous, incompressible Euler equations, should not be ignored. We are quick to point out that, unlike in [7], our mass density(r)is not upper bounded. The graphical picture is for one Cartesian direction only.

approximation, solution, non-homogeneous, Euler equation.