  An optimization framework is presented for minimizing the energy functional
developed around a generalized equation governing physical systems such as
fluid dynamics, particle transport, phase transition, and other related
systems. The convexity of the energy functional is investigated to derive the
necessary conditions for a smooth and global optimum solution. Furthermore, the
Total Variation (TV) regularization term is introduced to gain insights into
the solution space and convergence analysis of convection-dominated problems.
  We demonstrate the practical application of our method by applying it to some
selected examples such as the Boltzmann, Navier-Stokes, and Maxwell equations.
