A compatibly differenced total energy conserving form of SPH
  We describe a modified form of Smoothed Particle Hydrodynamics (SPH) in which
the specific thermal energy equation is based on a compatibly differenced
formalism, guaranteeing exact conservation of the total energy. We compare the
errors and convergence rates of the standard and compatible SPH formalisms on
analytic test problems involving shocks. We find that the new compatible
formalism reliably achieves the expected first-order convergence in such tests,
and in all cases improves the accuracy of the numerical solution over the
standard formalism.
