Correct Formula for Potential Energy (trixi-framework#2227)

* Correct Formula for Potential Energy

* Update examples/paper_self_gravitating_gas_dynamics/elixir_eulergravity_jeans_instability.jl

Co-authored-by: Hendrik Ranocha <[email protected]>

---------

Co-authored-by: Hendrik Ranocha <[email protected]>

Author: DanielDoehring

examples/paper_self_gravitating_gas_dynamics/elixir_eulergravity_jeans_instability.jl

@@ -136,7 +136,9 @@ function Trixi.analyze(::Val{:energy_potential}, du, u_euler, t,
         u_gravity_local = Trixi.get_node_vars(u_gravity, equations_gravity, dg, i, j,
                                               element)
         # OBS! subtraction is specific to Jeans instability test where rho0 = 1.5e7
-        return (u_euler_local[1] - 1.5e7) * u_gravity_local[1]
+        # For formula of potential energy see
+        # "Galactic Dynamics" by Binney and Tremaine, 2nd ed., equation (2.18)
+        return 0.5f0 * (u_euler_local[1] - 1.5f7) * u_gravity_local[1]
     end
     return e_potential
 end