En el artículo «MAGMA: a three-dimensional, Lagrangian magnetohydrodynamic code for merger applications» de S. Rosswog y D. Price comentan como introducir campos magnéticos en SPH. Las ecuaciones ya discretizadas quedan:
Ecuación de densidad:
$latex rho = sum_b m_b W(r-r_b,h)$
Ecuación del momento (conh: «grad-h» term, mag: magnetic force term, g: self-gravity and gravitational softening term)
$latex frac{d}{dt}v_{a,MHD} = frac{d}{dt} (v_{a,h}+v_{a,h,dis}+v_{a,g}+v_{a,mag}+v_{a,mag,dis})$
donde
$latex frac{d}{dt}v_{a,h} = -sum_b m_b big ( frac{P_a}{Omega_a rho_a^2} nabla_a W_{ab}(h_a)+ frac{P_b}{Omega_b rho_b^2} nabla_a W_{ab}(h_b) big )$
$latex frac{d}{dt}v_{a,h,dis} = $
$latex frac{d}{dt}v_{a,g} = -G sum_b m_b big [ frac{phi’_{ab}(h_a) + frac{phi’_{ab}(h_b)}{}}{2} big ]hat{e}_{ab}$
$latex -frac{G}{2} sum_b m_b big [ frac{zeta_a}{Omega_a} nabla_a W_{ab}(h_a) + frac{zeta_b}{Omega_b} nabla_a W_{ab}(h_b) big ]$
con
$latex phi’_{ab} = frac{partial phi}{partial|r_a-r_b|}$, $latex zeta_k := frac{partial h_k}{partial rho_k} sum_b m_b frac{partial phi_{kb}(h_k)}{partial h_k}$
y
$latex Omega_a := big ( 1- frac{partial h_a}{partial rho_a} cdot sum_b m_b frac{partial}{partial h_a} W_{ab}(h_a) big )$
$latex frac{d}{dt}v_{a,mag} = – sum_b frac{m_b}{mu_0} big { frac{B_a^2 / 2}{Omega_a rho_a^2} nabla_a W_{ab}(h_a) + frac{B_b^2 / 2}{Omega_b rho_b^2} nabla_a W_{ab}(h_b) }$
$latex + sum_b frac{m_b}{mu_0} big { frac{B_a(B_a cdot overline{nabla_a W_{ab}}) – B_b(B_b cdot overline{nabla_a W_{ab}})}{rho_a rho_b} big }$
con
$latex overline{nabla_a W_{ab}} = frac{1}{2} big [ frac{1}{Omega_a} nabla_a W_{ab}(h_a) + frac{1}{Omega_b} nabla_a W_{ab}(hb) big ]$
$latex frac{d}{dt}v_{a,mag,dis} = $
Ecuación de la energía (con h: «grad-h» term, AV: Artificial Viscosity term y C: Condutivity):
$latex frac{d}{dt}u_{a,MDH} = frac{d}{dt} (du_{a,h} + du_{a,AV} + du_{a,C})$
donde
$latex frac{d}{dt}u_{a,h} = frac{1}{Omega_a} frac{P_a}{rho_a^2} sum_b m_b v_{ab} cdot nabla_a W_{ab}(h_a) $
$latex frac{d}{dt}u_{a,AV} = $
$latex frac{d}{dt}u_{a,C} = $