Growth of gas bubbles in magmas may be modelled by a system of differential equations coupled with an advection-diffusion equation, and characterised by two relaxation times linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme which, unlike previously published schemes, preserves the total mass of the coupled system of equations. We also study the asymptotic behavior of the system of equations, when letting the relaxation times vary from zero to infinity, and show the numerical convergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments.