Numerical Implementation

In this section we motivate and describe our finite difference representation of spherically symmetric radiation hydrodynamics in numerical computations. The continuous equations of radiation hydrodynamics are unique. But the number of possible finite difference representations is large. We select ours according to the following requirements (with descending priority):

1. In the limit of high resolution, the finite difference representation should converge to the continuous equations.
2. The finite difference representation should lead to a numerically stable solution.
3. The finite difference representation should not be less accurate than simpler schemes in regions where simpler schemes are sufficient.
4. The computed evolution of the particle distribution function should imply an accurate evolution of expectation values like the particle density or the average particle energy.

In our discrete ordinates (S) method, we discretize the full transport equation at once. This takes care of requirement (1). In the first place, we choose the finite differencing such that the numerical scheme is stable according to requirement (2). Moreover, we adjust it such that requirement (3) is met with respect to the important diffusion limit. In the finite differencing of the free streaming limit, we focus on accurate particle number luminosities and accurate particle energy luminosities. The accuracy of the angular moments of the radiation field is less essential in this domain because the neutrinos cannot contribute to the supernova mechanism in regions far outside the neutrinospheres. More accurate angular information in the free streaming limit with the S method would require a very large number of angular bins, an adaptive grid in angle space Yamada_Janka_Suzuki_99, or a postprocessing step with a ray-tracing tool. All remaining degrees of freedom in the choice of the finite difference representation are used to optimize requirement (4). Of course, point (1) already guarantees that requirement (4) will successfully be approached if the resolution is sufficiently high. However, as we solve the transport equation at once, and with an implicit finite difference representation, resolution is computationally rather expensive. In the spirit of requirement (3), high resolution may only be required where the very details of the complete transport equation are relevant for the physical solution. In order to make low resolution results reliable, important physical laws must be represented accurately independent of the resolution setting. This includes the challenge of conserving lepton number and total energy in the simulations. Indeed, one might say that the power of a specific implementation of the S method is given by the extent to which requirements (3) and (4) can be satisfied at low resolution.

A variable Eddington factor method (VEF) is a different numerical approach than the S method. There, requirements (2), (3), and (4) are satisfied by construction. A series of simpler radiation moment equations are solved and combined with an Eddington factor to produce the solution to the equations of radiation hydrodynamics. The quality of the variable Eddington factor method is determined by the capability to meet requirement (1). It depends on the interplay between the different moment equations and the accuracy of the Eddington factor (which may require the numerical solution of a model Boltzmann equation Burrows_et_al_00,Rampp_Janka_02,Thompson_Burrows_Pinto_03). With careful implementation, both methods should fulfill requirements (1)-(4) and therefore lead to similar results Liebendoerfer_et_al_04.