Asymptotic preserving and positive schemes for radiation hydrodynamics

Abstract

In view of radiation hydrodynamics computations, we propose an implicit and positive numerical scheme that captures the diffusion limit of the two-moments approximate model for the radiative transfer even on coarses grids. The positivity of the scheme is equivalent to say that the scheme preserves the limited flux property. Various test cases show the accuracy and robustness of the scheme.

Introduction

The aim of this work is about accurate discretization of radiation hydrodynamics by means of eulerian finite volume methods. In this direction we study and discretize a non-linear two-moments model (2) for radiation in the Eulerian frame written in one space dimension. The model, called in the next M¹, derives from a maximum entropy principle as in [10], [27], [19], [26], [5], [1]. This model and the related ones are stiff models, the stiffness parameter is ε small. For ICF (inertial confinement fusion applications) ε ≈ 10⁻³, for example see [30]. There are some regions of the space with very little interactions between the radiation and the matter and other regions with strong interactions with the matter. The case of little interactions is called the streaming regime. For strong interaction it is possible to derive diffusion approximations of (2). This is called the diffusion regime. Among the numerical methods that have been proposed for the discretization of simplified radiation models, we distinguish between discretization of diffusion approximations and direct discretization of two-moment models. Concerning numerical method for the discretization of diffusion approximations we quote [18] in which a technique of flux limiting is discussed in detail. The technique of flux limiting somehow extends the domain of validity of these diffusion approximations to the streaming regime. We exhibit a new intermediate interaction regime, of pure hyperbolic type, for which the limit equation can be written in divergent form or in non-divergent form. We privilege the divergence form of this equation. This is different from the work of [9] where the non-divergent formulation is used for the radiation energy. This is important for the numerics since a non-divergent discretization may converge to wrong discontinuous solutions [17]. It has been noticed since a long time, in the work of Mihalas [24] for example, that a direct discretization of a two-moments models can help to get a better discretization for both streaming and diffusion regimes. It is also possible that higher moments models could treat more efficiently the anisotropy of radiation in dimension greater than one.

In this direction, a preliminary question is how to discretize “correctly” a two-moments model for the radiation in the non-relativistic case in dimension one, that is for ε small, and at the matter scale. That is, we desire to adapt the grid according to the length scale of the matter. From the point of view of photons the grid is coarse. Among the works dedicated to this specific problem we point out [21], [22], [23], [2]. We show on a simple example that freezing naively the Eddington factor during the time step gives a non-positive implicit linear discretization of the problem. This is based on the fact that the linear implicitation of the HLL solver is non-positive if the equation is non-linear, that is if the Eddington factor is not constant. The new method we propose has the following features:
Positivity of the scheme: the scheme guarantees the positivity in 1D

$$-E_{\mathrm{r}}⩽F_{\mathrm{r}}⩽E_{\mathrm{r}}$$

where E_r is the energy of radiation and F_r is the radiation flux. This is equivalent to say the scheme is flux limited

$$\frac{|F_{\mathrm{r}}|}{E_{\mathrm{r}}}⩽1\text{.}$$

The physical meaning of flux limitation is more natural with (1). This property is useful essentially in the streaming regime for which |F_r| ≈ E_r. This flux limiting has nothing to do with the numerical tricks used for the discretization of diffusion approximations of two-moments models.

• the equilibrium and non-equilibrium diffusion limits of the scheme are correct even for a coarse mesh,

• the numerical treatment of the purely hyperbolic limit (9) is compatible with the divergent form of the equation.

• the scheme is explicit for the hydrodynamic part and totally implicit for the radiation part. Therefore, the only CFL limitation comes from the hydrodynamic part.

The first three properties are true at the PDE level for the basic two-moments model. Since the basic two-moments model considered in this paper is highly non-linear (see (2)), a direct implicit discretization of this model may be tricky if the material velocity v is non-zero. We now describe the strategy we propose to obtain these properties at the numerical level. We think this approach is simpler.

The first ingredient to obtain all these properties is what we call a splitted approximation. Usually this kind of model is called a co-moving frame radiation hydrodynamics model, see [25], [20], [2]. The model consists in transporting the radiation entropy and the radiation entropy flux at velocity v, then to discretize with the matter at rest with v = 0. This splitted approximation has almost zero additional numerical cost. The maximal hyperbolic wave velocity of the splitted approximation is ${\displaystyle \frac{1}{\epsilon }}\pm v$ which slightly exceeds the physical maximal wave velocity ${\displaystyle \frac{1}{\epsilon }}$ which is the velocity of light in non-dimensional variables. This is shared with other co-moving frame radiation hydrodynamics model like [25], [20], [2]. We think that slight violation of the velocity of light in very rare cases is much better than diffusion approximations without flux limiting of the diffusion coefficient. Even with flux limiting of the diffusion coefficient it is difficult to guarantee a correct velocity for the front propagation in transparent materials. In this work we only consider an explicit discretization of the hydrodynamics part. On the other hand, the radiative step of the splitted model is discretized by an implicit method. The reason is the high speed of the radiation signal which is much greater than the mean velocity of the matter. Since we are interested in the time scale of the matter, the radiation must be treated implicitly.

The second ingredient lies in the discretization of the radiative step. The numerical scheme for the radiative step must be consistent with the diffusion approximation in opaque region and must preserve the flux limited property. It is well known, cf. [14], [15], [28], that in opaque regions standard finite volume schemes lead to a wrong coefficient of diffusion for coarse grids. It is illuminating to consider the hyperbolic heat equation

$${\mathrm{\partial }}_{t}u+\frac{1}{\epsilon }{\mathrm{\partial }}_{x}v=0\text{,}{\mathrm{\partial }}_{t}u+\frac{1}{\epsilon }{\mathrm{\partial }}_{x}v=-\frac{1}{{\epsilon }^2}u\text{.}$$

The asymptotic regime ε → 0⁺ is the heat equation ∂_tu − ∂_xxu = 0. The problem stressed in [14], [15], [28] is that a discretization with stable upwinded schemes of the hyperbolic equation has the asymptotic limit

$${\mathrm{\partial }}_{t}u-\left(1+\frac{C\mathrm{\Delta }x}{\epsilon }\right){\mathrm{\partial }}_{\mathit{xx}}u=0\text{,}C>0\text{.}$$

Therefore, the discretization of the hyperbolic equation on a coarse grid may have the limit

$${\mathrm{\partial }}_{\mathit{xx}}u=0$$

which is wrong. Last decade “asymptotic preserving schemes” (AP scheme), cf. [14], [15], [28], [12], have been developed in order to recover the correct diffusion asymptotic limit. Thus AP schemes seems to be the good tool for radiative two-moment models. But actual AP schemes suffer of some limitations. Semi-implicit AP scheme like those proposed in [15], [14] can handle non-linear system but are, in the best case, only positive and stable under a parabolic CFL condition and this not acceptable for our purpose since we do not want such time step limitation. The AP scheme proposed by Gosse in [12] has a better stability property, but it is difficult to extend this scheme for non-linear systems: Moreover, the method used in [12] seems quite impossible to extend for a resonant problem like the one we study in this work. Therefore, we have developed a new discretization of radiation part of the model. This implicit discretization is unconditionally positive, and has the correct diffusion limit. It is based on two basic schemes. The first one is a “relaxation” scheme, see for example [16], which has the advantage to transform the initial non-linear problem into a linear one: this is why we solve two linear systems at each time step. The second one is the AP scheme described [12], in order to obtain the right asymptotic diffusion limit with no time step limitation for the positivity. This discretization of the radiative step is an improved version of the one proposed in [7], it can handle more efficiently opaque and transparent zones. Concerning the cost of such a discretization, let us mention that in one dimension the matrix can be inverted by the a low cost algorithm like the LU one.

The plan is as follows. In Section 2, we study the M¹ model for the radiation and recall various diffusion approximations of this model. In Section 3, we derive what we call splitted approximations for the M¹ model. Numerical schemes for each part of the splitting are proposed and analyzed in Section 4. In Section 5, we show some numerical results to demonstrate the features of the scheme, with application to the full system of hydrodynamics coupled with radiation. Section 6 is the conclusion. We sketch future developments of the method.