The Simulation of Superluminous Supernovae Using the M1 Approach for Radiation Transfer

Supernovae (SN), i.e., bursts produced by the explosions of stars, whose luminosity increases over weeks and months even to the total luminosity of the host galaxy, are the subject of close study in astrophysics. They play a decisive role in the dynamics of the interstellar medium and its enrichment with chemical elements, the generation of cosmic rays, the formation of neutron stars, and for the measurement of the parameters of cosmological models of our universe. Not all issues with the mechanism of SN explosion and light production have been resolved, and in the last decade a new class of mysterious superluminous SN (SLSN), the maximum luminosity of which is an order of magnitude higher than the luminosity of previously known types of SN, has been discovered. One of the most possible significant applications of SLSN is the direct method of measuring cosmological distances (Blinnikov et al. 2012; Baklanov et al. 2013). The conventional method using Type Ia SN (SNIa) requires calibrations that are done in the local universe, but where it is applied at high redshifts, the properties of SN may be rather different.

Today there are several scenarios of SLSN, e.g., runaway production of pairs (pair-instability SN; Langer & El Eid 1986; Kasen et al. 2011; Kozyreva & Blinnikov 2015; Woosley & Heger 2015) and deposition of energy from magnetars (Kasen & Bildsten 2010; Woosley 2010; Barkov & Komissarov 2011; Inserra et al. 2013; Nicholl et al. 2013). However, the model for pair-instability SN, which is based on very massive pre-SN stars and large explosion energies, is able to explain only very slow light curves (Kozyreva & Blinnikov 2015). On the other hand, the magnetar scenario requires extreme physical properties of rapidly rotating, strongly magnetized neutron stars (period of rotation ∼1 ms and magnetic field ∼10¹⁵ G). According to one of the main scenarios of SLSN formation, the high luminosity of an SLSN is explained by a strong shock wave propagation through the dense circumstellar matter (a shock-interacting mechanism; Grasberg & Nadezhin 1986; Chugai et al. 2004; Woosley et al. 2007). This model has been widely investigated in 1D numerical simulation. In such a system a dense, geometrically thin shell forms and the light curve peak exceeds the value L ∼ 10⁴⁵ erg s⁻¹ (Chevalier & Irwin 2011; Moriya et al. 2013; Sorokina et al. 2016; Moriya et al. 2018). In a multidimensional case, the shell may be unstable and hence luminosity may change. The paper by Badjin et al. (2016) presents multidimensional simulations of this dense shell in the approximation of free-streaming radiation, for which only local properties of the medium matter. It was shown that the dense shell is unstable due to the Rayleigh–Taylor-like instability. Accounting for more accurate radiation transport can alter this result for several reasons. First, the formation of the dense shell is governed by radiative cooling when the matter is optically thin. For optically thick matter, another mode appears: the radiation-dominated Sedov stage. In SN, the thin mode follows after the thick one, so the final solution for the shock wave is unknown beforehand. Second, nonlocal radiation effects can change rates of instability growth and result in the stabilization of the shell. This is precisely why we investigate the shock-interacting scenario with the simulation of radiation hydrodynamics.

The front code (an updated version of the code front3d Glazyrin 2013) is developed in particular to be applied for multidimensional simulations of SLSN. The code has been recently upgraded with a radiation transfer module. Here we present several radiation tests that show the pros and cons of the M1 radiation transfer approximation. We use the M1 approximation for SLSN simulations and reproduce light curves that are usually calculated with the reliable and well-tested one-dimensional radiation code stella (Blinnikov et al. 2006).

This paper is organized as follows. In Section 2, we provide tests for the radiation module in front. In Section 3, we present simulations of SLSN for 1D and 2D cases.

Simulation of radiation transfer is numerically expensive; the equation for intensity is 7D—besides spatial coordinates and time, it contains beam directions and photon energies. Alongside the nontrivial physics of radiation–matter interaction, this leads to extremely high requirements on computational resources in many practical problems. In practice, simplified models are used, which are based on lower-dimensional equations; the original equation is integrated over angles, and sometimes over energy. As a result, the information about individual beams is lost (as well as the information on spectral properties for energy-integrated equation); the radiation field is described by integral characteristics like radiation energy density, radiation flux, etc. The simplest group of models (radiation diffusion) is based on the conjecture that radiation is nearly isotropic, which is a good approximation for small radiation mean free paths. A more advanced model, M1 approximation, also takes into account another regime—beam propagation. The model uses some reasonable interpolation for the Eddington tensor that continuously switches between diffusion and free-streaming regimes. Therefore, the model can be directly used to simulate problems with variable opacities in the medium: e.g., it is commonly used in astrophysics to simulate SN light curves, neutrino leakage, etc. But due to the simplifications in radiation field description in these models, two different beams interact with each other even without active radiation–matter interaction, which is absent when solving the original radiation transport equations.

Equations of radiation transport derived in M1 approximation are hyperbolic and physically close to hydrodynamics, so radiation is considered as a sort of fluid. The benefit of such an approach is that the equations can be easily incorporated into hydro codes. On the other hand, their solutions have some peculiarities that relate to hydrodynamics rather than radiation transport, as discussed below. This limits the applicability of the model for some physical problems.

The basic test suites used for verification of most radiation–hydrodynamics codes cannot reveal all the features of the various methods for implementing radiation transfer. In this paper, we present some additional tests that are designed to show the peculiarities of the M1 approximation of radiation transfer. SN light curve calculations present a class of radiation problems with outgoing rays. Thus, we pay special attention to problems with rays and discuss ray propagation in the context of the M1 model.

The equation of radiation transfer for intensity $I\left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)$ at frequency ν, direction n , moment of time t, and point x may be taken in the form (Mihalas & Mihalas 1984)

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{c}\displaystyle \frac{\partial I\left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)}{\partial t}+{\boldsymbol{n}}\cdot {\rm{\nabla }}I\left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)=\eta \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)\\ -(\kappa \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)+\sigma \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right))I\left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right),\end{array}\end{eqnarray} \tag{ 1 }$$

where ∇ is the nabla operator, c the speed of light, $\kappa \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)$ the absorption coefficient, $\sigma \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)$ the scattering coefficient, and $\eta \left({\boldsymbol{x}},t;{\boldsymbol{n}},\nu \right)$ the emission, which is divisible into thermal, η^T , and scattering, η^S , parts. After integrating Equation (1) and also integrating Equation (1) multiplied by the n -vector over the solid angle Ω and over the frequency, one may obtain the following moment equations (Mihalas & Auer 2001; Krumholz et al. 2007):

$$\begin{eqnarray}&&\displaystyle \frac{\partial U}{\partial t}+{\partial }_{i}{F}_{i}=-{{cG}}^{0},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{\partial {F}_{i}}{\partial t}+{\partial }_{j}{P}_{{ij}}=-{G}_{i},\end{eqnarray} \tag{ 3 }$$

where U, F, and P_ij are the frequency-integrated radiation energy density, flux, and pressure tensor, respectively:

$$\begin{eqnarray}&&U=\displaystyle \frac{1}{c}{\int }_{0}^{\infty }d\nu \oint {Id}{\rm{\Omega }},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{F}_{i}={\int }_{0}^{\infty }d\nu \oint {n}_{i}{Id}{\rm{\Omega }},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{P}_{{ij}}=\displaystyle \frac{1}{c}{\int }_{0}^{\infty }d\nu \oint {n}_{i}{n}_{j}{Id}{\rm{\Omega }},\end{eqnarray} \tag{ 6 }$$

and the right-hand side terms:

$$\begin{eqnarray}&&{G}^{0}=\displaystyle \frac{1}{c}{\int }_{0}^{\infty }d\nu \oint \left(\eta -(\kappa +\sigma )I\right)d{\rm{\Omega }},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{G}_{i}=\displaystyle \frac{1}{c}{\int }_{0}^{\infty }d\nu \oint {n}_{i}\left(\eta -(\kappa +\sigma )I\right)d{\rm{\Omega }}.\end{eqnarray} \tag{ 8 }$$

The set of Equations (2) and (3) can be coupled with the nonrelativistic hydrodynamics in the following way (Krumholz et al. 2007):

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\partial }_{i}\left(\rho {v}_{i}\right)=0,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\rho {v}_{i}\right)+{\partial }_{j}\left(\rho {v}_{i}{v}_{j}+p{\delta }_{{ij}}\right)={G}_{i},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\rho e+\rho \displaystyle \frac{{v}^{2}}{2}\right)+{\partial }_{i}\left(\left(\rho e+\rho \displaystyle \frac{{v}^{2}}{2}+p\right){v}_{i}\right)={{cG}}^{0},\end{eqnarray} \tag{ 11 }$$

where ρ, v, p, and e are the gas density, velocity, pressure, and internal energy, respectively, and δ_ij is the Kronecker delta.

The medium moves with velocity v, and all values in Equation (1) are measured in the lab (rest) frame, whereas we assume that both the absorption and scattering coefficients, as well as emission (both thermal and scattering), are isotropic in the comoving frame (which moves with the velocity v relative to the lab frame). We also assume that scattering is coherent in the comoving frame. Consequently, we must make the Lorentz transformation of the right-hand side part of the Equation (1) (Mihalas & Mihalas 1984). The G⁰ and G_i terms also require this transformation. In many situations, only terms to the order of v/c must be kept in G⁰ and G_i (Mihalas & Klein 1982), but there are some situations (e.g., nonequilibrium dynamic diffusion) in which the v²/c² terms are necessary (Krumholz et al. 2007). Moreover, the expressions for the frequency-integrated G⁰ and G_i valid to all orders in v/c are available (Mihalas & Auer 2001).

Here we use the expressions for G⁰ and G_i to the second order in v/c, and we will later demonstrate the situation where the second order is necessary. Moreover, we consider only the gray approximation where the absorption and scattering coefficients in the comoving frame are independent of frequency, so ${\kappa }_{0}\left(\nu \right)=\kappa =1/{l}_{\mathrm{abs}}$ and ${\sigma }_{0}\left(\nu \right)=\sigma =1/{l}_{\mathrm{scat}}$, where l is the free path and the subscript 0 means that the value is measured in the comoving frame. In this case, the expressions for G⁰ and G_i are the following (Krumholz et al. 2007):

$$\begin{eqnarray}\begin{array}{rcl}{G}^{0} & = & \kappa \left(U-{{aT}}^{4}\right)+\left(\sigma -\kappa \right)\displaystyle \frac{{v}_{i}{F}_{i}}{{c}^{2}}\\ & & +\ \displaystyle \frac{1}{2}\displaystyle \frac{{v}^{2}}{{c}^{2}}\left[\kappa \left(U-{{aT}}^{4}\right)-2\sigma U\right]-\sigma \displaystyle \frac{{v}_{i}{v}_{j}{P}_{{ij}}}{{c}^{2}},\end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\begin{array}{l}{G}_{i}=\left(\kappa +\sigma \right)\displaystyle \frac{{F}_{i}}{c}+\kappa \displaystyle \frac{{v}_{i}}{c}\left(U-{{aT}}^{4}\right)-\displaystyle \frac{\left(\kappa +\sigma \right)}{c}\\ \left({v}_{i}U+{v}_{j}{P}_{{ij}}\right)+\displaystyle \frac{1}{2}\displaystyle \frac{{v}^{2}}{{c}^{2}}\left(\kappa +\sigma \right)\displaystyle \frac{{F}_{i}}{c}+2\sigma \displaystyle \frac{{v}_{j}{F}_{j}{v}_{i}}{{c}^{3}},\end{array}\end{eqnarray} \tag{ 13 }$$

where T is the temperature of the gas and $a=8{\pi }^{5}{k}_{{\rm{B}}}^{4}/15{h}^{3}{c}^{3}$ is the radiation constant (k_B is the Boltzmann constant and h the Planck constant).

The set of Equations (2) and (3) is exact, but not closed—it requires knowledge about P_ij . If the tensor P_ij for some problem is known from above, the solution to this problem will be exact. The main idea of the M1 model is to prescribe a general relationship between P_ij , U, and F_i , and also to close the system (Levermore 1984; Dubroca & Feugeas 1999):

$$\begin{eqnarray}&&{P}_{{ij}}={D}_{{ij}}U,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{D}_{{ij}}=\displaystyle \frac{1-\xi }{2}{\delta }_{{ij}}+\displaystyle \frac{3\xi -1}{2}{n}_{i}{n}_{j},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\xi =\displaystyle \frac{3+4{f}^{2}}{5+2\sqrt{4-3{f}^{2}}},\quad f=\displaystyle \frac{| F| }{{cU}},\quad {n}_{i}=\displaystyle \frac{{F}_{i}}{| F| }.\end{eqnarray} \tag{ 16 }$$

The variant of M1 approximation described above is implemented in our astrophysical code front. We will calculate the following comparison test suite on three numerical codes: front, heracles (González et al. 2007), and shdom (Tominaga et al. 2015). The heracles code also uses the M1 approach, but there the set of Equations (2) and (3) is considered in the comoving frame and hence the realization is completely different. The shdom code, which is based on the original SHDOM algorithm (spherical harmonic discrete ordinate method) (Evans 1998; Pincus & Evans 2009), has the most sophisticated radiation transfer among these three codes. It solves time-dependent radiative transfer equations with angle-dependent intensity and scattering using spherical harmonics expansion. The SHDOM method allows one to correctly solve the radiation transfer equation in situations with highly anisotropic intensity and scattering and to save more information about the radiation field. However, this method has its own disadvantages. First of all, of course, it is much more computationally expensive than such methods as the M1 model or the diffusive approximation. It also has some radiation dispersion because of the intensity interpolation in the ray-tracing procedure (the short characteristics method is adopted). Furthermore, a propagating wave front is always smeared over time, because of the implementation of time dependence.

2.1. Radiating Sphere Test

In this test, similar to tests discussed in Strömgren (1939) and Spitzer (1998), we consider the typical structure when a hot, optically thick sphere radiates energy into the surrounding optically thin area. This situation is similar to that in SN when there is an outgoing flux from the optically thick medium, so it helps to check the flux from the surface. In a region with axial symmetry of dimensions L × L, L = 10¹⁰ cm, a sphere with radius R = 2 × 10⁹ cm is defined. Its center is located on the z-axis (z₁ = 0 cm). Inside the sphere, the free path l = 2 × 10⁸ cm (a medium with pure absorption is used), and hence the optical depth τ = R/l = 10 (optically thick region) and the matter temperature T_m = 3000 K (and hence the ${U}_{p}={{aT}}_{m}^{4}$), outside: l = 10¹² cm (τ = L/l = 0.01 in the optically thin region) and T_m = 10 K. The matter heat capacity is assumed to be infinite, so the matter temperature does not change during the simulation. In addition, we assume that the matter has constant zero velocity during the simulation. The radiation temperature T_r (which is defined from the blackbody approximation $U={{aT}}_{r}^{4}$) at the initial moment is equal to that of the matter: T_r = T_m . We use only the radiation module without calculating the hydrodynamics, which solves only the set of Equations (2) and (3) with all the terms in Equations (12) and (13). The thermal emission (aT⁴) term outside the sphere is negligible due to the small matter temperature, so it does not significantly change the total radiation flow. The outer boundary conditions are free (zero gradients) and the inner boundary is reflective (mirror).

For this setup, we may obtain the analytical solution for radiation flux, if we assume the free path l = 0 inside and l → ∞ outside the sphere:

$$\begin{eqnarray}F(r,z,t)=\left\{\begin{array}{ll}0, & \quad \mathrm{if}\,\sqrt{{r}^{2}+{z}^{2}}\geqslant R+{ct},\\ \sigma {T}^{4}\left[1-{\left(\frac{{r}^{2}+{z}^{2}+{\left({ct}\right)}^{2}-{R}^{2}}{2\sqrt{{r}^{2}+{z}^{2}}{ct}}\right)}^{2}\right], & \quad \mathrm{if}\,R+{ct}\gt \sqrt{{r}^{2}+{z}^{2}}\geqslant \sqrt{{R}^{2}+{\left({ct}\right)}^{2}},\\ \sigma {T}^{4}\frac{{R}^{2}}{{r}^{2}+{z}^{2}}, & \quad \mathrm{if}\,\sqrt{{R}^{2}+{\left({ct}\right)}^{2}}\gt \sqrt{{r}^{2}+{z}^{2}}\geqslant R,\\ 0, & \quad \mathrm{if}\,R\gt r,\end{array}\right.\end{eqnarray} \tag{ 17 }$$

where T = 3000 K and $\sigma =2{\pi }^{5}{k}_{{\rm{B}}}^{4}/15{h}^{3}{c}^{2}$ is the Stefan–Boltzmann constant. Because the radiation flux rapidly declines (generally ∼$1/\left({r}^{2}+{z}^{2}\right))$, the luminosity $L=4\pi \left({r}^{2}+{z}^{2}\right)| F|$ seems to be an educational quantity to compare. In the region between the ${R}_{1}=\sqrt{{R}^{2}+{({ct})}^{2}}$ and R₂ = R + ct, the L should decline. On the other hand, the luminosity should remain constant in the region between R and R₁.

The simulation results at time t = 0.2 s are presented in Figure 1. For all simulations in this article, we use rectangular and uniform grids. Grid parameters are presented in Table 1, as well as the relative timings for each code and time steps used (which were determined from Courant–Friedrichs–Lewy conditions and fixed for each calculation). All simulations were carried out on a workstation with a 2.8 GHz Intel Core i7 processor.

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Table 1. The Parameters of the Simulations

Simulation	Code	Geometry	Grid Resolution	Time Step	Simulation Time	CPU Time
Sphere test	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.2 s	1.7 hr
Sphere test	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	6.5 minutes
Sphere test	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	5.4 minutes
Beam test (absorption)	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.3 s	2.6 hr
Beam test (absorption)	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	9 minutes
Beam test (absorption)	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	2.1 minutes
Beam test (scattering ω = 0.5)	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.3 s	2.7 hr
Beam test (scattering ω = 0.5)	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	9 minutes
Beam test (scattering ω = 0.5)	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	2.1 minutes
Beam test (scattering ω = 1)	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.3 s	3 hr
Beam test (scattering ω = 1)	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	9 minutes
Beam test (scattering ω = 1)	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.3 s	2.1 minutes
Two sphere test	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.2 s	1.7 hr
Two sphere test	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	6.1 minutes
Two sphere test	front, FLD	RZ-cylindrical	2562	1.5 × 10−4 s	0.2 s	5.7 minutes
Two sphere test	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	5.3 minutes
Shadow test	shdom	RZ-cylindrical	2562	6.3 × 10−4 s	0.2 s	1.7 hr
Shadow test	front	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	6 minutes
Shadow test	front, FLD	RZ-cylindrical	2562	1.5 × 10−4 s	0.2 s	5.8 minutes
Shadow test	heracles	RZ-cylindrical	2562	3.9 × 10−4 s	0.2 s	7.5 minutes
Steady RS Ma = 2	front	1D Cartesian	512	3.9 × 10−16 s	10−10 s	3.5 hr
Steady RS Ma = 5	front	1D Cartesian	512	7.8 × 10−16 s	10−10 s	1.8 hr
Subcritical RS	front	1D Cartesian	512	1.3 × 10−3 s	3.8 × 104 s	36.6 hr
Supercritical RS	front	1D Cartesian	512	1.3 × 10−3 s	1.3 × 104 s	12.6 hr
Radiation advection test	front	1D Cartesian	256	2.6 × 10−14 s	10−10 s	6 s
SLSN	stella	R-spherical	Lagr 358	adaptive	60 days	3 hr
SLSN	front	R-spherical	8192	23.2 s	60 days	11.6 hr
SLSN	front	R-spherical	1024	181.6 s	100 days	11.4 minutes
SLSN	front	RZ-cylindrical	10242	95.8 s	100 days	604 hr

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The numerical solutions are generally in agreement with the analytical one: the luminosity between the radiating sphere and ${R}_{1}=\sqrt{{R}^{2}+{({ct})}^{2}}$ is approximately constant, outside R₂ it is equal to zero, and between R₁ and R₂ = R + ct it more or less follows the analytical solution. The outer edge of radiation flow is less smeared in front than in heracles and shdom due to the employment of the Godunov method with Riemann solvers for the treatment of radiation transfer. In this approach, the radiation propagates like the fluid flow with a rather sharp border. On the other hand, this leads to more pronounced filament-like numerical structures in the optically thin region in front than in heracles, which is more diffusive. The discussion of smearing will also be applicable for the tests described below.

2.2. Light Beam Test in the Absorptive Medium

This is a typical radiation test with beam propagation that has been widely discussed in, e.g., Richling et al. (2001) and Takahashi et al. (2013). In a region with axial symmetry (along the angle ϕ) of dimensions L × L, L = 10¹⁰ cm, the upward radiation flux is specified on the left half of the axis r. The shdom code solves the equation for intensity I, so the boundary conditions for the beam in the shdom simulation is the intensity inside the simulation domain. We set the incoming ($\cos \theta \geqslant 0$) intensity on the lower boundary of the z-axis: ${I}_{b}={I}_{0}\delta (\cos \theta )$, where the θ is the angle between the direction of the beam and the z-axis, δ—the Dirac delta function, and I₀ = 3 × 10¹⁰ erg cm⁻² s. The intensity outside the domain is calculated in the simulation and depends on the properties of the medium. First, we assume a medium with pure absorption. The free path l = 10¹⁰ cm across the whole simulation domain, so the optical depth of the entire region τ = L/l = 1. Thus, the intensity remains nonzero only along the z-axis, and so F = cU = I₀. In the front and heracles simulations, we set U and the upward F = cU at the left half of the r-axis for the beam. Other boundary conditions are free. In addition, we set the low matter temperature T_m = 10 K (the matter heat capacity is set to infinity) in the whole domain to avoid the cells with zero energy. The radiation temperature at the initial moment is equal to the matter temperature. We also assume that the matter has zero velocity, which does not change during the simulation. Here, we use only the radiation module without calculating the hydrodynamics.

The simulation results at time t = 0.3 s are presented in Figure 2.

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The beam front is less smoothed in front than in heracles and shdom. Nevertheless, the solutions from the different codes are in a good agreement.

2.3. Light Beam Test in the Medium with Scattering

The test under consideration can be expanded for the case of a medium with scattering. The setup, including the boundary conditions, is similar to that described in Section 2.2 aside from the opacity. Let us suppose that the absorption and scattering free paths in the whole simulation domain are equal: l_scat = l_abs = l = 2 × 10¹⁰ cm (so τ_scat = τ_abs = L/l = 1/2, and the total optical depth of the entire region τ = τ_scat + τ_abs = 1, hence the albedo ω = τ_scat/τ = 0.5).

The simulation results for the absorption and isotropic scattering case at time t = 0.3 s are presented in Figure 3.

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An obvious difference observed in the simulation with M1 approximation is the overheated region that is formed parallel to the axis r. Because such a region is observed both in front and in heracles, which use radically different numerical schemes for solving M1-approximation equations, we can see that this defect is not a numerical bug but inherent for the M1 approximation.

Moreover, the overheated region forms if we consider a pure isotropic scattering medium with no true absorption (l_scat = 10¹⁰ cm, l_abs → ∞). The simulation results at time t = 0.3 s are presented in Figure 4.

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Due to the scattering, the intensity in directions with cos θ < 0 becomes nonzero. Therefore, the radiation energy density that is proportional to the zero moment of the intensity will be greater than U₀ = I₀/c = 1 erg cm⁻³ near the base of the beam. This increase should grow with the growing albedo. It is clearly seen from the simulations that the radiation energy density near the base of the beam in shdom calculations with ω = 0.5 is greater than U₀ and is still greater in calculations with ω = 1. Moreover, the increase of the radiation energy density a little farther from the base of the beam should be even greater in the scattering media with high albedo; see, e.g., Barichello & Siewert (2000), where the classical searchlight problem by Chandrasekhar (1958) is investigated. We denote this increase as some kind of overheating, although this is just an enhancement of radiation energy density.

Undoubtedly, some differences should be related to different boundary conditions of the beam. It is nontrivial to set the U and F values on the boundary because the incoming flux should be consistent with scattered radiation (the M1 approximation uses the flux integrated over all angles, so it includes both incoming and outgoing rays). For a more detailed study of the overheating effect, we could change the setup in such a way that the boundary condition impact is minimized. This can be done by inserting a true absorption layer with large optical depth between the boundary and the scattering layer. In that case, scattered rays going back toward the boundary will be attenuated and the boundary flux should contain only incoming rays. The following 1D setup with a slab geometry implements this idea. On the left boundary of the region [0, 3x₀], where x₀ = 10¹⁰ cm, we set the incoming intensity ${I}_{b}={I}_{0}\delta (\cos \theta )$, where I₀ = 3 × 10¹⁰ erg cm⁻² s (so for the front calculations with the M1 approach, U₀ = I₀/c and F₀ = I₀). The whole region is divided into three layers [0, x₀], [x₀, 2x₀], and [2x₀, 3x₀], and we set a pure absorption medium for the first and third regions. In the middle region, there is scattering (we consider a different albedo ω = τ_scat/τ = 0.2, 0.5, 0.8, 1), and the effective free path for all regions is l = 5 × 10⁹ cm (so the total optical depth of each region is τ = x₀/l = 2). The other parameters of the simulation are the same as in the test described above. We use the 1D implementation of the discrete ordinate method (Warsa 2002). In addition, we use Monte Carlo simulations based on Whitney et al. (2013) as a more physically motivated simulation.

Here we pay attention only to stationary solutions (t = 100 s for front). The profiles of the solutions are presented in Figure 5.

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The profiles calculated by the discrete ordinate method and Monte Carlo do coincide, which provides a means to use them as the reference solution. For cases with low albedo (ω = 0.2, 0.5), there is no overheated region, and for those with high albedo (ω = 0.8, 1), the increase in the radiation energy near x/x₀ = 1 becomes more and more noticeable. The M1 approach qualitatively reproduces the appearance of this overheated region correctly (especially for ω = 1), but the form and the peak value of radiation energy density are wrong. The detailed consideration of this "overheating" effect deserves a separate investigation. The overheating due to beam collision will be clearly demonstrated in the following test.

2.4. Two Radiating Spheres Test

In order to clarify the collision effect in the M1 model, we present the two-spheres test. Let us consider the radiation between two heated spheres; with such a setup, we certainly can reproduce the collision of radiation fluxes (Stone 2015). This test is similar to the radiating sphere test in Section 2.1, but now there is another sphere whose center is located on the z-axis (z₂ = 7 × 10⁹ cm).

The analytical solution for this setup could be obtained with a sum of two solutions for each individual sphere (we again assume that the free path l = 0 inside and l → ∞ outside the sphere):

$$\begin{eqnarray}\begin{array}{c}{U}_{1,2}(r,z,t)=\left\{\begin{array}{ll}0, & \,{\rm{if}}\sqrt{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}}\geqslant R+{ct},\\ \displaystyle \frac{{{aT}}^{4}}{2}\left(1-\displaystyle \frac{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}+{\left({ct}\right)}^{2}-{R}^{2}}{2\sqrt{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}}{ct}}\right), & \,{\rm{if}}R+{ct}\gt \sqrt{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}}\geqslant \sqrt{{R}^{2}+{\left({ct}\right)}^{2}},\\ \displaystyle \frac{{{aT}}^{4}}{2}\left(1-\sqrt{1-\displaystyle \frac{{R}^{2}}{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}}}\right), & \,{\rm{if}}\sqrt{{R}^{2}+{\left({ct}\right)}^{2}}\gt \sqrt{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}}\geqslant R,\\ {{aT}}^{4}, & \,{\rm{if}}R\gt \sqrt{{r}^{2}+{\left(z-{z}_{1,2}\right)}^{2}},\end{array}\right.\end{array}\end{eqnarray} \tag{ 18 }$$

where T = 3000 K. To find the total radiation energy density U (and hence the radiation temperature T_r = (U/a)^1/4), we also have to take into account that the radiation energy density inside both spheres U_s = aT⁴ remains constant:

$$\begin{eqnarray}&&U(r,z,t)=\max \left\{{U}_{1}+{U}_{2},{{aT}}^{4}\right\}.\end{eqnarray} \tag{ 19 }$$

The simulation results at time t = 0.2 s are presented in Figure 6.

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This test directly provides the configuration of the radiation flux collision, which leads to the appearance of the overheated region between the two spheres due to the fluid-like behavior of radiation in the M1 approach. This effect is almost the same as in the beam-crossing problem, e.g., Weih et al. (2020). The results from shdom are in good agreement with the analytical solution. It is important to note that the analytical solution was derived under the assumption that spheres radiate independently, so the state of each sphere is not affected by the other sphere. In the setup used here, this is done by defining the heat capacity of the radiating medium to be infinite.

Such an overheating effect is absent in the more primitive (compared to the M1 approximation) flux-limited diffusion (FLD) approximation, where radiation is treated as a diffusive substance, not fluid-like. As a result, no overheated region appears, which is clearly seen in the simulations with FLD approximation in front. Here we use FLD in the whole simulation domain, both for optically thick and thin regions. The application of FLD approximation is not suited for optically thin regions. Our simulation only checks the appearance of the radiation energy spike (overheated region).

This peculiarity of the M1 approximation should be taken into account in multidimensional studies. For example, in the problem of dense shell instability in SLSN, which is considered before the shell breaks into a number of clumps, these clumps act as separate radiation sources, which could lead to a similar structure. Interpreting the results of the test, we can state that the possible appearance of overheated regions between clumps in a disrupted dense shell can have an artificial nature. Similar behavior of the numerical solution could be seen in other radiation problems, in which several sources of radiation are present.

2.5. Shadow Test

In real astrophysical objects, the structure of the medium can be complex, and the large optical depth regions can alternate with small optical depth ones. If the radiation flow propagating through a non-absorptive medium falls on a localized region with high absorption, a shadow should appear behind the region. To test this behavior, we use a modification of the radiating sphere test (Section 2.1) with an additional, optically thick sphere. This sphere has l = 10⁷ cm, R_s = 10⁹ cm (τ = R_s /l = 100), and T = 10 K, and its center is located near the center of the medium (r = 4 × 10⁹ cm, z = 4 × 10⁹ cm). Therefore, in this test we consider a region with large optical depth, which is surrounded by an optically thin medium like in Hayes & Norman (2003) and Jiang et al. (2012).

The simulation results at time t = 0.2 s are presented in Figure 7.

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Both the M1 model and the discrete ordinate method reproduce the shadow. Blurring of the shadow borders depends on the numerical realization of the model and the numerical diffusion of the scheme used. The outer edge of the radiation wave in front is less blurred than in heracles, and there are also filament-like structures in the optically thin region, just as in the test in Section 2.1. The solution using M1 approximation is good enough compared to shdom. For comparison we present the solution by FLD approximation in front, where the radiation diffuses too strongly behind the obstacle due to the nearly isotropic propagation approximation, and hence the shadow region does not form.

2.6. Radiation Shock Waves

To verify the front code for simulation of SLSN, we consider some standard tests of shock waves with radiation. Such tests are related to the regime of shock wave propagation in optically thick matter during the SLSN explosion. The structure of the front of these shock waves differs from the pure hydrodynamics case because radiation leads to a redistribution of energy between gas layers. Radiation transfers energy from the hot gas behind the shock wave to the incoming gas ahead of the shock and preheats it. Depending on the efficiency of the preheating (which in turn depends on the strength of the shock wave), two types of radiation shock waves can be distinguished: subcritical and supercritical. If the temperature immediately ahead of the shock front (inflow temperature) is less than the final temperature after the shock (behind the front and behind the relaxation region), then the shock wave is called subcritical. If these temperatures are equal, it is the critical case. Further strengthening of the shock wave cannot lead to the case where inflow temperature exceeds the temperature behind the front (because then the second law of thermodynamics would be violated); instead, energy is spent on heating thicker layers ahead of the front. The theory of such shock waves is considered in detail in Zeldovich & Raizer (1968).

We investigate the structure of the shock front and use the initial setup of steady radiative shock waves (Roth & Kasen 2015), which is based on semianalytical solutions by Lowrie & Edwards (2008). The strength of the shock wave depends on the Mach number Ma, so we consider two cases: Ma = 2 for a subcritical shock wave and Ma = 5 for a supercritical shock wave. The simulation results are presented in the figures below; Figure 8 corresponds to subcritical shock, and Figure 9 corresponds to supercritical shock.

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The solution obtained with M1 approximation is in a good agreement with the semianalytical solution.

Now we investigate the process of shock front formation. To do this, we consider the hydrodynamic flow over matter in which a small gradient of radiation temperature (and hence the radiation flux) is set. Reflection from one of the walls causes the propagation of a radiation shock wave. The same setup for the nonsteady case was used in Ensman (1994). In this test, we use both the hydrodynamics and radiation modules of front. The simulation results are presented in the figures below; Figure 10 corresponds to subcritical shock, and Figure 11 corresponds to supercritical shock.

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The structure of the shock wave coincides with the theory (Zeldovich & Raizer 1968). In the gas temperature, there is a peak called the Zeldovich peak. In the subcritical case, the gas temperature differs from the radiation one, but in the supercritical case, they are in equilibrium and begin to differ far away, behind the front. Also, the temperature immediately ahead of the shock front is equal to the final temperature after the shock, as expected.

The front structure in SLSN is supercritical, so this test validates the front code for this regime.

2.7. Radiation Advection Test

In such astrophysical objects as SN, the velocity of the medium can be large, so the numerical code has to carefully consider the transfer of radiation in a moving medium. To test the front code under these conditions, we investigate the evolution of the Gaussian profile of the radiation ${U}_{0}{(x,t=0)={U}_{g}\exp (-(x-{x}_{0})}^{2}/{a}_{g}^{2})$, (the profile is set in the comoving frame, U_g = 10¹¹ erg cm⁻³, x₀ = 1 cm, and a = 0.125 cm) in a pure scattering optically thick medium with free path l = 0.01 cm (τ = L/l = 200, where L = 2 cm is the size of the full 1D simulation domain [0, L]), moving with velocity v = 2 × 10⁹ cm s⁻¹, which does not change during the simulation (Roth & Kasen 2015).

We assume that the medium has a very low temperature, which does not change (infinite heat capacity), so the diffusion–advection equation gives the analytical solution for the radiation energy density in the comoving frame:

$$\begin{eqnarray}&&{U}_{0}(x,t)=\displaystyle \frac{{U}_{g}}{\sqrt{1+4{Dt}/{a}_{g}^{2}}}\exp \left(-\displaystyle \frac{{\left(x-({x}_{0}+{vt})\right)}^{2}}{{a}_{g}^{2}+4{Dt}}\right),\end{eqnarray} \tag{ 20 }$$

where D = cl/3. Then, we can transform this profile into the lab frame, assuming that because the medium is optically thick, the radiation flux in the comoving frame F₀ ≈ −cl∂_x P_0,00 and P_0,ij ≈ U₀ δ_ij /3 (Krumholz et al. 2007):

$$\begin{eqnarray}&&U={U}_{0}+2\displaystyle \frac{{v}_{i}{F}_{0,i}}{{c}^{2}}+\displaystyle \frac{1}{{c}^{2}}\left({v}^{2}{E}_{0}+{v}_{i}{v}_{j}{P}_{0,{ij}}\right).\end{eqnarray} \tag{ 21 }$$

This setup is an example of the nonequilibrium dynamic diffusion regime, and hence the v²/c² terms in Equation (12) for S⁰ are of the same order as the vF term (Krumholz et al. 2007). Here we use only the radiation module without calculating the hydrodynamics. We use two different simulations: the first one with only v/c terms, and the second one with both v/c and v²/c² terms on the right-hand sides of the radiation transfer equation to highlight the crucial importance of the v²/c² terms in such a regime. The simulation results are presented in Figure 12.

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The propagation and the smearing of the Gaussian profile of the radiation is correctly described by the M1 approximation with the treatment of the v²/c² terms (the profiles coincide with the analytical solution). If we consider only v/c terms, the numerical profiles deviate drastically from the analytical solution. This test verifies the front code for simulations with a large velocity of the medium.

The kinetic energy of expanding ejecta for typical SN exceeds the energy that is radiated. The addition of any mechanism that can effectively dissipate and radiate the kinetic energy is a way to significantly increase the luminosity. The feasible mechanism for this is the propagation of a radiative shock through an extended star's envelope. Stellar evolution theory gives several variants for the creation of such envelopes: extensive stellar winds prior to stellar explosions, etc. In this paper, we do not pay attention to the formation mechanism of such pre-SN environments and use some simple analytical expressions to set such dense envelopes in our simulations—see Figure 13. The envelope is constructed from a hydrostatic core, which has a polytropic structure with pressure–density relation p ∝ ρ^1.4, mass 1 M_⊙ (M_⊙ is the solar mass,) and radius 4 × 10¹⁴ cm. For r > 4 × 10¹⁴ cm, the density is assumed to have a power-law "wind" structure ρ ∝ r^−1.8, and the temperature is fixed as T = 3 × 10³ K as in Sorokina et al.(2016).

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Advanced 1D simulations of SLSN aim to reproduce fine details of real SLSN light curves (Blinnikov et al. 2006; Sorokina et al. 2016). Most of the radiated energy goes from the dense shell, which is formed during the interaction of a shock wave with the surrounding cloud. Multidimensional effects can play a significant role because in some cases the shell is unstable: it bends and disrupts (as was seen in Badjin et al. 2016), so the emitted radiation can change. To study this effect in multiple dimensions, we simplify the physical model from Sorokina et al. (2016) to reduce the computational cost. Nevertheless, such a simplified model correctly reproduces the effects of radiative shock formation and propagation, which is sufficient to study the question of stability. First, we ignore possible chemical composition stratification, which appears due to nucleosynthesis in SN explosions. The ejecta is uniformly filled with carbon/oxygen with the admixture of metals, as in the paper by Sorokina et al. (2016). The average atomic mass is μ = 1.75. Second, the full ionized ideal plasma equation of state with adiabatic index γ = 5/3 is used. Radiative properties of matter are set with constant pure absorption opacity κ = 0.2 cm² g⁻¹, which corresponds to Thomson scattering in fully ionized hydrogen-free plasma.

The general feature of the SLSN mechanism considered here is that the physical reason for the initial explosion energy (the central engine) does not play a key role; it leads to the formation of a strong shock, but does not influence the stage of its propagation in the cloud at which it radiates. Thus, we use a simple thermal bomb with an energy of a typical ordinary (but strong) SN to start the dynamics: a small region in the center is heated with E = 4 × 10⁵¹ erg during t = 10 s (this time is small on the light curve timescale).

The initial stage of SN dynamics, where a shock wave and general flow are formed, is simulated with stella code. This stage needs to account for self-gravity and is well simulated by the stella code; the results show that the contribution of gravitational energy to the total energy remains small after t_i. At some moment (t_i = 2.7 days) profiles of all hydrodynamic quantities are exported to be used as initial conditions for the front code, while the stella simulation continues. The front code does not account for gravity, which can be ignored after t_i . As far as stella having a more sophisticated radiation transport model, we can use its results to additionally verify the front results.

The comparison of 1D spherical simulations between front and stella is presented in Figures 14 and 15.

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The figures show the SN structure for moments when the dense shell is already formed; the shock wave is followed by a density peak, where the compression factor significantly exceeds the value (γ + 1)/(γ − 1) for the ordinary adiabatic strong shock wave. Such a structure is maintained for a long time (several days). Results for both codes are consistent with each other. Diversity is seen only on the temperature profiles, which are the most sensitive to radiation transport details. stella simulations show a faster preheat of the surrounding medium at early times, but the difference diminishes later. The post-shock temperature and density (r < r_{shock wave}) do not differ between simulations. The patterns of the shell dynamics also coincide.

Density in the post-shock region decreases, which signifies that more and more matter is involved in the motions around the shell; almost the entire mass of the cloud is concentrated in a geometrically thin layer. In stella, a Lagrangian numerical scheme is implemented, which contains an additional artificial viscosity that smears the front of the shock wave. The front code is based on the Eulerian scheme, and the front is smeared with the internal numerical viscosity of the scheme. The dense shell can be unstable in a multidimensional case (Badjin et al. 2016; see also Chevalier & Blondin 1995), which is a physical reason for shell smearing and spreading. In stella, a special term is implemented to control the thickness of the shell. This term is used to take into account multidimensional effects (shell spreading) in 1D simulations. It works in addition to the commonly known von Neumann artificial viscosity and has a similar form, but the total amount of kinetic energy is conserved. This term is proportional to the B_q parameter, which determines the strength of the shell smearing (a higher value of B_q corresponds to more powerful smearing). The exact value of B_q is quite uncertain and can be determined from comparison with multidimensional calculations. As a result, a thicker shell is less efficient at converting kinetic energy to radiation (Blinnikov et al. 1998). In our simulations, we use a default value B_q = 1 in stella during all simulation times, and it is clear from Figure 14 that the shell in stella is more smeared and propagates a little bit faster than in front after t ∼ 25 days. It is important to note that the Lagrangian mesh can move, while the Eulerian mesh is fixed, which is why the principal perturbation could flow out the border of the Eulerian mesh. To work around this problem, we have to add a low-density region in front that does not change the dynamics of the system. This region corresponds to the strong drop in the density profile near the outer boundary.

One of the important questions in the dynamics of SLSN is the location of the photosphere, as it directly affects the light curve. The location of the photosphere remains almost static until at least t ∼ 60 days and coincides in both codes. A detailed discussion about the role of the photosphere, as well as its accurate definition, will be given below.

The light curves generally agree (Figure 15) between the two codes. It is important that the peak value coincides in both simulations and exceeds 10⁴⁵ erg s⁻¹, which means that the model reproduces SLSN. It is important to note that these peaks were synchronized between the two numerical models. In front, the peak is slightly later. This shift is due to the finite speed of light; Eulerian simulation has an additional low-density region and radiation needs time to cross it. This region is optically thin, so it does not influence the absolute value of the flux. Such an effect is present because the light curve is reconstructed from flux values at the mesh boundary. The later decline (i.e., the tail of the light curve) is faster in stella than in front due to the additional smearing by B_q , which is one of the parameters of the SN model. This smearing affects only the tail, and the reduction of B_q leads to a slower decline of the light curve (Moriya et al. 2013), so the tail in stella will be more consistent with front if we use a lower value of B_q .

At the next step, after the 1D simulations, we present the 2D results from front using RZ-cylindrical geometry (with axial symmetry). Initial conditions are set in the same way as for the 1D simulations. During multidimensional dynamics, the spherical symmetry of the shell breaks down. For the light curve, the crucial question is this moment of breakdown and its influence on radiation flux; in many studies the light curve is calculated with a spherically symmetric approximation. The simulation results are presented in the figures below; Figure 16 corresponds to t = 10 days, Figure 17 corresponds to t =50 days, and Figure 18 corresponds to t = 100 days.

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The symmetry starts to break down at t_p ≈ 50 days, when perturbations become evident on the shell's radial scale. These perturbations continue to grow during the subsequent expansion of the shell. The instability has the same nature as thoroughly discussed in Badjin et al. (2016) for SN remnants. Shell perturbation in the region near the Z-axis is the most notable and has a numerical origin—it is known as a "carbuncle" (see, e.g., Liou 2000). This perturbation is localized only near the axis and does not influence other parts of the shell.

Though the shell loses spherical symmetry, the general direction of its propagation remains radial. The radiation flux also preserves its radial component as the dominant one in F . Therefore, the effect of overheating due to radiation flux collision (discussed in Section 2.4), inherent to the M1 model, is negligible in this simulation of SLSN. This conclusion may be incorrect for later stages of SN remnant evolution, when the shell breaks into separate clumps, but these stages are not considered here. Also, other scenarios of SN explosion that contain configurations with an initial nonspherical dense environment, like a dense equatorial disk (Margutti et al. 2019; Leung et al. 2020), should be considered using the M1 model with precautions, keeping in mind the effect of nonphysical overheating.

On average (for 2D cylindrical geometry, we take mean values on spherical radii R = (r² + z²)^1/2) the dynamics of the dense shell center of mass coincides in 1D and 2D simulations—see Figure 19. For t ≲ 10 days the density distribution contains a peak that is geometrically "thick," which corresponds to the Sedov–Taylor stage. For t ≳ 10 days, the dense shell is formed due to the process of extreme cooling: most of the matter is gathered in a geometrically thin region. The shell in 2D simulations is thicker due to the averaging procedure, and the small peak that appears at t = 25 days has a numerical nature—it is a trace of the "carbuncle" perturbation. Further, for t > t_p , the shell structure is governed by the instability: it is gradually spreading. As discussed above, this effect is taken into account into the 1D stella simulation by the B_q parameter.

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The simulations presented above allow us to calculate light curves and compare them for different geometries. The M1 model of radiation transport that we use includes radiation flux F as a dynamical field. In 1D simulations we can use the flux value on the outer boundary of the mesh F_B to calculate the instantaneous luminosity: $L=4\pi {R}_{{\rm{B}}}^{2}{F}_{{\rm{B}}}$, where R_B is the outer radius of the mesh. We will denote this method as "direct 1D." For 2D simulations, the application of such a method is not so trivial; one should integrate the flux over the rectangular-shaped boundary, taking into account the angular distribution of radiation at each point. In this paper we use a simpler approach—a photosphere method—and we limit our light curve calculation to t < t_p , when the solution is close to the spherically symmetrical one. This method integrates an optical depth $\tau (R)={\int }_{R}^{\infty }\kappa (x)\rho (x){dx}$ from the boundary to the center of the star over some radial direction (the same approach was used in Suzuki & Maeda 2017). As far as we consider time moments for which the solution is close to spherical, the optical depth function τ(R) should be independent of the chosen direction. The photosphere position is defined as τ(R_ph) = 2/3 (Sobolev 1969). It is important to note that this definition comes from the assumption of a geometrically thin atmosphere. For the different cases, the value τ(R_ph) varies, but remains of the order of unity (Mihalas 1978; Baschek et al. 1991). This difference is small for our simple model; because of constant absorption opacity, the optical depth grows very fast with the distance from the boundary to the center due to the power law for the density, which for t < 50 days is almost the same as at the initial moment. By definition, R_ph is a radius from which photons escape to infinity, and therefore, it determines the value of the luminosity; we assume that at this point the radiation has an isotropic blackbody spectrum, and outer layers R > R_ph do not absorb or scatter the radiation. Thus, the total luminosity is $L=4\pi \sigma {T}_{\mathrm{ph}}^{4}{R}_{\mathrm{ph}}^{2}$, where σ is the Stefan–Boltzmann constant, and T_ph is the radiation temperature at the photosphere point. For convenience of calculation, and to exclude numerical axis effects, we integrate the depth over the diagonal in the 2D simulations (r = z in cylindrical coordinates).

For 1D simulations we apply both direct and photosphere methods, and for 2D simulations only the photosphere method. Despite the difference in methods, the light curves coincide well with each other—see Figure 20. The difference in the peak value of luminosity is about 14% between the direct method and photosphere one for 1D and is about 2% for the photosphere method for 1D and 2D. The difference in the value at t = 50 days is about 14% and 7%, respectively. The peaks obtained by different methods were also synchronized in time. From Figure 14, it is clearly seen that the photosphere location is closer to the center than the outer boundary, and it takes some time for light to propagate through this distance.

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The important result is that for our simplified model of SLSN, the increasing parts of the light curve coincide for the 1D and 2D simulations. This is due to the fact that the peak value of the light curve has been achieved before the moment when multidimensional instability develops, so the solution has spherical symmetry and is fully described even in 1D. In addition, as it is clearly seen from Figure 14, the photosphere location is far away from the shell and hence the parameters on it do not change due to possible perturbations. The light curve is determined by the temperature of the photosphere, and the temperature depends on the power of the preheating from the radiation that is born on the shell that is optically thin. In our simple model, even if there are small perturbations on the shell, the temperature has time to level out due to its long propagation through the purely absorptive medium with large optical depth between the shell and the photosphere. These conclusions, though achieved for a realistic model, are very important for the direct method of measuring cosmological distances, which is based precisely on the growth period of the SLSN luminosity (Baklanov et al. 2013) and relies on the parameters of the photosphere (especially on the velocity). In a model with more sophisticated κ, the outer cold matter can be optically thin, and hence the photosphere will be close to the thin layer, also due to the scattering. On top of this, the thin layer could lose its spherical symmetry long before this, which would change the parameters of the shell and hence of the photosphere.

This paper considers the shock-interacting mechanism of SLSN formation. This mechanism requires accurate simulation of radiation–matter interaction, especially in several dimensions. In order to study these objects, we develop our radiation–hydrodynamics code, front, which uses M1 approximation for radiation transfer. We have tested our code on several commonly used radiation–hydrodynamics problems and by direct comparison with more accurate simulations using shdom code. In addition, we have presented some new tests that uncover a peculiarity of the M1 approximation: the appearance of unphysical overheated regions. We describe configurations that lead to such regions in numerical solutions.

We have also presented simulations of SLSN. We use the scenario where the exploded star is surrounded by a dense cloud. The radiative shock running over the cloud forms a dense, geometrically thin shell that effectively radiates away its kinetic energy. As a result, the luminosity of the star grows by a factor of ∼100. This scenario has been thoroughly studied in a 1D approximation with the stella code, which implements a sophisticated radiation transport model. For multidimensional studies, we have presented a simplified problem setup that reproduces shell formation and general dynamics. It is studied with both stella and front and gives consistent results. The setup is relevant for the SLSN scenario with a dense cloud: the peak value of the light curve in our simulations exceeds 10⁴⁵ erg s⁻¹ at t ≈ 15 days after explosion.

The simplified SLSN setup allows us to study the multidimensional dynamics of the shell. The shell becomes unstable and loses its spherical symmetry for t > 50 days in our model, but this does not influence the light curve at early epochs t < 50 days. We have calculated the light curve with two different methods, and this allows us to conclude that the parameters on the photosphere, which define the light curve, do not differ in the 1D and 2D cases. Within this model, we have validated the direct method of measuring cosmological distances (Baklanov et al. 2013), but this conclusion is not general because it is based on a simplified problem setup. Simulations with more detailed physics may shift the moment of instability domination as well as the photosphere location, and the instability can influence the light curve at earlier epochs. The detailed investigation of these multidimensional effects will be the subject of our following studies.

The authors are grateful to the anonymous referee for important suggestions and to Feodor Blinnikov for careful reading and correcting the manuscript. This research has been supported in part by the RFBR-JSPS bilateral program (19-52-50014) and the Grant-in-Aid for Scientific Research of the JSPS (15H05440). S.G. and D.S. are partly supported by the RSF 19-12-00229 grant in the development of physical models in front. E.U. is grateful to the RSF 21-11-00362 grant supporting the design of the parallel version of M1 radiation transport in the front code. S.B. acknowledges the support of the RSF 19-12-00229 project in SN light curve modeling.