High order positivity-preserving discontinuous Galerkin methods for radiative transfer equations

Transcription

1 High order positivity-preserving discontinuous Galerkin methods for radiative transfer equations Daming Yuan 1, Juan Cheng 2 and Chi-Wang Shu 3 Abstract The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. n the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes, however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric flexibility, suitability for h- and p-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. n this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in [33] and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes. Keywords: positivity-preserving; high order accuracy; radiative transfer equation; discontinuous Galerkin (DG) scheme; discrete-ordinate method 1 ntroduction The radiative transfer equation describes the interaction of photons with a scattering and absorbing background medium, which has wide applications in many areas such as astrophysics, inertial confinement fusion, optical molecular imaging, shielding, infrared and visible light in space and the atmosphere, just to name a few. The radiative transfer equation is an integro-differential equation with six independent variables for a three spatial dimensional and time dependent problem. The high dimensionality and the presence of integral coupling terms bring a serious challenge to solve the equation numerically. 1 nstitute of Applied Physics and Computational Mathematics, Beijing 194, China. damingyuan915@163.com. Research is supported in part by NSFC grant Additional support is provided by China Postdoctoral Science Foundation 214M nstitute of Applied Physics and Computational Mathematics, Beijing 194, China. cheng juan@iapcm.ac.cn. Research is supported in part by NSFC grants and Division of Applied Mathematics, Brown University, Providence, R shu@dam.brown.edu. Research is supported in part by AFOSR grant F and NSF grant DMS

2 Over the past few decades, several techniques for solving this kind of equations have been introduced, which include the Monte Carlo method, the discrete-ordinate method (DOM), the spherical harmonics method, the spectral method, the finite difference method, the finite volume method and the finite element method. Among these methods, the discrete-ordinate method has received particular attention in the literature due to its relatively high accuracy, flexibility, and relatively low computational cost. The discrete-ordinate method discretizes the solid angle with a set of ordinate directions. The integration over the solid angle that appears in the radiative transfer equation is evaluated by means of a weighted summation over the ordinate directions (numerical quadrature), where the specified weights are determined through algebraic and geometrical relationships [1, 8]. n this paper, we focus on the discrete-ordinate discontinuous Galerkin method for solving the radiative transfer equation, which is among the most flexible numerical methods in discrete-ordinate formulations for the radiative transfer equation. The discontinuous Galerkin (DG) finite element method was first introduced by Reed and Hill [27] in 1973 to solve the steady linear neutron transport (radiative transfer) equation, in which the numerical solution is allowed to be discontinuous across the cell boundaries. This feature makes the DG finite element method a local method, that is, it is possible to construct a set of small linear systems approximating the governing equation in each cell to avoid assembling and solving a large, global linear system. Soon after, in [15], theoretical properties of the DG method including stability and error estimates were provided. Later, Cockburn et al. [5, 4, 3, 6] established a framework to easily solve nonlinear time dependent problems, such as the compressible Euler equations of gas dynamics, using explicit, nonlinearly stable high order Runge-Kutta time discretizations [29] and DG discretization in space with exact or approximate Riemann solvers as interface fluxes and total variation bounded (TVB) nonlinear limiters [28] to achieve non-oscillatory properties for strong shocks. The DG method has many advantages such as high order accuracy, geometric flexibility, suitability for h- and p-adaptivity, extremely local data structure, high parallel efficiency and a good theoretical foundation for stability and error estimates. t is particularly powerful for convectiondominated problems, in which the solutions develop discontinuities or sharp fronts. The DG method has been widely used in many convection-dominated equations such as neutron/photon (radiative) transfer equation studied in this paper, Euler and Navier-Stokes equations for compressible gas dynamics, shallow water equations, KDV equations and so on. n the transport community, second order DG method using piecewise linear polynomials has been employed predominantly to solve the discrete-ordinate transfer equation. Starting from the pioneering work [27] mentioned above, a piecewise linear function representation was used for three-dimensional unstructured tetrahedral meshes in [32, 24] and a trilinear representation was used for three-dimensional hexahedral meshes in [32]. n the neutron/photon transport area, limited research has been carried out using elements of higher polynomial degree, which include DG method 2

3 up to order 4 developed for the steady transport equation by using hierarchical basis functions in [3, 21] and the quadratic DG method used for the neutron transport in spherical geometry [17, 23]. Robustness of numerical methods has attracted an increasing interest in the community of computational science. One mathematical aspect of robustness for numerical methods is the positivitypreserving property. t is known that under certain conditions, which are satisfied by almost all physical problems, the discrete-ordinate radiative transfer equations have nonnegative solutions whenever the source terms and the boundary conditions (and, for time-dependent problems, also the initial conditions) are nonnegative [7, 18]. For a good numerical method, it should ideally also yield a nonnegative solution. Especially in multidimensional problems, the appearance of negative solution could slow the convergence rate of the iterative processes, and sometimes may also cause a complete failure of convergence of the acceleration procedure. For time dependent problems, negative solution may lead to numerical instabilities. Furthermore, negative radiative intensity is a physically unrealistic solution which is difficult to be accepted by physicists. A scheme for the radiative transfer equation is called positivity-preserving if it can always produce nonnegative solution for nonnegative source term and boundary condition (and, for time dependent problems, also nonnegative initial condition). n this paper we use the word positivity loosely which is the same as nonnegativeness. Several studies exist in the literature on this issue, with various ways of ensuring positive intensities being proposed. The step scheme, which is the counterpart of the upwind scheme in computational fluid dynamics, is proved to be positivity-preserving but is only first order accurate and introduces excessive numerical smearing [2]. The diamond scheme reduces the numerical smearing, but negative intensities may appear. These negative intensities may be eliminated by using the negative intensity fix-up procedure, that is, setting them to zero. However, spatial oscillation and physically unrealistic intensities may still occur. The other existing positivity-preserving schemes include the variable-weight scheme which combines the step and the diamond schemes by a variable weight [1, 19], the linear exponential discontinuous finite-element method [31], the step and linear adaptive methods [22], the step characteristic scheme [12] and the linear characteristic scheme [11] which is nonnegative as long as the projected scattering source and projected outflow boundary fluxes remain positive which can be guaranteed by a rotational fix-up procedure. The positive intensities criteria for purely absorbing media is proposed by Fiveland in [9]. The linear discontinuous Galerkin finite-element method with the set-to-zero fix-up technique is proposed more recently in [2]. The procedures mentioned above are either only first or second order accurate, or use non-polynomial nonlinear procedures which require iterative procedures to obtain the solution even for the system inside each cell, or rely on the characteristic procedure and hence are difficult to be generalized to multi-dimensions. For solving convection-dominated equations, such as Euler equations of compressible gas dynamics, recently Zhang and Shu developed a general framework which relies on a simple scaling 3

4 limiter and can be applied to Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory (WENO) finite volume schemes of arbitrary order of accuracy on arbitrary meshes to ensure the positivity-preserving property without affecting the originally designed high order accuracy [33, 34, 35]. n this paper, we focus on designing a high order positivity-preserving DG method for solving the steady and unsteady discrete-ordinate radiative transfer equations in Cartesian coordinates. Differently from the explicit schemes for Euler equations and other convection dominated equations, the scheme we consider here is an implicit or iterative type, thus the above mentioned methodology of positivity-preserving scaling limiter proposed by Zhang and Shu can not be applied directly. n fact, if we adopt a similar positivity-preserving scaling limiter in the DG method for these radiative transfer equations, degeneracy of accuracy may happen for third and higher order schemes (see the Appendix of this paper). Here, instead, we develop a combined technique of the scaling positivity-preserving limiter and a rotational positivity-preserving limiter which can be used to solve the radiative transfer equations by implicit or iterative DG methods. This new limiter is simple to implement, does not affect convergence to weak solutions (Lax-Wendroff theorem), and can be theoretically proved to preserve positivity and to maintain the originally designed high order accuracy both in one and two spatial dimensions. One- and two-dimensional numerical tests for these positivity-preserving DG schemes are provided to demonstrate their effectiveness. An outline of the rest of this paper is as follows. n Section 2, we describe the radiative transfer equation and its DG discretization for the steady and unsteady discrete-ordinate radiative transfer equation. n Section 3, we discuss the methodology to construct positivity-preserving DG schemes for the radiative transfer equation in one spatial dimension. n Section 4, we present a positivitypreserving DG scheme in two spatial dimensions. n Section 5, numerical examples are given to demonstrate the good performance of these DG schemes. We give concluding remarks in Section 6. 2 The radiative transfer equation and its DG discretization 2.1 The radiative transfer equation The radiative transfer equation is the mathematical statement of the conservation of photons. The Eulerian derivation leads to the so-called integro-differential form of the radiative transfer equation. More details can be found in [25]. We first consider a steady-state, one-group, isotropically-scattering transfer equation Ω (r,ω) + σ t (r,ω) = σ s (r,ω)dω + q(r,ω) (2.1) 4π where (r,ω) is the radiative intensity in the direction Ω and the spatial position r, S is the unit sphere, σ s is the scattering coefficient of the medium, σ t is the extinction coefficient of the 4 S

5 medium due to both absorption and scattering (that is, σ t σ s ), and q(r,ω) is a given source term. For two spatial dimensional problems, the position vector r = (x,y) D R 2 and the vector Ω is usually described by a polar angle β measured with respect to a fixed axis in space and a corresponding azimuthal angle ϕ. f we introduce µ = cos β, we may denote dr = dxdy, dω = sin βdβdϕ = dµdϕ. To solve the radiative transfer equation numerically, we must discretize the spatial variables and the angular variables to obtain a system of simultaneous equations. n the discrete-ordinate method (DOM), the radiative transfer equation (2.1) is solved for a finite number of directions spanning the total solid angles of the unit sphere around a point in space, and integrals over solid angles are replaced by a numerical quadrature. For each discrete direction Ω m,l = (ζ m,η l ), m = 1,...,M, l = 1,...,L where M,L are the numbers of directions in ζ and η respectively where ζ = sin β cos ϕ = 1 µ 2 cos ϕ,η = sin β sinϕ = 1 µ 2 sinϕ. The equation (2.1) becomes a spatial differential equation which is written in Cartesian coordinates as ζ m m,l (r) x + η l m,l (r) y + σ t m,l (r) = σ s 4π m,l ω m,l m,l (r) + q(r,ζ m,η l ), (2.2) where m,l (r) = (r,ζ m,η l ) is the radiative intensity in the direction (ζ m,η l ), ω m,l is the quadrature weight with m,l ω m,l = 4π (in this paper we assume ω m,l > for all m,l, which is correct for all the quadratures that we use in the numerical tests), and S (r,ζ,η)dζdη m,l ω m,l (r,ζ m,η l ). n most applications of the DOM, S N or T N quadratures are used [13]. More details can be found in Section 5 when we give numerical examples The one-dimensional steady radiative transfer equation The steady transfer equation in one-dimensional planar geometry can be described as follows, µ (x,µ) x + σ t (x,µ) = σ s (x,µ)dµ + q(x,µ), a x b, 1 µ 1, (2.3) where (x, µ) is the radiative intensity in the direction µ and the spatial position x. The boundary condition for the equation (2.3) is specified as (a,µ) = l (µ), < µ 1; (b,µ) = r (µ), 1 µ < (2.4) where l and r are the prescribed radiative intensity on the left and the right boundaries, respectively. For each discrete direction m, one obtains a spatial differential equation as follows, µ m m (x) x + σ t m (x) = σ s 2 M m =1 ω m m (x) + q m (x), m = 1,...,M (2.5) 5

6 where M is the number of directions, µ m is the direction cosines along the x-coordinate of the direction m, ω m > is the quadrature weight with m ω m = 2 and m (x) = (x,µ m ) is the radiative intensity in the direction m (x,µ)dµ M m =1 ω m m (x) The one-dimensional unsteady radiative transfer equation We assume the range of the time variable as < t T, then the unsteady isotropically-scattering transport problem in planar geometry is described as follows, 1 (x,µ,t) c t +1 + µ (x,µ,t) x + σ t (x,µ,t) = σs 2 1 (x,µ,t)dµ + q(x,µ,t), a x b, 1 µ 1, < t T (2.6) where c is the speed of photon. For the above unsteady radiative transfer equation, we need to specify the boundary condition as (a,µ,t) = l (µ,t), < µ 1, t T; (b,µ,t) = r (µ,t), 1 µ <, t T (2.7) and the initial condition as (x,µ,) = (x,µ). (2.8) Similarly, the discrete-ordinate approximation for the unsteady radiative transfer equation in planar geometry can be written as 1 m (x,t) m (x,t) + µ m + σ t m (x,t) = σ s c t x 2 M m =1 ω m m (x,t) + q m (x,t), m = 1,...,M. (2.9) 2.2 The DG method for the discrete-ordinate radiative transfer equation n this paper, we employ the DG method to discretize the spatial variables of the discrete-ordinate radiative transfer equations. Here we first take the one-dimensional radiative transfer equation as an example to show the form of the DG discretization for this kind of equations. The specific form of the DG scheme for the two-dimensional radiative transfer equation will be given in Section 4. Without loss of generality, we denote S i = [x i 1/2,x i+1/2 ] (i = 1,,N x ) as a subdivision of [a,b] with a = x 1/2 < x 3/2 < < x Nx+1/2 = b, x i = x i+1/2 x i 1/2 and h = max 1 i Nx ( x i ). We define the finite-element space consisting of the following piecewise polynomials V k h = {h m(x) L 2 (a,b) : h m(x) Si = (x) P k (S i ), S i,i = 1,,N x } where P k (S i ) denotes the set of polynomials of degree up to k defined in the cell S i. t is noted that functions in V k h may be discontinuous across cell boundaries. Due to the discontinuous nature of the spatial approximation, functions h m(x) V k h are doublevalued at interior nodes (cell boundaries) x i+1/2 for i = 1,,N x 1. Consider a node x i+1/2 6

7 separating two cells S i and S i+1. For the convenience of the following discussion, we will use the notation (x) to denote the polynomial solution of h m inside the cell S i. The left and right values of h m (x) at the node x i+1/2 are therefore given by respectively. h m (x i+1/2 ) = (x i+1/2 ), h m (x+ i+1/2 ) = +1(x i+1/2 ), (2.1) The DG method for the one-dimensional steady radiative transfer equation We consider a given direction µ m, and only illustrate the case of µ m >, as a similar procedure can be repeated for µ m <. By applying the upwind principle to determine the numerical flux at the cell boundaries, the DG method for solving (2.5) is defined as follows: find the unique function m(x) h Vh k such that, for all the test functions bh (x) Vh k where bh (x) Si = b i (x) P k (S i ), S i,i = 1,,N x, we have S i ( µ m m h (x)(bh ) (x) + σ t m h (x)bh (x))dx + µ m m h (x i+1/2 )bh (x i+1/2 ) = σ s S i 2 φ i (x)b h (x)dx + S i q m (x)b h (x)dx + µ m m h (x i 1/2 )bh (x + i 1/2 ), i.e. where S i ( µ m (x)b i (x) + σ t (x)b i (x))dx + µ m (x i+1/2 )b i (x i+1/2 ) = σ s S i 2 φ i (x)b i (x)dx + S i q m (x)b i (x)dx + µ m 1 (x i 1/2 )b i (x i 1/2 ) φ i (x) = (2.11) M ω m m,i(x). (2.12) m = The DG method for the unsteady radiative transfer equation The DG method, with backward Euler time discretization, for solving the unsteady DOM transfer equation (2.9) is similar to the steady state scheme (2.11). When the n-th time step solution n (for all m = 1,,M and i = 1,,N x ) is known, we would like to find polynomials n+1 P k (S i ), for all m = 1,,M and i = 1,,N x, such that S i ( µ m n+1 (x)b i (x) + σ t n+1 (x)b i(x))dx + µ m n+1 (x i+1/2)b i (x i+1/2 ) = σ s S i 2 φ n+1 i (x)b i (x)dx + S i q (x)b i (x)dx + µ m 1 n+1 (x i 1/2)b i (x i 1/2 ) (2.13) where σ t = σ t + 1 c t n, q (x) = q m (x,t n+1 )+ 1 c t n n (x), and tn = t n+1 t n is the time step size. We use backward Euler in order to avoid the extreme constraint on the time step for explicit time stepping due to the high speed c. Of course, higher order implicit time stepping methods can also be used, but our discussion in this paper is restricted to first order backward Euler time stepping. 7

8 2.3 The solution algorithm for the DG method The discrete set of algebraic equations in the DOM-DG schemes such as (2.11) and (2.13) is usually solved by an iteration method in an optimal sweeping order. This is usually referred to as the grid sweeping algorithm. For a specific discrete direction, the optimal marching procedure starts from a cell located at a corner of the computational domain. We determine the corner where the calculation begins for each specific discrete direction by the sign of the direction cosines such as µ m for onedimensional problems and (ζ m,η l ) for two-dimensional problems under consideration in a way that the upstream cell boundaries lie on the boundary of the domain. For example, when ζ m >, η l >, the sweeping starts from the bottom-left corner cell, whose left and bottom cell boundaries coincide with the inflow boundary of the domain where the intensity function is prescribed. The discrete equations for all the remaining cells are solved successively in the direction of the orientation of the direction cosines, so that the intensities at the upstream boundaries of the cell we are computing can be obtained either from the boundary conditions or from the calculations performed in the previously computed cells. Without the coupling integral terms, this marching procedure provides the DG solution in all cells just in one sweep, which is a major advantage of the DG methods. For iterative methods used to solve the discretized transfer equation with the coupling integral terms, one of the widely used methods is the so-called source iteration (S) method [16], which is defined for solving the DG scheme (2.11)-(2.12) as follows: When the l-th iteration solution (l) (for all m = 1,,M and i = 1,,N x ) is known, we compute (l+1), for i = 1,,N x (in this order when µ > ), and for each fixed i, running though m = 1,,M to solve S i ( µ m (l+1) S i σ s 2 φ ( ) i (x)b i (x) + σ t (l+1) (x)b i (x))dx + µ m (l+1) (x i+1/2 )b i (x i+1/2 ) = (x)b i (x)dx + S i q m (x)b i (x)dx + µ m (l+1) 1 (x i 1/2)b i (x i 1/2 ) (2.14) with φ ( ) i (x) = m =1,,M ω m ( ) m,i(x) (2.15) where ( ) m,i(x) is taken as (l+1) m,i (x) if it is already available, otherwise it is taken as (l) m,i(x). Since (l+1) 1 (x i 1/2) (for i = 1 this is taken as the given boundary condition) and the other (l + 1)-th iteration solution needed on the right hand side of (2.14) have already been computed in the sweep, the S solver (2.14) is completely local in cell S i, thus can be very efficiently computed. The initial iteration values () can be determined arbitrarily (e.g. by the boundary conditions). The source iteration process continues until a prescribed convergence criterion is satisfied, in our numerical experiments this is taken as when the maximum residue is less than n the S method, each ordinate is solved independently while the couplings between different ordinates are deferred to the integral term involving φ ( ) i (x), which uses a mixture of information from both (l + 1)-th (when available) and l-th iterations. 8

9 Similarly, the S method solving the DG scheme for the unsteady radiative transfer equation (2.13) can be described as follows S i ( µ m n+1,(l+1) S i σ s 2 φ n+1,( ) i (x)b i (x) + σ t n+1,(l+1) (x)b i (x))dx + µ m n+1,(l+1) (x i+1/2 )b i (x i+1/2 ) = (x)b i (x)dx + S i q (x)b i (x)dx + µ m n+1,(l+1) 1 (x i 1/2 )b i (x i 1/2 ). (2.16) 3 High order positivity-preserving DG scheme for the discreteordinate radiative transfer equation in one spatial dimension Generally, higher order approximations for radiative intensity may provide more accurate solutions but artifacts might appear such as negativeness of the solutions. n this section, we first discuss how to design a high order positivity-preserving DG scheme for both the steady radiative transfer equation and the unsteady radiative transfer equation in one spatial dimension. n the next section, we will propose a high order positivity-preserving DG scheme for the two-dimensional radiative transfer equation. 3.1 High order positivity-preserving DG scheme for the one-dimensional steady discrete-ordinate radiative transfer equation We denote G i = {x i 1/2 = ˆx 1 i, ˆx2 i,, ˆxN 1 i, ˆx N i = x i+1/2 } as the N-point Gauss-Lobatto quadrature points in the cell S i and ŵ α > (α = 1,2,,N) as the corresponding quadrature weights, where N could be chosen as the smallest integer satisfying 2N 3 k. However, in this paper, in order to make the rotational limiter simpler, we choose N = k + 1 so that the k-th degree polynomial solution can be completely and uniquely determined by its values at these Gauss-Lobatto points. For a polynomial (x), denote by Ī its cell average in S i (from now on, we denote by f the cell average of the function f), then we have Ī = 1 (x)dx = x i S i N ŵ α (ˆx α i ). We aim to develop a high order positivity-preserving DG scheme for solving the discrete-ordinate steady radiative transfer equation (2.5). That is, if we know the source term, the values of at the domain boundary, { (l) (ˆx α i ), α, } and { (l+1) (ˆx α i ),α = 1,,N} in all the upstream cells of the cell S i are nonnegative, then we would like to limit the DG solution (l+1) (x) computed by (2.14) to obtain a new polynomial Î(l+1) (x) such that Î(l+1) (ˆx α i ) are nonnegative for all α = 1,,N. This of course also implies that the cell average of Î(x) is nonnegative. Furthermore, the limiting procedure should not affect the accuracy of the scheme, i.e., Î(l+1) (x) (l+1) (x) Ch k+1 when the exact solution is smooth. Here and below, C is a constant independent of h, which 9 α=1

10 may take different values in different locations. After the limited polynomial Î(x) is obtained, it will be relabeled as (l+1) (x) before moving to the next direction m + 1 or to the next cell, i.e. S i+1 for the case µ m >. Consider again the case of µ m >. f we take the test function b i (x) = 1 in (2.14), then we obtain σ t i.e., (l+1) S i σ t Ī (l+1) (x)dx + µ m (l+1) (x i+1/2 ) = σ s φ ( ) i (x)dx + q m (x)dx + µ m (l+1) 1 2 (x i 1/2), S i S i x i + µ m (l+1) (x i+1/2 ) = σ s 2 From the above assumption, we know q, ( ) φ i x i + q x i + µ m (l+1) 1 (x i 1/2). (3.1) φ ( ) i and (l+1) 1 (x i 1/2), then by (3.1) and the mean value theorem, we can deduce that the following convex combination (l+1) (ξ) = µ m µ m + σ t x i (l+1) (x i+ 1 2 ) + σ t x i µ m + σ t x i Ī (l+1), (3.2) where ξ [x i 1,x 2 i+ 1]. Thus it is also easy to see that at least one of (l+1) 2 nonnegative. (x i+ 1 2 ) and Ī(l+1) We remark that (3.2) states that the original DG solution obtained by (2.14), without being limited yet, is nonnegative at least at one point in the cell. This is crucial for the success of the limiter to be introduced later. n the work of Zhang and Shu [33, 35], the original DG solution has a nonnegative cell average, then a simple scaling limiter can be applied to bring the whole polynomial at the desired Gauss-Lobatto points to be nonnegative, without sacrificing the original high order accuracy. An obvious idea here would be to adopt a similar scaling limiter, which can be described as follows, where Î (l+1) λ = min{ (x) = λ( (l+1) (l+1) (ξ) (l+1) } (ξ) z i,1 (x) (l+1) (ξ)) + (l+1) (ξ), (3.3), z i = min α=1,...,n (,(l+1) (ˆx α i )). Then we can easily verify that Î(l+1) (ˆx α (l+1) i ),α = 1,,N and therefore also Î = 1 S i Î (l+1) (x)dx = N α=1 ŵαî(l+1) (ˆx α i ). This is exactly the scaling limiter used in Zhang and Shu [33, 35], with the cell average Ī(l+1) replaced by (l+1) (ξ). This appears to be just a small change, as both Ī(l+1) and (l+1) (ξ) are particular point values of the DG polynomial solution (l+1) (x) at different points inside the cell S i. t is proved in [33, 35] that the scaling limiter with Ī(l+1) maintains the original high order accuracy. Unfortunately, the same scaling limiter (3.3) with (l+1) (ξ) defined by (3.2) can only guarantee the original second order accuracy for the piecewise linear k = 1 case, but may lead to possible degeneracy of the original high order accuracy for k 2. For a more detailed discussion on this issue, we refer to the Appendix of this paper. 1 x i is

11 n order to keep the high order accuracy of the method as well as the positivity-preserving property of the radiative intensity, we adopt an alternative positivity-preserving limiter which will be illustrated in the following subsections. As shown above, at least one of Ī(l+1) and (l+1) (x i+ 1) is non-negative. The limiting strategy depends on which one is non-negative. f Ī(l+1), then the same scaling limiter as introduced in [33, 35] is employed which will be introduced in subsection 3.1.1, otherwise a rotational limiter is applied which will be described in subsection The scaling limiter f Ī(l+1) with, we apply the scaling limiter [33] to modify (l+1) (x) as follows λ = min { Î (l+1) (x) = λ( (l+1) Ī (l+1) Ī (l+1) } z i,1 (x) Ī(l+1) ) + Ī(l+1) (3.4), z i = min α=1,...,n (,(l+1) (ˆx α i )). (3.5) This scaling limiter can keep the original high order of accuracy of the unlimited polynomial, as proved in [33]. Here we only state the conclusion in the following proposition. Proposition 1. (Zhang and Shu [33]) Assume (l+1) (x) is a k-th degree polynomial defined on cell S i which approximates a smooth function (x) to (k + 1)-th order accuracy, and Ī(l+1), then the limited polynomial Î (l+1) (x) defined by (3.4) and (3.5) achieves positivity Î(l+1) (ˆx α i ) for α = 1,...,N and maintains the same (k + 1)-th order accuracy for approximating (x) The rotational limiter First, we recall a few notations about the rotational transformation. For simplicity of notations, we denote the end point x i+1/2 as x c and (x i+1/2 ) as c. Similarly, for any points x,x S i, the values of the radiative intensity at these points are denoted as and respectively. As shown in Figure 3.1, let the point P(x,) be rotated clockwise for an angle θ around the point C(x c, c ), which is called the center of rotation, and reach the point Q(x, ). We also denote AB as the line segment between the points A and B and AB as the Euclidean length of AB, respectively. The rotational transformation can be written as a vector multiplied by a matrix calculated from the angle θ as follows, [ x ] [ c = M x x c ] [ c + [ ] cos θ sinθ where the rotational matrix M is defined as. sinθ cos θ Suppose =, then it is easy to verify that the value of θ can be computed by the following formula x c ] 2 (3.6) θ = arccos 2a2 b 2 2a 2, (3.7) 11

12 θ C After rotating Q P Before rotating Figure 3.1: The rotational transformation where a 2 = (x c x) 2 + ( c ) 2 and b 2 = (x c a 2 c 2 x) Now, just like in the scaling limiter case, we assume (x) is a k-th degree polynomial defined in the cell S i which approximates a smooth function (x) to (k + 1)-th order accuracy, and (x i+1/2 ). We would like to obtain a limited polynomial Î(x) with rotation, such that Î (ˆx α i ) for α = 1,...,N, while maintaining the same (k + 1)-th order accuracy for approximating (x). For the convenience of description, we call ˆx α i to be a negative Gauss-Lobatto point if (ˆx α i ) <. The rotational limiter algorithm: 1. For each negative Gauss-Lobatto point ˆx α i in the cell S i, compute the rotation angle θ α by (3.7) so that the point (ˆx α i, (ˆx α i )) is rotated around (ˆxN i, (ˆx N i )) clockwise to reach the new point (ˆx α i,). f a particular ˆx α i is not a negative Gauss-Lobatto point, then we set θ α =. 2. Taking θ = max α=1,,n 1 θ α, we rotate the polynomial (x) to obtain Ǐ(x) by the rotational transformation (3.6) with θ = θ. Then it is easy to see that Ǐ(ˆx α i ) for all α = 1,...,N The final modified polynomial Î(x) is the interpolation polynomial at all the N = k + 1 Gauss-Lobatto points which are determined by Î(ˆx α i ) = Ǐ(ˆx α i ), α = 1,,N 1 and Î (ˆx N i ) = (ˆx N i ). 12

13 From the definition of the rotational limiter, we can clearly see that Î(ˆx α i ) for all α = 1,...,N. That is, Proposition 2. Î(ˆx α i ) is nonnegative for all α = 1,,N, i.e., the rotational limiter is positivitypreserving. Next we will show that the above described rotational limiter can maintain the original high order accuracy. First we introduce the following lemma. Lemma 1. Suppose the k-th degree polynomial h (x) is a (k + 1)-order accurate approximation of the smooth function (x) in the cell S i. As shown in Figure 3.2, assume the Gauss-Lobatto point P(x,) with < (here = h (x) is a short notation) in the interval AB rotates clockwise by angle θ ( PCQ = θ), around the point C(x c, c ) with c > to reach the point Q(x,). Suppose AB = h (A = x i 1/2, B = x i+1/2 ), then we have tan θ 2 Chk (3.8) Proof: Suppose the point O is the foot of the perpendicular projection of P to AB. We first show that OQ Ch k+1. Let x = ˆx α i be one of the Gauss-Lobatto points in the interval AB, then OB = x c x C 1 h, for example C 1.35 if N = 5. An essential observation is that BQ = CQ 2 CB 2 = CP 2 CB 2 = (x c x) 2 + ( c ) 2 c 2 and OQ = BQ OB, then 2 c OQ = (xc x) 2 + ( c ) 2 c 2 + (x c x). (3.9) Since c >, <, we have, for constants C, C 2 >, c c = h (x c ) h (x) h (x c ) (x c ) + (x c ) (x) + (x) h (x) C h k+1 + d dx (ξ)(x c x) C 2 h where ξ [x,x c ]. Also, since = h (x) < and (x), we have (x) = h (x) (x) C 3 h k+1 for some constant C 3. Therefore, the numerator of the coefficient to on the right side of (3.9) satisfies 2 c + 2 c C 4 h 13

14 A Q(x,) O C(x c, c) B R P(x,) Figure 3.2: Sketch for the rotation and the denominator of the coefficient to on the right side of (3.9) satisfies (xc x) 2 + ( c ) 2 2 c + (x c x) x c x C 1 h Hence the coefficient itself is bounded by a constant C 5, which, by (3.9), implies where C 6 = C 3 C 5. OQ C 6 h k+1 t remains to show that tan θ 2 Chk. Let the point R be the midpoint of PQ as shown in Figure 3.2. Since OQ C 6 h k+1 and PO = C 3 h k+1, we have PR C 7 h k+1 where C 7 = 1 2 (C3 ) 2 + (C 6 ) 2. Then where C = C 7 /C 1. This completes the proof. tan θ 2 = PR RC < PR OB C 7h k+1 C 1 h Chk, Theorem 1. Assume (x) is a k-th degree polynomial defined in the cell S i which approximates a smooth function (x) to (k + 1)-th order accuracy, and (x i+1/2 ), then the limited polynomial Î (x) defined through Ǐ(x) by the procedure above, where Ǐ(x) is obtained by rotating the polynomial (x) around the point C(ˆx N i, (ˆx N i )) clockwise by the angle θ described above, achieves positivity Î(ˆx α i ) for α = 1,...,N and maintains the same (k + 1)-th order accuracy for approximating (x). Proof: n the transformation (3.6), we take x = ˆx α i,x = ˆx α i, x c = ˆx N i [ ] [ ][ Ǐ (ˆx α i ) cos θ sin θ ˆx α = (ˆx α i ) (ˆx N i ) i sin θ cos θ ˆx α i ˆxN i After a simple manipulation, the above equation can be rewritten as follows and get, ] [ (ˆx + N i ) ˆx N i ]. Ǐ (ˆx α i ) (ˆx α i ) = cos θ ( (ˆx α i ) (ˆx α i ))+(cos θ 1)((ˆx α i ) (ˆx N i ))+sin θ (ˆx α i ˆx N i ), 14 (3.1)

15 ˆx α i ˆx α i = sinθ ( (ˆx α i ) (ˆx α i )) sin θ ((ˆx α i ) (ˆx N i ))+(cos θ 1)(ˆx α i ˆx N i ). (3.11) From the equality (3.11), we can obtain (ˆx α i ) (ˆx N i ) = 1 sin θ (ˆx α i ˆx α i ) ( (ˆx α i ) (ˆx α i )) + cos θ 1 sinθ (ˆx α i ˆx N i ). Substituting the above expression of (ˆx α i ) (ˆx N i ) into the equality (3.1), we obtain Ǐ (ˆx α i ) (ˆx α i ) = (ˆx α i ) (ˆx α i ) + 2tan θ 2 (ˆxα i ˆx N i ) + tan θ 2 (ˆx α i ˆx α i ). (3.12) By using the result of Lemma 1, it is straightforward to prove that Ǐ(x ) m (ˆx α i ) Chk+1. Since Î(ˆx α i ) = Ǐ(ˆx α i ), we also have Î(ˆx α i ) (ˆxα i ) Chk+1 for all α = 1,...,N, which implies that Î(x) approximates the function (x) with (k + 1)-th order accuracy in S i. This completes the proof. The easiest way to implement the rotational limiter is through the values of the limited polynomial Î(x) at the N = k + 1 Gauss-Lobatto points, as described above. This would involve a Lagrangian basis set (consisting of basis functions which achieve the value 1 at one Gauss-Lobatto point and at other Gauss-Lobatto points). f other basis functions are used, a change of coefficients under different basis sets is needed. We emphasize that neither the DG method itself nor the rotational limiter depends on the particular choice of basis functions for the implementation. We now summarize the limiting procedure to obtain a high order positivity-preserving scheme for solving (2.14) as follows. Here we assume that the values of radiative intensity at the boundary and the cell average of the extra source term q i are all positive. f (l) (ˆx α i ), α,i,m and (l+1) (ˆx α i ) for all α,i,m in the upstream cells, then we have φ ( ) and from (3.1)-(3.2) we know at least one of Ī(l+1) and (l+1) (x i+1/2 ) is nonnegative. Then, f Ī(l+1) to obtain Î(l+1), the scaling limiter (3.4)-(3.5) is employed to modify the DG polynomial (l+1) (x) (x); Otherwise, we must have (l+1) (x i+1/2 ), then the rotational limiter algorithm is applied on (l+1) (x) to obtain Î(l+1) (x). Remark 1. The procedure for the case of µ m < can be obtained symmetrically. Remark 2. Clearly, if the scaling limiter is used, the cell average of the DG polynomial is not changed, hence conservation is automatic. f the rotational limiter is used, the cell average is 15

16 changed (in fact, the rotational limiter is used only if Ī(l+1) <, while the cell average after limiting is nonnegative, hence the cell average must have changed). This would appear to be a problem to conservation. However, the crucial property which helps us is that the limited polynomial Î(l+1) (x) and the original polynomial (l+1) (x) share the same value at x i+1/2 (or at x i 1/2 for the µ m < case). Therefore, the difference between the two Riemann sums approximating the weak formulation S i (x)ψ x (x)dx with a smooth function ψ(x): D m = i Î (l+1) ψ x (x i ) x i i Ī (l+1) ψ x (x i ) x i, is bounded by D m = i A ( Î (l+1) Ī(l+1) )ψ x (x i ) x i Ch i A ( Ch ( Ch i A Î (l+1) Î (l+1) TV (Î(l+1) Î(l+1) (x i+1/2 ) + (l+1) (x i+1/2 ) Ī(l+1) Î(l+1) (x i+1/2 ) + i A ) ) + TV ( (l+1) ), (l+1) (x i+1/2 ) Ī(l+1) ) where A is the set of cells in which the rotational limiter is applied. Therefore, this difference goes to zero when the mesh size h, provided both (l+1) and Î(l+1) have bounded total variation. That is, if the numerical solution converges with bounded total variation towards a function, then is a weak solution of the original equation and will thus have the correct discontinuity location and strength. This is to say that our limited scheme satisfies the classical Lax-Wendroff theorem [14], which is the main purpose of using conservative schemes. Remark 3. We could actually also take the number of Gauss-Lobatto points N < k + 1 as long as 2N 3 k (this is possible when k 3) to save cost for the limiter. Positivity can still be achieved. The order of accuracy can be maintained when we take the limited polynomial Î (x) to interpolate Î(ˆx α i ) = Ǐ(ˆx α i ),α = 1,...,N 1 and Î(ˆx N i ) = (ˆx N i ), and to be closest to the original (x) in the L 2 -norm (least square) subject to such interpolation properties. For simplicity of presentation we do not pursue this route further in this paper. 3.2 High order positivity-preserving DG scheme for the unsteady radiative transfer equation The high order positivity-preserving DG scheme proposed in the previous subsection for the steady radiative transfer equation can be easily extended to the fully discrete unsteady radiative transfer equation with backward Euler time discretization. n fact, comparing equation (2.14) with equation (2.16), we find that they are the same except that σ t, q (x) and (l+1) (x) are replaced by σ t, q (x) 16

17 Figure 4.1: The inflow and outflow boundaries for the direction Ω m,l = (ζ m,η l ) with ζ m > and η l > in the rectangular cell S i,j. and n+1,(l+1) (x), respectively. Thus the same procedure can be applied, and we do not repeat the details. 4 High order positivity-preserving DG scheme for solving the radiative transfer equation in two spatial dimensions 4.1 The DG method for the steady radiative transfer equation in two spatial dimensions Consider the steady radiative transfer equation in two spatial dimensions (2.2) with the domain D = [a,b] [c,d] and the rectangular mesh of D = i=1,,nx,j=1,,n y S i,j with S i,j = [x i 1/2,x i+1/2 ] [y j 1/2,y j+1/2 ] as shown in Figure 4.1. For simplicity, we only illustrate how to implement the limiter in the direction (ζ m,η l ) with ζ m > and η l >, that is, its outflow boundary S + i,j = Γ out1 Γ out2 and inflow boundary S i,j = Γ in1 Γ in2 (see Figure 4.1) can be written as follows, Γ in1 = x i 1/2 [y j 1/2,y j+1/2 ], Γ in2 = [x i 1/2,x i+1/2 ] y j 1/2, Γ out1 = [x i 1/2,x i+1/2 ] y j+1/2, Γ out2 = x i+1/2 [y j 1/2,y j+1/2 ]. The implementation for the other three cases can be obtained symmetrically. The DOM equation of (2.2) solved by the source iteration method can be written as (l+1) m,l (r) ζ m x (l+1) m,l (r) + η l y + σ t (l+1) m,l (r) = σ s 4π φ( ) (r) + q(r,ζ m,η l ) (4.1) where φ ( ) (r) = M,L m,l =1 ω m,l ( ) m,l (r). Similarly as the case in one spatial dimension, ( ) m,l (r) is taken as (l+1) m,l (r) if it has already been obtained, otherwise, it is taken as (l) m,l (r). 17

18 The DG method for the equation (4.1) in a rectangular cell S i,j can be written as S i,j ( ζ m (l+1) m,l;i,j (x,y) b i,j(x,y) x η l (l+1) m,l;i,j (x,y) b i,j(x,y) y + σ t (l+1) m,l;i,j (x,y)b i,j(x,y))dxdy m,l;i,j (x,y)b i,j(x,y)dx + ζ m +η l Γ out1 (l+1) Γ out2 (l+1) m,l;i,j (x,y)b i,j(x,y)dy = S i,j σ s 4π φ( ) i,j (x,y)b i,j(x,y)dxdy + S i,j q m,l (x,y)b i,j (x,y)dxdy+ ζ m Γ in1 (l+1) m,l;i 1,j (x,y)b i,j(x,y)dy + η l Γ in2 (l+1) m,l;i,j 1 (x,y)b i,j(x,y)dx (4.2) where m,l;i,j (x,y) is the DG solution polynomial in the cell S i,j, and b i,j (x,y) is a test function. Both m,l;i,j (x,y) and b i,j (x,y) are polynomials of degree at most k. 4.2 The high order positivity-preserving DG scheme for the two-dimensional steady radiative transfer equation Taking b i,j (x,y) = 1, the DG method (4.2) gives σ t x j y j Ī (l+1) m,l;i,j + η l x i Ī (l+1) m,l;i,j (y j+1/2) + ζ m y j Ĩ (l+1) m,l;i,j (x i+1/2) = σ s 4π x i y φ( ) j i,j + x i y j q m,l;i,j + ζ m y j Ĩ (l+1) m,l;i 1,j (x i 1/2) + η l x i Ī (l+1) m,l;i,j 1 (y j 1/2) where, for any function p, we denote 1 p i,j = x i y j xi+1/2 yj+1/2 x i 1/2 y j 1/2 p i,j (x,y)dxdy, (4.3) p i,j (y j +1/2) = 1 xi+1/2 p i,j (x,y x j +1/2)dx, j = j 1,j, i x i 1/2 p i,j(x i +1/2) = 1 yj+1/2 p i y,j(x i +1/2,y)dy i = i 1,i. j y j 1/2 That is, we use ( ) to denote the cell averaging operator in the x-direction, ( ) to denote the cell averaging operator in the y-direction, and ( ) to denote the two dimensional cell averaging operator in the cell S i,j. n the cell S i,j, by the mean value theorem, there exists (ξ,ν) S i,j such that (l+1) m,l;i,j (ξ,ν) = σ t x i y j Ī (l+1) m,l;i,j + η l x i Ī (l+1) m,l;i,j (y j+1/2) + ζ m y j Ĩ (l+1) m,l;i,j (x i+1/2). (4.4) σ t x i y j + η l x i + ζ m y j Suppose the source term q m,l (x,y) and (l+1) m,l;i,j (x,y) at the domain boundary are nonnegative, and the values of the DG polynomials (l) m,l;i,j (x,y) and (l+1) m,l;i,j (x,y) in the upstream cells (which have already been updated) at the Gauss-Lobatto points are also nonnegative (which is achieved by using the positivity-preserving limiter described below in the upstream cells), then we know that φ ( ) i,j, q m,l;i,j, Ĩ(l+1) m,l;i 1,j (x i 1/2), Ī(l+1) m,l;i,j 1 (y j 1/2) are all nonnegative, thus we can see that (l+1) m,l;i,j (ξ,ν) (l+1) defined by (4.4) is nonnegative by (4.3), i.e., at least one term among Ī m,l;i,j, Ĩ(l+1) m,l;i,j (x i+1/2) and Ī (l+1) m,l;i,j (y j+1/2) is nonnegative. Again, we emphasize that this nonnegative result is a property of the DG scheme which is valid before the limiter is applied in the current cell S i,j. 18

19 We denote the Gauss-Lobatto points in the cell S i,j as G i,j = G i G j, where G i = {x i 1/2 = ˆx 1 i, ˆx2 i,, ˆxN 1 i, ˆx N i = x i+1/2 }, G j = {y j 1/2 = ŷj 1,ŷ2 j,,ŷn 1 j,ŷj N = y j+1/2 }. For convenience, we denote the Gauss-Lobatto points in G i,j as ˆr α 1,α 2 i,j = (ˆx α 1 i,ŷ α 2 j ). Next, in order to obtain a nonnegative solution, we will perform either the positivity-preserving scaling limiter or the positivity-preserving rotational limiter on m,l;i,j (x,y), depending on which is (l+1) nonnegative among Ī m,l;i,j, Ĩ(l+1) m,l;i,j (x i+1/2) and Ī(l+1) m,l;i,j (y j+1/2) The positivity-preserving scaling limiter in two spatial dimensions f Ī m,l;i,j is nonnegative, we will employ the scaling limiter proposed in [33], which can be described as Î (l+1) (l+1) (l+1) m,l;i,j (x,y) = λ((l+1) m,l;i,j (x,y) Ī m,l;i,j ) + Ī m,l;i,j (4.5) with Ī (l+1) λ = min m,l;i,j Ī (l+1) m,l;i,j z i,j,1, z i,j = min ˆr α 1,α 2 i,j G i,j ( (l+1) m,l;i,j (ˆrα 1,α 2 i,j ),). (4.6) Then for all ˆr α 1,α 2 i,j,α 1,α 2 = 1,..,N, it is easy to check that Îm,l;i,j(ˆr α 1,α 2 i,j ) is nonnegative. This scaling limiter maintains the original (k + 1)-th order accuracy, as proved in [33] The positivity-preserving rotational limiter in two spatial dimensions f Ī m,l;i,j is negative, then at least one of Ĩ(l+1) m,l;i,j (x i+1/2) and Ī(l+1) m,l;i,j (y j+1/2) should be nonnegative by (4.3). n this case, the limiting procedure consists of a one-dimensional scaling limiter on the relevant cell boundary followed by a two-dimensional rotational limiter around this cell boundary. For simplicity, we only illustrate how to implement the limiting procedure when Ĩ(l+1) m,l;i,j (x i+1/2). First we modify the polynomial m,l;i,j (x,y) Vh k(s i,j) as follows. At the right boundary of the cell x = x i+1/2, we apply the one dimensional scaling limiter to obtain Ǐ (l+1) m,l;i,j (x i+1/2,y) = λ( (l+1) m,l;i,j (x i+1/2,y) Ĩ(l+1) m,l;i,j (x i+1/2)) + Ĩ(l+1) m,l;i,j (x i+1/2) (4.7) where the parameter λ is determined as Ĩ (l+1) λ = min m,l;i,j (x i+1/2) Ĩ (l+1) m,l;i,j (x i+1/2) z i,j,1, z i,j = min ( m,l;i,j (x ŷ α 2 i+1/2,ŷ α 2 j ),). (4.8) j G j This determines the modified polynomial Ǐ(l+1) m,l;i,j (x,y) at the right boundary of the cell x = x i+1/2, which is positive at the Gauss-Lobatto points along this cell boundary Ǐ (l+1) m,l;i,j (x i+1/2,ŷ α 2 j ), α 2 = 1,2,,N. (4.9) 19

20 We then take the difference of Ǐ(l+1) m,l;i,j (x,y) and (l+1) m,l;i,j (x,y) at the Gauss-Lobatto points along the right boundary of the cell x = x i+1/2 : d α2 = Ǐ(l+1) m,l;i,j (x i+1/2,ŷ α 2 j ) (l+1) m,l;i,j (x i+1/2,ŷ α 2 j ), α 2 = 1,2,...,N. Clearly, we have d α2 = O(h k+1 ), α 2 = 1,2,...,N, (4.1) since the one-dimensional scaling limiter does not affect the order of accuracy [33]. We now modify the values of (l+1) m,l;i,j at the other Gauss-Lobatto points as Ǐ (l+1) m,l;i,j (ˆxα 1 i,ŷ α 2 j ) = (l+1) m,l;i,j (ˆxα 1 i,ŷ α 2 j ) + d α2, α 1 = 1,...,N; α 2 = 1,...,N. (4.11) Finally, the modified polynomial Ǐ(l+1) m,l;i,j (x,y) is determined by the unique interpolation polynomial in Q k satisfying (4.11). Clearly, the modified polynomial Ǐ (l+1) m,l;i,j (x,y) satisfies positivity at the Gauss-Lobatto points along the right boundary of the cell x = x i+1/2 (see (4.9)), and is O(h k+1 ) close to the original polynomial (l+1) m,l;i,j (x,y) (see (4.1)). To guarantee the positivity-preserving of the radiative intensity at all Gauss-Lobatto points ˆr α1,α2 i,j G i,j, we need to further apply the one-dimensional rotational limiter algorithm defined in subsection to Ǐ(l+1) m,l;i,j (x,y), along each line y = ŷα 2 j, as follows. The two-dimensional rotational limiter algorithm: 1. Take each point (ˆx N i,ŷα 2 j ), for α 2 = 1,,N, which lies on Γ out2, as the rotational center, and apply the one-dimensional rotational limiter algorithm along the line y = ŷ α 2 j, with rotational angle θ α 2 m,l;i,j, to obtain the modified values at all the Gauss-Lobatto points along this line and the modified one-dimensional polynomial Î(l+1) m,l;i,j (x,ŷα 2 j ). We then have Î (l+1) m,l;i,j (ˆxα 1 i,ŷ α 2 j ), α 1 = 1,,N. 2. The final limited polynomial Î (l+1) m,l;i,j (x,y) is the unique interpolation polynomial with the values at all Gauss-Lobatto points (ˆx α 1 i,ŷ α 2 j ) with α 1 = 1,,N, α 2 = 1,,N, as obtained in step 1 above. Remark 4. t is straightforward to prove that this limiter maintains the original high order accuracy, since we are only applying the one-dimensional rotational limiter along each line. The implementation of the positivity-preserving limiter is simple. Specifically the flowchart for the 2D positivity-preserving limiter is as follows. Again, we list the algorithm flowchart only for the case of ζ m >,η l >, as the other three cases can be obtained symmetrically. 2

21 1. f Ī m,l;i,j then perform the scaling limiter (4.5)-(4.6) on m,l;i,j (x,y) to obtain Îm,l;i,j(x,y); 2. else if Ĩ m,l;i,j (x i+1/2 ) then perform (4.7)-(4.8) and (4.11) on m,l;i,j (x,y) to obtain Ǐm,l;i,j(x,y) first, and then perform the two-dimensional rotational limiter algorithm to obtain Îm,l;i,j(x,y); 3. else if Īm,l;i,j(y j+1/2 ) then perform the similar procedure as the second case above with the role of x and y switched to obtain Îm,l;i,j(x,y). Finally the polynomial Î(l+1) m,l;i,j (x,y) is nonnegative at all Gauss-Lobatto points in the cell S i,j and hence the cell-average of Îm,l;i,j(x,y) is also nonnegative. The original high order accuracy is maintained. Remark 5. Similarly as for the one-dimensional high order positivity-preserving DG scheme, the two-dimensional high order positivity-preserving DG scheme proposed for the steady radiative transfer equation can also be easily extended to the unsteady radiative transfer equation. We do not repeat the details here. 5 Numerical results n this section, we perform numerical experiments in one- and two-dimensions to validate the properties of high order accuracy and positivity-preserving of our DG schemes. Regarding the discrete-ordinate quadrature rule, we adopt the Legendre-Chebyshev P N -T N quadrature [13] in which the µ-levels are equal to the roots of the Gauss-Legendre quadrature, and the azimuthal angles are determined from the roots of the orthogonal Chebyshev (T N ) polynomials. To be more specific, S 8 and P 8 -T 8 are used for all the following one-dimensional and the two-dimensional tests with non-zero scattering terms respectively, unless otherwise stated. Example 1. (The accuracy test of the DG schemes for the one-dimensional steady radiative transfer equation) n this test, we solve the absorbing-scattering radiative transfer problem described by the equation (2.3) with σ t = 1, σ s = 1, q(x,µ) = 4πµ 3 cos 3 πxsin πx + σ t (µ 2 cos 4 πx + a) σ s (a + cos4 πx 3 ). Here a = 1 14 is a small positive constant which is used to ensure the source term to be nonnegative. The computational domain is D = [,1]. The boundary condition is given as follows { (,µ) = µ 2 + a, if µ >, (1,µ) = µ 2 + a, if µ <. For this problem, we have the exact solution given as (x,µ) = µ 2 cos 4 πx + a. 21

22 .8 exact p1 4 without limiter p1 4 with limiter exact p1 4 without limiter p1 4 with limiter x Figure 5.1: The comparison of the radiative intensity simulated by the P 1 DG scheme without and with the positivity-preserving limiter on a 4 uniform grid. Left: The radiative intensity in the whole domain and directions; Right: The radiative intensity in the zoomed region in the direction µ =.963. x Figure 5.1 shows the comparison of simulated by the P 1 DG scheme without and with the positivity-preserving limiter using 4 uniform cells. From the figures, we can observe that negative values do occur in the solution of the P 1 DG scheme without the positivity-preserving limiter, while the P 1 DG scheme with the positivity-preserving limiter produces nonnegative results. The errors and orders of accuracy for the {P 1,P 2,P 3,P 4 } DG schemes without the positivity-preserving limiter and with the positivity-preserving limiter are shown in Tables respectively. n these tables, we also list the percentage of the cells where the positivity-preserving limiter is enacted during the computation. We can clearly see from these tables that the DG schemes with the positivity-preserving limiter can achieve the same designed order of accuracy as the DG schemes without the positivity-preserving limiter both in the L 2 and L norms, while the DG schemes with the positivity-preserving limiter can also keep the positivity of the radiative intensity. Table 5.1: Errors of the P 1 DG scheme for the 1D steady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.293E-2.279E-1.297E-2.279E E E E E E E E E E E E E Example 2. (The accuracy test of the DG schemes for the one-dimensional unsteady radiative transfer equation) 22

23 Table 5.2: Errors of the P 2 DG scheme for the 1D steady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.257E-3.251E-2.257E-3.251E E E E E E E E E E E E E Table 5.3: Errors of the P 3 DG scheme for the 1D steady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.21E-4.19E-3.313E-4.457E E E E E E E E E E E E E To simulate the one-dimensional unsteady transfer equation (2.9), the same domain and parameters σ t, σ s and a are taken as those in the previous example. The source term is given as q(x,µ,t) = 4πµ 2 cos 3 π(x + t)sin π(x + t)( 1 c + µ) + σ t(µ 2 cos 4 π(x + t) + a) σ s (a + cos4 π(x+t) 3 ). c = The initial condition is (x,µ,) = µ 2 cos 4 πx + a. The boundary conditions are given as { (,µ,t) = µ 2 cos 4 πt + a, if µ >, (1,µ,t) = µ 2 cos 4 π(1 + t) + a, if µ <. The exact solution for this model is (x,µ,t) = µ 2 cos 4 π(x + t) + a. The final computational time is t =.1. Since our DG schemes are designed implicitly, there is no limitation on the time step for the stability requirement. But as the time derivatives are discretized by the Euler backward time stepping in our DG schemes, the schemes are high order accurate in space and but only first order accurate in time. n order to verify the spatial accuracy of the DG schemes with our limiter, we choose a small time step t = 1 3 in order to make the spatial error dominate. For this problem, the DG schemes without the positivity-preserving limiter Table 5.4: Errors of the P 4 DG scheme for the 1D steady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.129E-5.125E-4.188E-5.199E E E E E E E E E E E E E

24 .8 exact p1 4 without limiter p1 4 with limiter.15 exact p1 4 without limiter p1 4 with limiter E x Figure 5.2: The comparison of the radiative intensity simulated by the P 1 DG scheme without and with the positivity-preserving limiter on a 4 uniform grid. Left: The radiative intensity in the whole domain and directions; Right: The radiative intensity in the zoomed region in the direction µ = x do produce negative results. Figure 5.2 shows the comparison of simulated by the P 1 DG scheme without and with the positivity-preserving limiter using 4 uniform cells. The errors and orders of accuracy for the {P 1,P 2,P 3,P 4 } DG schemes without and with the positivity-preserving limiter are shown in Tables respectively. We observe that the order of accuracy is maintained for the DG schemes after the application of the positivity-preserving limiter, as expected. Table 5.5: Errors of the P 1 DG scheme for the 1D unsteady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.293E-2.289E-1.297E-2.289E E E E E E E E E E E E E Example 3. (The accuracy test of the DG schemes for the two-dimensional steady radiative transfer equation simulating the purely absorbing model) n this test, we solve the two-dimensional steady radiative transfer equation (2.2) with σ t = 1,σ s =, q =. The computational domain is [,1] [,1]. ζ =.5,η =.1. The boundary condition is (x,) =, (,y) = sin 6 (πy). 24

25 Table 5.6: Errors of the P 2 DG scheme for the 1D unsteady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.257E-3.251E-2.257E-3.251E E E E E E E E E E E E E Table 5.7: Errors of the P 3 DG scheme for the 1D unsteady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.22E-4.212E-3.314E-4.457E E E E E E E E E E E E E n this case, the problem has the exact solution given as follows, (x,y) = {, y < η ζ x, sin 6 (π(y η σ t ζ x))e ζ, else. (5.1) For this problem, numerically negative radiative intensity appears if the positivity-preserving limiter is not used in the high order DG schemes. Figure 5.3 shows the contours of the radiative intensity simulated by the P 1 DG scheme using 4 4 uniform cells and the cells where the positivitypreserving limiter has been enacted during the simulation. The errors and orders of accuracy for the {P 1,P 2,P 3,P 4 } DG schemes without and with the positivity-preserving limiter are listed in Tables respectively. The percentage of the cells that require the usage of the positivitypreserving limiter is listed in these tables as well. From these tables, we can see that the expected order of accuracy for the positivity-preserving DG schemes has been achieved, both in L 2 -norm and L -norm, as expected from our theoretical results. Example 4. (The accuracy test of the DG schemes for the two-dimensional steady radiative trans- Table 5.8: Errors of the P 4 DG scheme for the 1D unsteady radiative transfer equation without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x error order error order error order error order limited cells(%) 1.129E-5.131E-4.189E-5.199E E E E E E E E E E E E E

26 Figure 5.3: The radiative intensity for the purely absorbing model simulated by the P 1 DG scheme with the positivity-preserving limiter on a 4 4 uniform grid. The white points represent the cells where the positivity-preserving limiter has been enacted during the computation. Table 5.9: Errors of the P 1 DG scheme for the 2D steady radiative transfer equation simulating the purely absorbing model without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order limited cells(%) 1.424E-3.94E-1.453E-3.91E E E E E E E E E E E E E fer equation simulating the absorbing-scattering model) n this test, we solve the two-dimensional steady radiative transfer equation (2.2) with all nonezero parameters which describes an absorbing-scattering process. To be more specific, we take σ t = 1,σ s = 1, and q = ζη(ζπ cos πxsin πy + ηπ sin πxcos πy + sin πxsin πy). The computational domain is [,1] [,1]. The boundary condition is the radiative intensity with unity value at all the four walls, that is, (x,) = 1,η > ; (x,1) = 1,η < ; (,y) = 1,ζ > ; (1,y) = 1,ζ <. For this absorbing-scattering radiative transfer problem, we can obtain its exact solution as follows, (ζ,η,x,y) = 1 + ζη sin πxsin πy. (5.2) The errors and orders of accuracy for the {P 1,P 2,P 3,P 4 } DG schemes are listed in Tables respectively. This example is mainly used to test the accuracy performance of our schemes 26

27 Table 5.1: Errors of the P 2 DG scheme for the 2D steady radiative transfer equation simulating the purely absorbing model without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order limited cells(%) 1.591E-4.132E-1.59E-4.132E E E E E E E E E E E E E Table 5.11: Errors of the P 3 DG scheme for the 2D steady radiative transfer equation simulating the purely absorbing model without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order limited cells(%) 1.86E-5.187E-2.113E-4.187E E E E E E E E E E E E E for solving the highly coupled 2D absorbing-scattering model. On the other hand, this is not a highly demanding example in terms of positivity-preserving, as the original DG schemes without the positivity-preserving limiter can already maintain positivity in most cases. From all these tables, we can see the desired order accuracy both in the L 2 -norm and L -norm for the radiative intensity, which demonstrates the high order property of the DG schemes to simulate this kind of highly coupled radiative transfer model. Next, we will test the positivity-preserving performance of our DG schemes when they are used to solve problems with discontinuities. n all the following tests, the radiative intensity does become negative numerically, especially in the region where the values of the radiative intensity should be equal to, if the DG schemes without the positivity-preserving limiter are adopted. Also notice that, since we have not used any non-oscillatory limiters such as the total variation bounded (TVB) Table 5.12: Errors of the P 4 DG scheme for the 2D steady radiative transfer equation simulating the purely absorbing model without positivity-preserving limiter with positivity-preserving limiter L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order limited cells(%) 1.893E-6.193E-3.919E-6.193E E E E E E E E E E E E E

28 Table 5.13: Errors of the P 1 and P 2 DG schemes for the 2D steady radiative transfer equation simulating the absorbing-scattering model P 1 P 2 L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order 1.55E-3.121E-1.283E-4.956E E E E E E E E E E E E E-5 3. Table 5.14: Errors of the P 3 and P 4 DG schemes for the 2D steady radiative transfer equation simulating the absorbing-scattering model P 3 P 4 L 2 norm L norm L 2 norm L norm N x = N y error order error order error order error order 1.122E-5.552E-4.46E-7.216E E E E E E E E E E E E E limiters [28, 5, 3] or the weighted essentially non-oscillatory (WENO) limiter [26, 36, 37], there are some localized spurious oscillations near the discontinuities in the numerical solution, which are not eliminated by the positivity-preserving limiter if they are not near zero. Example 5. (The positivity-preserving test of the DG schemes for the two-dimensional steady radiative transfer equation simulating the transparent model) This problem is a two-dimensional unity square enclosure with a transparent medium which is described by the equation (2.2) with σ t =,σ s =, q =. ζ =.7,η =.7. The computational domain is [,1] [,1]. A 4 4 uniform grid is used in the computation. The boundary condition is (x,) =, (,y) = 1. For this problem, it has the exact solution given as follows, (x,y) = {, y < η ζ x, 1, else. (5.3) n this test, negative solution will appear if we do not adopt the positivity-preserving limiter in the DG schemes with higher than first order, while the DG schemes with the positivity-preserving limiter can always maintain the nonnegative solution. Figure 5.4 plots the contours of the radiative intensity simulated by the {P 1,P 2,P 3,P 4 } DG schemes with the positivity-preserving limiter respectively. n the pictures, we mark the cells where the positivity-preserving has been enacted by discrete white points as well. Figures show the comparison of the radiative intensity 28

29 Figure 5.4: The contours of the radiative intensity for the transparent model simulating by the DG schemes with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. The white points represent the cells where the positivity-preserving limiter has been enacted during the computation. cut along the line y =.5 and x =.5 simulated by the DG schemes without and with the positivity-preserving limiter respectively. We can clearly see that the DG schemes without the positivity-preserving limiter produce negative solutions while the positivity of the radiative intensity can be kept well for the DG schemes with the positivity-preserving limiter. Also, higher order DG schemes produce more accurate solutions than the lower order DG schemes. Example 6. (The positivity-preserving test of the DG schemes for the two-dimensional steady radiative transfer equation simulating the purely absorbing model) We test the schemes on the purely absorbing model which is expressed by the equation (2.2) with σ t = 1,σ s = and q =. The computational domain is [,1] [,1]. ζ =.7,η =.7. The boundary condition is (x,) =, (,y) = 1. 29

30 1 exact p1 4 without limiter p1 4 with limiter 1 exact p2 4 without limiter p2 4 with limiter x exact p3 4 without limiter p3 4 with limiter x exact p4 4 without limiter p4 4 with limiter x Figure 5.5: The comparison of the radiative intensity cut along the line y =.5 for the transparent model simulated by the DG schemes without and with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. x 3

31 exact p1 4 without limiter p1 4 with limiter exact p2 4 without limiter p2 4 with limiter y exact p3 4 without limiter p3 4 with limiter y exact p4 4 without limiter p4 4 with limiter y y Figure 5.6: The comparison of the radiative intensity cut along the line x =.5 for the transparent model simulated by the DG schemes without and with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. 31

32 The exact solution for this example can be described as follows, {, y < η ζ (x,y) = x, e σ t x ζ, else. (5.4) Figures show the results of {P 1,P 2,P 3,P 4 } DG schemes on a 4 4 uniform grid individually. To be more specific, Figure 5.7 depicts the contours of the radiative intensity simulated by the {P 1,P 2,P 3,P 4 } DG schemes with the positivity-preserving limiter respectively, where the cells in which the positivity-preserving limiter has been enacted during the computation are marked by the discrete white points. n Figures , the comparison of the radiative intensity cut along the lines y =.5 and x =.5 obtained by the DG schemes without and with the positivitypreserving limiter is presented respectively. From these figures, we can observe that the positivitypreserving limiter is necessary for the DG schemes to produce positive solutions and the limiter can also maintain the good resolution of the DG schemes. Example 7. (The positivity-preserving test of the DG schemes for the two-dimensional steady radiative transfer equation simulating the absorbing-scattering model) n this problem, we test the schemes on the absorbing-scattering model described by the equation (2.2) with σ t = 1,σ s = 1 and q =. The computational domain is [,1] [,1]. The boundary condition is set as follows, (x,) =,η > ; (x,1) =,η < ; (,y) = 1 cos(4πy),ζ > ; (1,y) =,ζ <. The exact solution for this problem cannot be obtained explicitly. For the purpose of comparison, we take the numerical solution by the P 4 DG schemes with the positivity-preserving limiter on a 8 8 uniform grid as a reference solution. We implement the test on a 4 4 uniform grid by using the different order DG schemes without and with the positivity-preserving limiter. The contours of the radiative intensity in the (ζ, η) = (.2578,.168) angular direction simulated by the {P 1,P 2,P 3,P 4 } DG schemes with the positivity-preserving limiter are given in Figure 5.1. n the pictures, the cells where the positivity-preserving limiter has taken effect during the computation are marked by discrete white points as well. Figure 5.11 shows the comparison of the radiative intensity cut along the cells near the left boundary in the (ζ,η) = (.3256,.786) angular direction simulated by the {P 1,P 2 } DG schemes without and with the positivity-preserving limiter individually. From these figures, we notice that negative solution will appear if the high order DG schemes without the positivity-preserving limiter are used, while the high order DG schemes with the positivity-preserving limiter can produce nonnegative solutions with good resolution. 32

33 Figure 5.7: The contours of the radiative intensity for the purely absorbing model simulated by the DG schemes with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. The white points represent the cells where the positivity-preserving limiter has been enacted during the computation. 33

34 1 exact p1 4 without limiter p1 4 with limiter 1 exact p2 4 without limiter p2 4 with limiter x x 1 exact p3 4 without limiter p3 4 with limiter 1 exact p4 4 without limiter p4 4 with limiter x Figure 5.8: The comparison of the radiative intensity cut along the line y =.5 for the purely absorbing model simulated by the DG schemes without and with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. x 34

35 exact p1 4 without limiter p1 4 with limiter exact p2 4 without limiter p2 4 with limiter y exact p3 4 without limiter p3 4 with limiter y exact p4 4 without limiter p4 4 with limiter y y Figure 5.9: The comparison of the radiative intensity cut along the line x =.5 for the purely absorbing model simulated by the DG schemes without and with the positivity-preserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. 35

36 Figure 5.1: The contours of the radiative intensity for the absorbing-scattering model in the (ζ, η) = (.2578,.168) angular direction simulated by the DG schemes with the positivitypreserving limiter on a 4 4 uniform grid. Top left: P 1 ; Top right: P 2 ; Bottom left: P 3 ; Bottom right: P 4. The white points represent the cells where the positivity-preserving limiter has been enacted during the computation. reference p1 4 without limiter p1 4 with limiter reference p2 4 without limiter p2 4 with limiter.6 reference p2 4 without limiter p2 4 with limiter y y Figure 5.11: The comparison of the radiative intensity cut along the cells near the left boundary in the (ζ, η) = (.3256,.786) angular direction for the absorbing-scattering model simulated by the DG schemes without and with the positivity-preserving limiter. Left: P 1 ; Middle: P 2 ; Right: the zoomed region by P 2. y 36