A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer

Olbrant, Edgar; Hauck, Cory D.; Frank, Martin

doi:10.1016/j.jcp.2012.03.002

Cited by 48 publications

(79 citation statements)

References 52 publications

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“…Although the realizability limiter plays some role in dampening spurious oscillations in numerical solutions, further dampening is needed [23]. We use the standard TVBM corrected minmod limiter proposed in [6].…”

Section: Slope Limitingmentioning

confidence: 99%

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

Alldredge

Schneider

2015

Journal of Computational Physics

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We implement a high-order numerical scheme for the entropy-based moment closure, the so-called M N model, for linear kinetic equations in slab geometry. A discontinuous Galerkin (DG) scheme in space along with a strong-stability preserving Runge-Kutta time integrator is a natural choice to achieve a third-order scheme, but so far, the challenge for such a scheme in this context is the implementation of a linear scaling limiter when the numerical solution leaves the set of realizable moments (that is, those moments associated with a positive underlying distribution). The difficulty for such a limiter lies in the computation of the intersection of a ray with the set of realizable moments. We avoid this computation by using quadrature to generate a convex polytope which approximates this set. The halfspace representation of this polytope is used to compute an approximation of the required intersection straightforwardly, and with this limiter in hand, the rest of the DG scheme is constructed using standard techniques. We consider the resulting numerical scheme on a new manufactured solution and standard benchmark problems for both traditional M N models and the so-called mixed-moment models. The manufactured solution allows us to observe the expected convergence rates and explore the effects of the regularization in the optimization.

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Section: Slope Limitingmentioning

confidence: 99%

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

Alldredge

Schneider

2015

Journal of Computational Physics

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“…which are non-negative provided the first condition in (38) holds. Next, we consider the μ dimension.…”

Section: Spherical Symmetry Flat Spacetime (1d X + 1d P)mentioning

confidence: 99%

“…which are non-negative if the second condition in (38) holds. Therefore, the CFL condition in (25) becomes (38).…”

Section: Spherical Symmetry Flat Spacetime (1d X + 1d P)mentioning

confidence: 99%

“…We use the analytical solution to specify incoming radiation on the boundary ∂ D and simulate the evolution from t = 0 to t = 1, using the bound-preserving CFL conditions given in Eq. (38), with s 1 = s 2 = 1/2 to set the time step. To evaluate the accuracy and the convergence rates of the different DG schemes, we compute the L 1 -error norm…”

Section: Test With Smooth Analytic Solutionmentioning

confidence: 99%

“…Eq. (38). However, for many applications the origin must be included in the computational domain, even though this may introduce significant numerical errors.…”

Section: Test With Smooth Analytic Solutionmentioning

confidence: 99%

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Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

Endeve

Hauck

Xing

et al. 2015

Journal of Computational Physics

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We extend the positivity-preserving method of Zhang and Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time integration. Special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f ; i.e., f ∈ [0, 1]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments -including one example in spherical symmetry adopting the Schwarzschild metric -demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.

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Kinetic and Moment Models for Cell Motion in Fiber Structures

Borsche

Klar

Schneider

2019

Active Particles, Volume 2

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A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer

Cited by 48 publications

References 52 publications

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

Kinetic and Moment Models for Cell Motion in Fiber Structures

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