Conservative high order positivity-preserving discontinuous Galerkin methods for linear hyperbolic and radiative transfer equations. (English) Zbl 1407.65196
Summary: We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in [D. Yuan et al., SIAM J. Sci. Comput. 38, No. 5, A2987–A3019 (2016; Zbl 1351.65105)]. The DG methods in Yuan et al. [loc. cit.] can maintain positivity and high order accuracy, but they rely both on the scaling limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys. 229, No. 23, 8918–8934 (2010; Zbl 1282.76128)] and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax-Wendroff type theorem is proved in Yuan et al. [loc. cit.], guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space \(P^k\) of piecewise polynomials of degree at most \(k\). A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu [loc. cit.] can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on \(P^k\) or \(Q^k\) spaces (piecewise \(k\)th degree polynomials or piecewise tensor-product \(k\)th degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu [loc. cit.]. Computational results are provided to demonstrate the good performance of our DG schemes.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35B09 | Positive solutions to PDEs |
| 35R09 | Integro-partial differential equations |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 85A25 | Radiative transfer in astronomy and astrophysics |
Keywords:
positivity-preserving; discontinuous Galerkin method; conservative schemes; linear hyperbolic equation; radiative transfer equation; high-order accuracyReferences:
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