Title: An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations
In this article, we introduce an invariant‐region‐preserving (IRP) limiter for thep‐system and the corresponding viscousp‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to thep‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscousp‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes. more »« less
Bressan, A; Lewicka, M; Wang, D.; Zheng, Y.X.
(Ed.)
In this paper we review the algorithm development in high order methods for some conservation laws. The emphasis is on our recent contribution in the study of two model classes: Fokker-Planck-type equations and hyperbolic conservation law systems. For the former we will review free-energy-satisfying and positivity-preserving schemes. For the later we will review the general invariant-region-preserving (IRP) limiter, and its application to high order methods for multi-dimensional hyperbolic systems of conservation laws.
Peng, Manting; Sun, Zheng; Wu, Kailiang
(, Mathematics of Computation)
Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess bothscale-invariantandevolution-invariantproperties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate fornonlinearDG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure isnonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control.It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques.Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.
Chu, R; Endeve, E; Hauck, C; Mezzacappa, A; Messer, B
(, Journal of physics Conference series)
We develop implicit-explicit (IMEX) schemes for neutrino transport in a background material in the context of a two-moment model that evolves the angular moments of a neutrino phase-space distribution function. Considering the upper and lower bounds that are introduced by Pauli’s exclusion principle on the moments, an algebraic moment closure based on Fermi-Dirac statistics and a convex-invariant time integrator both are demanded. A finite-volume/first-order discontinuous Galerkin(DG) method is used to illustrate how an algebraic moment closure based on Fermi-Dirac statistics is needed to satisfy the bounds. Several algebraic closures are compared with these bounds in mind, and the Cernohorsky and Bludman closure, which satisfies the bounds, is chosen for our IMEX schemes. For the convex-invariant time integrator, two IMEX schemes named PD-ARS have been proposed. PD-ARS denotes a convex-invariant IMEX Runge-Kutta scheme that is high-order accurate in the streaming limit, and works well in the diffusion limit. Our two PD-ARS schemes use second- and third-order, explicit, strong-stability-preserving Runge-Kutta methods as their explicit part, respectively, and therefore are second- and third-order accurate in the streaming limit, respectively. The accuracy and convex-invariance of our PD-ARS schemes are demonstrated in the numerical tests with a third-order DG method for spatial discretization and a simple Lax-Friedrichs flux. The method has been implemented in our high-order neutrino-radiation hydrodynamics (thornado) toolkit. We show preliminary results employing tabulated neutrino opacities.
In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.
Yang, X; Zhao, J; He, X.
(, Journal of computational and applied mathematics)
In this paper, we consider numerical approximations for the viscous Cahn–Hilliard equa- tion with hyperbolic relaxation. This type of equations processes energy-dissipative struc- ture. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed ‘‘Invariant Energy Quadratization’’ approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy, and efficiency of the proposed schemes.
Jiang, Yi, and Liu, Hailiang. An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations. Numerical Methods for Partial Differential Equations 35.1 Web. doi:10.1002/num.22274.
@article{osti_10064750,
place = {Country unknown/Code not available},
title = {An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations},
url = {https://par.nsf.gov/biblio/10064750},
DOI = {10.1002/num.22274},
abstractNote = {In this article, we introduce an invariant‐region‐preserving (IRP) limiter for thep‐system and the corresponding viscousp‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to thep‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscousp‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes.},
journal = {Numerical Methods for Partial Differential Equations},
volume = {35},
number = {1},
publisher = {Wiley Blackwell (John Wiley & Sons)},
author = {Jiang, Yi and Liu, Hailiang},
}
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