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Register a new userHigh order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations
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Tao Xiong
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the work cite{Peng2020stability}, in which stability enhanced high order AP DG methods are proposed for linear transport equations, we propose to pernalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. In the diffusive limit, a hyperbolic, namely $Delta t=mathcal{O}(h)$ where $Delta t$ and $h$ are the time step and mesh size respectively, instead of parabolic $Delta t=mathcal{O}(h^2)$ time step restriction is obtained, which is also free from the photon mean free path. The main new ingredient is that we further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations from an outer iterative loop. Our scheme is shown to be asymptotic preserving and asymptotically accurate. Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is of high order, effective and efficient.
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This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong stability preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are applicable to the Euler system with general equation of state. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions.
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin-Bona-Mahony equation, the Camassa-Holm equation, the Korteweg-de Vries equation and the nonlinear Schrodinger equation are discussed. Some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods. Numerically, we observe that certain choices of numerical fluxes in the discussed class may help achieve better accuracy compared with the commonly used ones including the central fluxes.
Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the methods efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.
In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear $L^2$-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.
We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.
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