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Register a new userStable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
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Added by
Jan Glaubitz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre-Gauss-Lobatto quadrature. The theoretical discussion serves to re-contextualize stable filtering results for finite difference methods into the DG setting. It is shown that the stability of the filtering is equivalent to a particular contractivity condition borrowed from the analysis of so-called transmission problems. As such, the temporal stability proof relies on the fact that the underlying spatial discretization of the problem possesses a semi-discrete bound on the solution. Numerical tests are provided to verify and validate the underlying theoretical results.
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.
This paper studies high-order accurate entropy stable nodal discontinuous Galerkin (DG) schemes for the ideal special relativistic magnetohydrodynamics (RMHD). It is built on the modified RMHD equations with a particular source term, which is analogous to the Powells eight-wave formulation and can be symmetrized so that an entropy pair is obtained. We design an affordable fully consistent two-point entropy conservative flux, which is not only consistent with the physical flux, but also maintains the zero parallel magnetic component, and then construct high-order accurate semi-discrete entropy stable DG schemes based on the quadrature rules and the entropy conservative and stable fluxes. They satisfy the semidiscrete entropy inequality for the given entropy pair and are integrated in time by using the high-order explicit strong stability preserving Runge-Kutta schemes to get further the fully-discrete nodal DG schemes. Extensive numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our schemes. Moreover, our entropy conservative flux is compared to an existing flux through some numerical tests. The results show that the zero parallel magnetic component in the numerical flux can help to decrease the error in the parallel magnetic component in one-dimensional tests, but two entropy conservative fluxes give similar results since the error in the magnetic field divergence seems dominated in the two-dimensional tests.
We present unconditionally energy stable Runge-Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved without imposing any restriction on time steps and meshes. We achieve this in two steps. First, taking advantage of the penalty free DG method introduced by Liu and Yin [J Sci. Comput. 77:467--501, 2018] for spatial discretization, we reformulate an extended linearized ODE system by the energy quadratization (EQ) approach. Second, we apply an s-stage algebraically stable RK method for temporal discretization. The resulting fully discrete DG schemes are linear and unconditionally energy stable. In addition, we introduce a prediction-correction procedure to improve both the accuracy and stability of the scheme. We illustrate the effectiveness of the proposed schemes by numerical tests with benchmark problems.
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