An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

Zhu, Hongqiang; Qiu, Jianxian; Qiu, Jing-Mei

doi:10.1007/s10915-017-0440-9

Cited by 11 publications

(7 citation statements)

References 30 publications

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“…As time evolves, the solution quickly rolls up with smaller and smaller spatial scales so on any fixed grid, the full resolution will be lost eventually. This problem is a classic benchmark for demonstrating the effectiveness of a new scheme so it has been tested for many schemes such as the high order nonsplitting SL WENO scheme [34], the DG method in [24,36,39] and the spectral element method in [18,35]. We first show surface plots of numerical solutions for this problem at T = 8 in Figure 13, where the solution is rolled up in a very small scale.…”

Section: Meshmentioning

confidence: 99%

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Cai,

Qiu,

Yang

2020

Preprint

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We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017], which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. In the SLDG method, depending on the velocity field, upstream cells could be of arbitrary shape. Thus, a more sophisticated approximation to sides of the upstream cells is required to get high order approximation. For example, quadratic-curved (QC) quadrilaterals were proposed to approximate upstream cells for a third-order spatial accuracy in a swirling deformation example. In this paper, for linear advection problems, we propose a more general formulation, named the ELDG method. The scheme is formulated based on a modified adjoint problem for which the upstream cells are always quadrilaterals, which avoids the need to use QC quadrilaterals in the SLDG algorithm. The newly proposed ELDG method can be viewed as a new general framework, in which both the classical Eulerian Runge-Kutta DG formulation and the SL DG formulation can fit in. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

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Section: Meshmentioning

confidence: 99%

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Cai,

Qiu,

Yang

2020

Preprint

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“…Eventually, with a fixed mesh, the full resolution of the structures will be lost for any methods. This problem has been tested by the high-order Eulerian finite difference ENO/WENO method in [16,27], the high-order SL WENO scheme in [13,42,52], the DG method in [37,58,61] and the spectral element method in [23,54]. We solve this problem up to T = 8 and present the surface plots of for both SLDG methods in Fig.…”

Section: D Incompressible Euler Equations and The Guiding Center Vlamentioning

confidence: 99%

Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations

Cai

Guo

Qiu

2020

Commun. Appl. Math. Comput.

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Transport problems arise across diverse fields of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order deterministic transport solvers that enjoy advantages of both the SL approach and the DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerically their performance. In particular, we make a comparison between the splitting and nonsplitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we offer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations.

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“…We let k = 0.5, which will create a Kelvin-Helmholtz instability [18], which is well studied numerically by many authors before (e.g. see [22,4]).…”

Section: Nonlinear Vlasov Dynamicsmentioning

confidence: 99%

High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics

Cai

Boscarino

Qiu

2019

Preprint

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In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

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An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

Cited by 11 publications

References 30 publications

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations

High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics

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