Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier–Stokes equations

Cheng, Jian; Yue, Huiqiang; Yu, Song; Liu, Tiegang

doi:10.1016/j.jcp.2018.02.031

Cited by 8 publications

(5 citation statements)

References 33 publications

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“…where q  denotes the numerical heat flux across the interface. Based on the DG method, many approaches, such as the interior penalty method (Douglas et al 1976;Arnold 1982), symmetric interior penalty method (Hartmann & Houston 2008), local DG method (Cockburn & Shu 1998), the first Bassi-Rebay (BR1) scheme (Bassi & Rebay 1997), the second Bassi-Rebay (BR2) scheme (Bassi et al 2005), compact DG method (Peraire & Persson 2008), hybridizable DG method (Cockburn et al 2009;Nguyen et al 2009), recovery-based DG method (van Leer et al 2007, rDG method (Luo et al 2010), and the DDG method (Liu & Yan 2009, 2010Cheng et al 2016Cheng et al , 2017Cheng et al , 2018, have been proposed to deal with the problems requiring the evaluation of the solution derivatives at a cell interface. In our work, we attempt to employ the DDG method to evaluate our heat flux in Equation (65) because of its simplicity in implementation and efficiency in computational cost.…”

Section: Ddg Discretization Of the Heat Fluxmentioning

confidence: 99%

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

Liu

Feng

Zhang

et al. 2023

ApJS

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The present study aims to apply an implicit high-order reconstructed discontinuous Galerkin (DG) scheme (rDG(P 1 P 2)) to simulate the steady-state solar corona. In this scheme, a piecewise quadratic polynomial solution, P 2, is obtained from the underlying piecewise linear DG solution, P 1, by least-squares reconstruction with a weighted essentially nonoscillatory limiter. The reconstructed quadratic polynomial solution is then used for the computation of the fluxes and source terms. In addition, an implicit time integration method with large time steps is considered in this work. The resulting large linear algebraic system of equations from the implicit discretization is solved by the cellwise relaxation implicit scheme which can make full use of the compactness of the DG scheme. The code of the implicit high-order rDG scheme is developed in Fortran language with message passing interface parallelization in Cartesian coordinates. To validate this code, we first test a problem with an exact solution, which confirms the expected third-order accuracy. Then we simulate the solar corona for Carrington rotations 2167, 2183, and 2210, and compare the modeled results with observations. We find that the numerical results basically reproduce the large-scale observed structures of the solar corona, such as coronal holes, helmet streamers, pseudostreamers, and high- and low-speed streams, which demonstrates the capability of the developed scheme.

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Section: Ddg Discretization Of the Heat Fluxmentioning

confidence: 99%

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

Liu

Feng

Zhang

et al. 2023

ApJS

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“…Then, it is combined with the simple numerical flux formula of the direct DG method to approximate the gradient of conserved variables. In summary, the new DDGIC method defines multiple individual diffusion processes, each of which is combined to calculate the viscous numerical fluxes for 2-D compressible NS equations as (21) + ∇(ρu) h • ξ (22) + ∇(ρv) h • ξ (23) ∇ρ h • ξ (31) + ∇(ρu) h • ξ (32) + ∇(ρv) h • ξ (33) ∇ρ h • ξ (41) + ∇(ρu) h • ξ (42) + ∇(ρv) h • ξ (43)…”

Section: The New Ddgic Scheme Formulation For 2-d Compressible Navier...mentioning

confidence: 99%

“…Similarly, the interface correction term can be simplified and computed as follows (21) + (ρu) h ξ (22) + (ρv) h ξ (23) • ∇φ j (x, y) ρ h ξ (31) + (ρu) h ξ (32) + (ρv) h ξ (33) • ∇φ j (x, y) ρ h ξ (41) + (ρu) h ξ (42) + (ρv) h ξ (43) + E h ξ (44) • ∇φ j (x, y)…”

Section: The New Ddgic Scheme Formulation For 2-d Compressible Navier...mentioning

confidence: 99%

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A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

Danis,

Yan

2021

Preprint

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We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined and approximated by the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P 1 , P 2 , P 3 and P 4 polynomial approximations are performed to verify the optimal (k + 1) th high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number.

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“…For the viscous term, we compute it by using the direct DG (DDG) method, which was first introduced by Liu and Yan to solve diffusion problems by applying the direct weak formulation for solutions of parabolic equations and then successfully extended by Cheng et al for the discretization of the viscous and heat fluxes in the two‐ and three‐dimensional compressible Navier‐Stokes equations. Due to the numerical flux defined by the DDG method that is simple, compact, conservative, and consistent, it provides an attractive advantage in the discretization of the viscous terms of the Navier‐Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

Zhong

Cheng

Liu

2019

Numerical Methods in Fluids

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Summary A high‐order discontinuous Galerkin (DG) method is proposed in this work for solving the two‐dimensional steady and unsteady incompressible Navier‐Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax‐Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.

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Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier–Stokes equations

Cited by 8 publications

References 33 publications

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

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