A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

Xia, Yidong; Liu, Xiaodong; Luo, Hong

doi:10.1002/fld.4057

Cited by 24 publications

(7 citation statements)

References 54 publications

(75 reference statements)

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“…In addition, we demonstrated an ability to solve for primitive variables, chosen on the requirement of solvability (better conditioning) of the underlying physics. This is in contrast to our previous rDG efforts [32,31,38,34,33,39,40,59,60,35,36,58], which used conservation variables as a solution vector. In the context of the application of interest, we are looking for all-speed flow capabilities, with phase change (melting/solidification).…”

Section: Introductionmentioning

confidence: 88%

Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

Luo

Weston

Anderson

et al. 2016

Journal of Computational Physics

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A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.

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Section: Introductionmentioning

confidence: 88%

Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

Luo

Weston

Anderson

et al. 2016

Journal of Computational Physics

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“…The LU-SGS preconditioner was first developed by Jameson and Yoon 9 on structured grids, and later extended to unstructured meshes by several authors. 23, 19, 22 Luo et al 13,12,14,16 developed and validated a matrix-free version of GMRES+LU-SGS , and GMRES+LU-SGS was used in the context of rDG formulation 29,30,27 and is also employed in this work for comparison with NKA+LU-SGS. Other efficient preconditioners could be considered as well, e.g., Persson et al 20 and Zahr et al 31 utilized Newton-GMRES with an ILU0 preconditioner for the DG discretization of the Navier-Stokes equations.…”

Section: More Details On Nkamentioning

confidence: 99%

“…Wang et al 26 developed a six-stage fourth-order implicit Runge-Kutta scheme for the time-accurate DG solution to the Euler equations, and Xia et al 27 used a four-stage third-order version for the WENO reconstructed DG formulation of the compressible Navier-Stokes equations. This class of Runge-Kutta schemes is termed as ESDIRK by Bijl et al, 2 standing for Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta.…”

Section: Temporal Discretizationmentioning

confidence: 99%

Application of Nonlinear Krylov Acceleration to a Reconstructed Discontinuous Galerkin Method for Compressible Flows

Wang

Cheng

Berndt

et al. 2016

46th AIAA Fluid Dynamics Conference

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A variant of Anderson Mixing, namely the Nonlinear Krylov Acceleration (NKA), is presented and implemented in a reconstructed Discontinuous Galerkin (rDG) method to solve the compressible Euler and Navier-Stokes equations on hybrid grids. A nonlinear system of equations as a result of a fully implicit temporal discretization at each time step is solved using the NKA method with a lower-upper symmetric Gauss-Seidel (LU-SGS) preconditioner. The developed NKA method is used to compute a variety of flow problems and compared with a well-known Newton-GMRES method to demonstrate the performance of the NKA method. Our numerical experiments indicate that the NKA method outperforms its Newton-GMRES counterpart for transient flow problems, and iscomparable to Newton-GMRES for steady cases, and thus provides an attractive alternative to solve the system of nonlinear equations arising from the rDG approximation.

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“…For the higher-order ESDIRK temporal scheme (Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta) [27,40,41]…”

Section: Geometric Conservation Law (Gcl)mentioning

confidence: 99%

A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids

Wang

Luo

2021

Adv. Aerodyn.

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A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.

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A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

Cited by 24 publications

References 54 publications

Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

Application of Nonlinear Krylov Acceleration to a Reconstructed Discontinuous Galerkin Method for Compressible Flows

A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids

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