A New Formulation of Hyperbolic Navier-Stokes Solver based on Finite Volume Method on Arbitrary Grids

Li, Lingquan; Lou, Jialin; Luo, Hong; Nishikawa, Hiroaki

doi:10.2514/6.2018-4160

Cited by 12 publications

(11 citation statements)

References 37 publications

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“…These solvers will be particularly useful for allowing parameters (e.g., α g ) to be defined locally to adapt the gradients to local flow features while keeping different values in the Jacobian/preconditioner for robust convergence. A preliminary study indicates that the Newton solver is much more robust than the IDC-IGG and JFNK solvers, but requires a very efficient linear solver (e.g., multigrid) to minimize the overall computing time; it seems suggest that the hyperbolic Navier-Stokes formulation is better suited since the linear relaxation converges rapidly by the reduced condition number (due to the elimination of second derivatives) [26,27,28,29,30,31].…”

Section: Discussionmentioning

confidence: 99%

“…However, in this case, the residual Jacobian consists of 12×12 blocks for 4 conservative variables and 8 primitive-variable gradient components (20×20 blocks in three dimensions). The size of the discrete problem is equivalent to the P 1 discontinuous Galerkin method, where the solution gradients are introduced as additional discrete unknowns, and to the hyperbolic Navier-Stokes method [26,27,28,29,30,31], where the solution gradients are introduced as additional unknowns in the differential-equation-level. Clearly, it would require much more memory to store the Jacobian matrix although it is quite feasible and can be very effective for iterative convergence.…”

Section: Implicit Defect-correction (Idc) and Igg Solvermentioning

confidence: 99%

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An Implicit Gradient Method for Cell-Centered Finite-Volume Solver on Unstructured Grids

Nishikawa

2019

AIAA Scitech 2019 Forum

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In this paper, we investigate an implicit gradient method for a second-order cell-centered finite-volume method on unstructured grids. In the implicit gradient method, solution gradients are obtained by solving a global system of linear equations, but they can be computed iteratively along with an implicit finite-volume solver iteration for the Euler or Navier-Stokes equations. The cost of the implicit gradient computation is thereby made comparable to or even cheaper than that of explicit gradient methods such as a least-squares method. Furthermore, a known stability issue with a face-neighbor gradient stencil is effectively circumvented by expanding the gradient stencil to the entire domain. We discuss the relative performance of the implicit gradient method and a least-squares method for various inviscid and viscous flow computations on unstructured grids.

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Section: Discussionmentioning

confidence: 99%

Section: Implicit Defect-correction (Idc) and Igg Solvermentioning

confidence: 99%

An Implicit Gradient Method for Cell-Centered Finite-Volume Solver on Unstructured Grids

Nishikawa

2019

AIAA Scitech 2019 Forum

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“…The 3 rd and 5 th order accurate polynomials for the left and right boundaries are given by Eqs. (32) and (33).…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…[30], to construct a numerical dissipation matrix, the flux Jacobian does not have to be exactly derived, and the coefficients can be frozen and thus not needed to be differentiated. This type of formulation is very useful for developing high-order methods [31,32]. Note that the preconditioned formulation is necessary, for whatever variables are chosen, to apply the hyperbolic method to variable-coefficient and nonlinear equations because T r is not a constant and thus cannot be included in the flux vectors [30].…”

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confidence: 99%

First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

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In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs. [J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

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“…Thus by solving the hyperbolic system (1.3) to steady state, we can obtain the solution to an elliptic problem at arbitrary precision. This idea has been successfully applied to develop finite volume-type methods, e.g., [20][21][22][23], as well as discontinuous Galerkin-type methods, e.g., [24,25], that approximate the solution of parabolic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics

Schlottke-Lakemper

Winters

Ranocha

et al. 2021

Journal of Computational Physics

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One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime using the same explicit high-order discontinuous Galerkin method we use for the flow solution. The flow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupled via the source terms. A key benefit of our approach is that it allows the reuse of existing explicit hyperbolic solvers without modifications, while retaining their advanced features such as non-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kutta stage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physics simulations. After verifying the expected order of convergence for single-physics and multi-physics setups, we validate our approach by a simulation of the Jeans gravitational instability. Furthermore, we demonstrate the full capabilities of our numerical framework by computing a self-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinement for the entire coupled system.

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A New Formulation of Hyperbolic Navier-Stokes Solver based on Finite Volume Method on Arbitrary Grids

Cited by 12 publications

References 37 publications

An Implicit Gradient Method for Cell-Centered Finite-Volume Solver on Unstructured Grids

An Implicit Gradient Method for Cell-Centered Finite-Volume Solver on Unstructured Grids

First order hyperbolic approach for Anisotropic Diffusion equation

A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics

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