Generalized coordinate transformation and gas-kinetic scheme

Pan, Liang; Xu, Kun

doi:10.1016/j.jcp.2015.02.010

Cited by 7 publications

(3 citation statements)

References 34 publications

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“…In smooth flow region, the hydrodynamic scale physics corresponding to the multi-dimensional central difference discretization will contribute mainly in the kinetic flux function, and accurate Navier-Stokes solution can be obtained once the flow structure is well resolved. Based on the unified coordinate transformation, a moving-mesh gas-kinetic scheme has been developed [18,28]. With the discretization of particle velocity space, a unified gas-kinetic scheme (UGKS) has been developed for the flow study in entire Knudsen number regimes from rarefied to continuum ones [41,26,13].…”

Section: Introductionmentioning

confidence: 99%

A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

Pan

2015

Commun. Comput. Phys.

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In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for highorder accuracy in the Discontinuous Galerkin (DG) method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. We believe that the current scheme is one of the most robust and accurate third-order compact schemes for both smooth and discontinuous viscous and heat conducting flow simulations. It can be faithfully used from the incompressible limit to the hypersonic flow computations. Many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique. Even with the increasing of computational cost in the evaluation of a multidimensional time-dependent gas distribution function at a cell interface, the current scheme is still efficient. Also, due to its multidimensional property of including both derivatives of flow variables in the normal and tangential directions of a cell interface, the viscous flow solution, especially those with vortex structure, can be accurately captured. With the same stencil of a second order scheme, numerical tests clearly demonstrate that the current compact third-order scheme is as robust as well-developed second-order shock capturing schemes, but provides much more accurate numerical solutions than the second order counterparts. 1

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Section: Introductionmentioning

confidence: 99%

A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

Pan

2015

Commun. Comput. Phys.

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“…With the discretization of particle velocity space, a unified gas-kinetic scheme (UGKS) has been developed for the flow study in entire Knudsen number regimes from rarefied to continuum ones [44,29,43]. Recently, with the incorporation of high-order initial reconstruction, high-order gas-kinetic schemes have been constructed [24,28,30]. The flux evaluation in the scheme is based on the moments of a time and space dependent gas distribution function evolved from an initially piece-wise discontinuous polynomials of macroscopic flow variables around a cell interface, where high-order spatial and temporal derivatives of flow variables are coupled nonlinearly.…”

Section: Introductionmentioning

confidence: 99%

A third-order gas-kinetic scheme for three-dimensional inviscid and viscous flow computations

Pan

2015

Computers & Fluids

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“…If carefully implemented, the MMPDE ( eq:mmpde-ch eq:mmpde-ch 68) can be quite effective (see, e.g. :03,JIN200668,ChT:08,XuN:13,Zho:13,ZhC:13,PAN2015207[53,54,55,56,20,40,57] and references therein). The design of moving mesh methods on the sphere is a relatively recent subject of research.…”

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confidence: 99%

Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes

Arpaia

Ricchiuto²

2020

Journal of Computational Physics

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We consider the numerical approximation of the Shallow Water Equations (SWEs) in spherical geometry for oceanographic applications. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we restate all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.

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Generalized coordinate transformation and gas-kinetic scheme

Cited by 7 publications

References 34 publications

A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

A third-order gas-kinetic scheme for three-dimensional inviscid and viscous flow computations

Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes

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