Design principles for bounded higher-order convection schemes – a unified approach

Waterson, Nicholas; Deconinck, H.

doi:10.1016/j.jcp.2007.01.021

Cited by 223 publications

(229 citation statements)

References 58 publications

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“…It corresponds to the maximal possible extent with Van Leer's j ¼ 1 3 scheme [27], within Sweby's TVD domain [25]. Its accuracy properties are known to be good [29]. Formally, in 1D, this MUSCL approach is thirdorder accurate [11].…”

Section: Cell-face State Construction and Time Integrationmentioning

confidence: 99%

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

Kreeft

Koren

2010

Journal of Computational Physics

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a b s t r a c tA new formulation of Kapila's five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.

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Section: Cell-face State Construction and Time Integrationmentioning

confidence: 99%

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

Kreeft

Koren

2010

Journal of Computational Physics

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“…According to Leonard [3,12], a bounded high resolution second and/or third order accurate scheme (in general, non-linear) within the CBC region must pass through points O(0, 0), Q(0.5, 0.75), P (1,1) and with inclination of 0.75 at Q. Passing through Q will provide second order accuracy and passing through Q with a slope of 0.75 will give third order accuracy.…”

Section: Summary Of the Topus Scheme And Its Modificationmentioning

confidence: 99%

“…where r f is a local smoothness measure satisfying Sweby's monotonicity preservation condition (see, for example, [1]) when it tends to zero, and it is given by…”

Section: Summary Of the Topus Scheme And Its Modificationmentioning

confidence: 99%

“…High resolution upwind schemes, an extension of the monotonicity preserving first-order upwind scheme by the use of non-linear limiters, have successful been employed for the simulation of a variety of PDEs; since advection of scalars to non-linear conservation laws (see, for instance, [1]). However, their adaptation to PDEs for predicting flow field in the presence of shocks or steep gradients is not so common in the literature and, in particular, their application to incompressible free surface flows at high Reynolds numbers is hindered by the moving boundary.…”

Section: Introductionmentioning

confidence: 99%

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Simulation results and applications of an advection bounded scheme to practical flows

Ferreira¹,

Queiroz²,

Candezano³

et al. 2012

Comput. Appl. Math.

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Abstract. This paper reports experiments on the use of a recently introduced advection bounded upwinding scheme, namely TOPUS (Computers & Fluids 57 (2012) 208-224), for flows of practical interest. The numerical results are compared against analytical, numerical and experimental data and show good agreement with them. It is concluded that the TOPUS scheme is a competent, powerful and generic scheme for complex flow phenomena.Mathematical subject classification: Primary: 06B10; Secondary: 06D05.

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“…There are many explicit FD schemes in the literature. Nonetheless, owing to the very strict Courant-Friedrichs-Lewy (CFL) constraint, there exists a stability upper bound for the normalized value of diffusivity, and also a mixture of upwind and central schemes for scalar advection is usually necessary [7]. A direct consequence of such a treatment is the increased numerical diffusion and loss of exact Galilean invariance.…”

Section: Introductionmentioning

confidence: 99%

A lattice Boltzmann approach for solving scalar transport equations

Zhang¹,

Fan²,

Chen³

2011

Phil. Trans. R. Soc. A.

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A lattice Boltzmann (LB) approach is presented for solving scalar transport equations. In addition to the standard LB for fluid flow, a second set of distribution functions is introduced for transport scalars. This LB approach fully recovers the macroscopic scalar transport equation satisfying an exact conservation law. It is numerically stable and scalar diffusivity does not have a Courant-Friedrichs-Lewy-like stability upper limit. With a sufficient lattice isotropy, numerical solutions are independent of grid orientations. A generalized boundary condition for scalars on arbitrary geometry is also realized by a precise control of surface scalar flux. Numerical results of various benchmarks are presented to demonstrate the accuracy, efficiency and robustness of the approach.

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Design principles for bounded higher-order convection schemes – a unified approach

Cited by 223 publications

References 58 publications

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

Simulation results and applications of an advection bounded scheme to practical flows

A lattice Boltzmann approach for solving scalar transport equations

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