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Berend Van Wachem - One of the best experts on this subject based on the ideXlab platform.
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TVD differencing on three dimensional unstructured meshes with monotonicity preserving correction of mesh skewness
Journal of Computational Physics, 2015Co-Authors: Fabian Denner, Berend Van WachemAbstract:Total variation diminishing (TVD) Schemes are a widely applied group of monotonicity-preserving advection differencing Schemes for partial differential equations in numerical heat transfer and computational fluid dynamics. These Schemes are typically designed for one-dimensional problems or multidimensional problems on structured equidistant quadrilateral meshes. Practical applications, however, often involve complex geometries that cannot be represented by Cartesian meshes and, therefore, necessitate the application of unstructured meshes, which require a more sophisticated discretisation to account for their additional topological complexity. In principle, TVD Schemes are applicable to unstructured meshes, however, not all the data required for TVD differencing is readily available on unstructured meshes, and the solution suffers from considerable numerical diffusion as a result of mesh skewness. In this article we analyse TVD differencing on unstructured three-dimensional meshes, focusing on the non-linearity of TVD differencing and the extrapolation of the virtual upwind node. Furthermore, we propose a novel monotonicity-preserving correction method for TVD Schemes that significantly reduces numerical diffusion caused by mesh skewness. The presented numerical experiments demonstrate the importance of accounting for the non-linearity introduced by TVD differencing and of imposing carefully chosen limits on the extrapolated virtual upwind node, as well as the efficacy of the proposed method to correct mesh skewness.
Fabian Denner - One of the best experts on this subject based on the ideXlab platform.
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TVD differencing on three dimensional unstructured meshes with monotonicity preserving correction of mesh skewness
Journal of Computational Physics, 2015Co-Authors: Fabian Denner, Berend Van WachemAbstract:Total variation diminishing (TVD) Schemes are a widely applied group of monotonicity-preserving advection differencing Schemes for partial differential equations in numerical heat transfer and computational fluid dynamics. These Schemes are typically designed for one-dimensional problems or multidimensional problems on structured equidistant quadrilateral meshes. Practical applications, however, often involve complex geometries that cannot be represented by Cartesian meshes and, therefore, necessitate the application of unstructured meshes, which require a more sophisticated discretisation to account for their additional topological complexity. In principle, TVD Schemes are applicable to unstructured meshes, however, not all the data required for TVD differencing is readily available on unstructured meshes, and the solution suffers from considerable numerical diffusion as a result of mesh skewness. In this article we analyse TVD differencing on unstructured three-dimensional meshes, focusing on the non-linearity of TVD differencing and the extrapolation of the virtual upwind node. Furthermore, we propose a novel monotonicity-preserving correction method for TVD Schemes that significantly reduces numerical diffusion caused by mesh skewness. The presented numerical experiments demonstrate the importance of accounting for the non-linearity introduced by TVD differencing and of imposing carefully chosen limits on the extrapolated virtual upwind node, as well as the efficacy of the proposed method to correct mesh skewness.
Liang Cheng - One of the best experts on this subject based on the ideXlab platform.
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a refined r factor algorithm for TVD Schemes on arbitrary unstructured meshes
International Journal for Numerical Methods in Fluids, 2016Co-Authors: Di Zhang, Chunbo Jiang, Liang Cheng, Dongfang LiangAbstract:© 2016 John Wiley & Sons, Ltd. A refined r-factor algorithm for implementing total variation diminishing (TVD) Schemes on arbitrary unstructured meshes, referred to henceforth as a face-perpendicular far-upwind interpolation scheme for arbitrary meshes (FFISAM), is proposed based on an extensive review of the existing r-factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing r-factor algorithms. The performance of the FFISAM, implemented in 10 classical TVD Schemes, is evaluated against four two-dimensional pure-advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the 10 classical TVD Schemes.
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A review on TVD Schemes and a refined flux-limiter for steady-state calculations
Journal of Computational Physics, 2015Co-Authors: Di Zhang, Chunbo Jiang, Dongfang Liang, Liang ChengAbstract:This paper presents an extensive review of most of the existing TVD Schemes found in literature that are based on the One-step Time-space-coupled Unsteady TVD criterion (OTU-TVD), the Multi-step Time-space-separated Unsteady TVD criterion (MTU-TVD) and the Semi-discrete Steady-state TVD criterion (SS-TVD). The design principles of these Schemes are examined in detail. It is found that the selection of appropriate flux-limiters is a key design element in developing these Schemes. Different flux-limiter forms (CFL-dependent or CFL-independent, and various limiting criteria) are shown to lead to different performances in accuracy and convergence. Furthermore, a refined SS-TVD flux-limiter, referred to henceforth as TCDF (Third-order Continuously Differentiable Function), is proposed for steady-state calculations based on the review. To evaluate the performance of the newly proposed scheme, many existing classical SS-TVD limiters are compared with the TCDF in eight two-dimensional test cases. The numerical results clearly show that the TCDF results in an improved overall performance.
Wendar Guo - One of the best experts on this subject based on the ideXlab platform.
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high resolution TVD Schemes in finite volume method for hydraulic shock wave modeling
Journal of Hydraulic Research, 2005Co-Authors: Gwofong Lin, Jihnsung Lai, Wendar GuoAbstract:High-resolution total variation diminishing (TVD) Schemes in the framework of the finite volume method are presented and evaluated for hydraulic shock wave modeling. Three approximate Riemann solvers, namely the FVS, Roe and Osher Schemes, are extended to high-resolution TVD Schemes based on the direct MUSCL-Hancock (DMH) slope limiter approach. The TVD Schemes are then used to develop numerical models to compute water depth and velocity. The numerical models developed are then verified through simulations of the dam-break flows, the oblique hydraulic jump, and the shock-on-shock interaction. The numerical models with TVD Schemes are capable of capturing discontinuous shock waves without any spurious oscillation. A comparison of numerical efficiency shows that the Osher-DMH scheme coupled with van Leer limiter performs the best among the proposed TVD Schemes. Applications of the Osher-DMH scheme to flows of partial dam-break experiments have shown that the simulated water depths agree well with the measur...
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performance of high resolution TVD Schemes for 1d dam break simulations
Journal of The Chinese Institute of Engineers, 2005Co-Authors: Gwofong Lin, Jihnsung Lai, Wendar GuoAbstract:Abstract The performance of high‐resolution total variation diminishing (TVD) Schemes for simulating dam‐break problems are presented and evaluated. Three robust and reliable first‐order upwind Schemes, namely FVS, Roe and HLLE Schemes, are extended to six second‐order TVD Schemes using two different approaches, the Sweby flux limiter approach and the direct MUSCL‐Hancock slope limiter. For idealized dam‐break flows, comparisons of the simulated results with the exact solutions show that the flux vector splitting (FVS) scheme coupled with the direct MUSCL‐Hancock (DMH) slope limiter approach has the best numerical performance among the presented Schemes. Application of the FVS‐DMH scheme to a dam‐break experiment with sloping dry bed shows that the simulated water depths agree well with the measured.
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finite volume component wise TVD Schemes for 2d shallow water equations
Advances in Water Resources, 2003Co-Authors: Gwofong Lin, Jihnsung Lai, Wendar GuoAbstract:Abstract Four finite-volume component-wise total variation diminishing (TVD) Schemes are proposed for solving the two-dimensional shallow water equations. In the framework of the finite volume method, a proposed algorithm using the flux-splitting technique is established by modifying the MacCormack scheme to preserve second-order accuracy in both space and time. Based on this algorithm, four component-wise TVD Schemes, including the Liou–Steffen splitting (LSS), van Leer splitting, Steger–Warming splitting and local Lax–Friedrichs splitting Schemes, are developed. These Schemes are verified through the simulations of the 1D dam-break, the oblique hydraulic jump, the partial dam-break and circular dam-break problems. It is demonstrated that the proposed Schemes are accurate, efficient and robust to capture the discontinuous shock waves without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. The simulated results also show that the LSS scheme has the best numerical accuracy among the Schemes tested.
Eleuterio F Toro - One of the best experts on this subject based on the ideXlab platform.
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weno Schemes based on upwind and centred TVD fluxes
Computers & Fluids, 2005Co-Authors: V A Titarev, Eleuterio F ToroAbstract:Abstract In this paper we propose to use second-order TVD fluxes, instead of first-order monotone fluxes, in the framework of finite-volume weighted essentially non-oscillatory (WENO) Schemes. We call the new improved Schemes the WENO-TVD Schemes. They include both upwind and centred Schemes on non-staggered meshes. Numerical results suggest that our Schemes are superior to the original Schemes used with first-order monotone fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.
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on the use of TVD fluxes in eno and weno Schemes
2003Co-Authors: V A Titarev, Eleuterio F ToroAbstract:Very high order methods, such as ENO/WENO methods [21, 30, 19], Runge-Kutta Discontinuous Galerkin Finite Element Methods [12] and ADER methods [54,46] often use high order (e.g. fifth order) polynomial reconstruction of the solution of a lower (first) order monotone flux as the building block. In this paper we propose to use second order TVD fluxes in the framework of such methods and apply the principle to the finite-volume ENO, WENO and MPWENO Schemes. We call the new improved Schemes the ENO-TVD, WENO-TVD and MPWENO-TVD Schemes respectively. They include both upwind and centred Schemes with non-stuggered meshes. Numerical results suggest that our Schemes are superior to original Schemes with first order fluxes. This is especially so for long time evolution problems containing both smooth and non-smooth features.
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centred TVD Schemes for hyperbolic conservation laws
Ima Journal of Numerical Analysis, 2000Co-Authors: Eleuterio F Toro, S J BillettAbstract:New first- and high-order centred methods for conservation laws are presented. Convenient TVD conditions for constructing centred TVD Schemes are then formulated and some useful results are proved. Two families of centred TVD Schemes are constructed and extended to nonlinear systems. Some numerical results are also presented.
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high order and TVD Schemes for non linear systems
1999Co-Authors: Eleuterio F ToroAbstract:This chapter is concerned with TVD upwind and centred Schemes for nonlinear systems of conservation laws that depend on time t, or a time-like variable t, and one space dimension x. The upwind Schemes are extensions of the Godunov first order upwind method of Chap. 6 and can be applied with any of the Riemann solvers presented in Chap. 4 (exact) and Chaps. 9 to 12 (approximate); they can also be used with the Flux Vector Splitting flux of Chap. 8. The centred Schemes are extensions of the First Order Centred (FORCE) method presented in Chap. 7. All the TVD Schemes are in effect the culmination of work carried out in all previous chapters, particularly Chap. 13, where the TVD concept was developed in the context of simple scalar problems. The Schemes are presented in terms of the time-dependent one dimensional Euler equations for ideal gases, which are introduced in Chap. 1 and studied in detail in Chap. 3. Applications to other systems may be easily accomplished. Techniques for extending the methods to systems with source terms, as for reactive flows for instance, are given in Chap. 15 and to multidimensional systems in Chap. 16.
