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Sin-chung Chang - One of the best experts on this subject based on the ideXlab platform.
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Courant Number and Mach Number Insensitive Ce/Se Euler Solvers
2013Co-Authors: Sin-chung ChangAbstract:It has been known that the space-time CE/SE method can be used to obtain 1D, 2D, and 3D steady and unsteady o w solution with Mach Number ranging from 0:0028 to 10. However, it is also known that a CE/SE solution may become overly dissipative when Mach Number is very small. As an initial attempt to remedy this weakness, new 1D Courant Number and Mach Number insensitive CE/SE Euler solvers are developed using several key concepts underlying the recent successful development of Courant Number insensitive CE/SE schemes. Numerical results indicate that the new solvers are capable of resolving crisply a contact discontinuity embeded in a o w with the maxmum Mach Number = 0:01.
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Explicit Von Neumann Stability Conditions for the C-Tau Scheme: A Basic Scheme in the Development of the Ce-Se Courant Number Insensitive Schemes
2013Co-Authors: Sin-chung ChangAbstract:As part of the continuous development of the space-time conservation element and solution element (CE-SE) method, recently a set of so called “Courant Number insensitive schemes” has been proposed. The key advantage of these new schemes is that the numerical dissipation associated with them generally does not increase as the Courant Number decreases. As such, they can be applied to problems with large Courant Number disparities (such as what commonly occurs in Navier-Stokes problems) without incurring excessive numerical dissipation. A basic scheme in the development of the Courant Number insensitive schemes is the so called “c-τ scheme”. It is a solver of the PDE ∂u ∂t + a ∂u ∂x =0 where a �= 0 is a constant. At each space-time staggered mesh points (j, n), the c-τ scheme is formed by u n = 1 �
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Validation and verification of Courant Number insensitive CE/SE method for transient viscous flow simulations
Mathematics and Computers in Simulation, 2008Co-Authors: Balaji Shankar Venkatachari, Gary C. Cheng, Bharat K. Soni, Sin-chung ChangAbstract:In this paper, we report an extension of the space-time conservation element-solution element (CE/SE) framework-based viscous flow solver. With the accuracy of solution obtained through the use of a CE/SE-based solver closely related to the CFL Number disparity across the mesh, a new formulation to make the solution insensitive to CFL Number disparity is herein presented. The capability of the developed solver is then validated through simulation of 2D problems such as driven cavity, external flow over a flat plate, laminar flow over a square cylinder, etc. Investigations are also conducted to verify the sensitivity of results to grid spacing and mesh structure.
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Validation and Verification of CE/SE Method based Courant Number Insensitive Transient Viscous Flow Solver
41st AIAA ASME SAE ASEE Joint Propulsion Conference & Exhibit, 2005Co-Authors: Balaji Shankar Venkatachari, Gary C. Cheng, Sin-chung ChangAbstract:In this paper, we rigorously validate and verify the recently developed Space-Time Conservation Element/Solution Element (CE/SE) based Courant Number Insensitive transient viscous flow solver. Although, the CE/SE framework in two spatial dimension was originally designed for triangular meshes, in recent times it has been extended to utilize polygon shaped meshes also. However the effect of use of quadrilateral meshes on the Courant Number Insensitive scheme (CNIS) has not been well documented and tested. Therefore, in this paper we propose to verify and compare the effect of using triangular and quadrilateral meshes on the numerical accuracy through use of benchmark test cases such as laminar flow over a flat plate, laminar flow over a thin flat plate (splitter plate) and laminar flow over a square cylinder. The numerical results and their comparison with analytical and experimental results are also presented. Possible future work is also discussed.
Laurent Stainier - One of the best experts on this subject based on the ideXlab platform.
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Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions
International Journal for Numerical Methods in Engineering, 2019Co-Authors: Adrien Renaud, Thomas Heuzé, Laurent StainierAbstract:In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multi-dimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant Number for a given space discretization. Though the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant Number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.
Adrien Renaud - One of the best experts on this subject based on the ideXlab platform.
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Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions
International Journal for Numerical Methods in Engineering, 2019Co-Authors: Adrien Renaud, Thomas Heuzé, Laurent StainierAbstract:In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multi-dimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant Number for a given space discretization. Though the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant Number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.
Balaji Shankar Venkatachari - One of the best experts on this subject based on the ideXlab platform.
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Validation and verification of Courant Number insensitive CE/SE method for transient viscous flow simulations
Mathematics and Computers in Simulation, 2008Co-Authors: Balaji Shankar Venkatachari, Gary C. Cheng, Bharat K. Soni, Sin-chung ChangAbstract:In this paper, we report an extension of the space-time conservation element-solution element (CE/SE) framework-based viscous flow solver. With the accuracy of solution obtained through the use of a CE/SE-based solver closely related to the CFL Number disparity across the mesh, a new formulation to make the solution insensitive to CFL Number disparity is herein presented. The capability of the developed solver is then validated through simulation of 2D problems such as driven cavity, external flow over a flat plate, laminar flow over a square cylinder, etc. Investigations are also conducted to verify the sensitivity of results to grid spacing and mesh structure.
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Validation and Verification of CE/SE Method based Courant Number Insensitive Transient Viscous Flow Solver
41st AIAA ASME SAE ASEE Joint Propulsion Conference & Exhibit, 2005Co-Authors: Balaji Shankar Venkatachari, Gary C. Cheng, Sin-chung ChangAbstract:In this paper, we rigorously validate and verify the recently developed Space-Time Conservation Element/Solution Element (CE/SE) based Courant Number Insensitive transient viscous flow solver. Although, the CE/SE framework in two spatial dimension was originally designed for triangular meshes, in recent times it has been extended to utilize polygon shaped meshes also. However the effect of use of quadrilateral meshes on the Courant Number Insensitive scheme (CNIS) has not been well documented and tested. Therefore, in this paper we propose to verify and compare the effect of using triangular and quadrilateral meshes on the numerical accuracy through use of benchmark test cases such as laminar flow over a flat plate, laminar flow over a thin flat plate (splitter plate) and laminar flow over a square cylinder. The numerical results and their comparison with analytical and experimental results are also presented. Possible future work is also discussed.
B.p. Leonard - One of the best experts on this subject based on the ideXlab platform.
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Note on the von Neumann stability of explicit one-dimensional advection schemes
Computer Methods in Applied Mechanics and Engineering, 1994Co-Authors: B.p. LeonardAbstract:There is a wide-spread belief that most explicit one-dimensional advection schemes need to satisfy the condition that the Courant Number, c = uΔt/Δx, must be less than or equal to 1, for stability in the von Neumann sense. This puts severe limitations on the time-step in high-speed, fine-grid calculations and is an impetus for the development of implicit schemes, which often require less restrictive time-step conditions for stability, but are more expensive per time-step. However, it turns out that, if explicit schemes are formulated in a consistent flux-based conservative finite-volume form, von Neumann stability analysis does not place any restriction on the allowable Courant Number. Any explicit scheme that is stable for c < 1, with a complex amplitude ratio, G(c), can be easily extended to arbitrarily large Courant Number. The complex amplitude ratio is then given by exp(—ιNθ)G(Δc), where N is the integer part of c, and Δc = c - N (
