The Method of Space-Time Conservation Element and Solution Element—A New Approach for Solving the Navier-Stokes and Euler Equations

Chang, Sin-Chung

doi:10.1006/jcph.1995.1137

Cited by 468 publications

(355 citation statements)

References 4 publications

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“…Note that the same relation also holds for many 2D and 3D CE/SE schemes [7,8,41]. (c) As shown in [3], the two amplification factors of the a scheme are identical to those of the leapfrog scheme. As a result, the a scheme is non-dissipative and it is stable if |ν| < 1 (see the additional discussions given in Sec.…”

mentioning

confidence: 69%

“…(f) In addition to the non-dissipative a scheme, as will be shown, there is a family of its dissipative extensions in which only the less stringent conservation condition Eq. (2.11) is assumed [3]. Because Eq.…”

Section: The a Schemementioning

confidence: 99%

“…Partly for this reason, extensions of the c scheme have been used extensively. (b) For the a-scheme, it is shown in [3] that the principal and spurious amplification factors per ∆t, respectively, are (λ + ) 2 and (λ − ) 2 with…”

Section: The A-scheme and The C Schemementioning

confidence: 99%

“…To serve as a preliminary for future development, here we shall briefly review an extension of the c scheme which was introduced as a remedy for this deficiency [3,35].…”

Section: The W-α Scheme-a Special Wiggle-suppressing Schemementioning

confidence: 99%

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Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Chang¹

2005

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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IntrodutionThe space-time conservation element and solution element (CE/SE) method is a high-resolution and genuinely multidimensional method for solving conservation laws . Its nontraditional features include: (i) a unified treatment of space and time; (ii) the introduction of conservation elements (CEs) and solution elements (SEs) as the vehicles for enforcing space-time flux conservation; (iii) a novel time marching strategy that has a space-time staggered stencil at its core and, as such, fluxes at an interface can be evaluated without using any interpolation or extrapolation procedure (which, in turn, leads to the method's ability to capture shocks without using Riemann solvers); (iv) the requirement that each scheme be built from a non-dissipative core scheme and, as a result, the numerical dissipation can be controlled effectively; and (v) the fact that mesh values of the physical dependent variables and their spatial derivatives are considered as independent marching variables to be solve for simultaneously. Note that CEs are nonoverlapping space-time subdomains introduced such that (i) the computational domain can be filled by these subdomains; and (ii) flux conservation can be enforced over each of them and also over the union of any combination of them. On the other hand, SEs are space-time subdomains introduced such that (i) the boundary of each CE can be divided into several component parts with each of them belonging to a unique SE; and (ii) within a SE, any physical flux vector is approximated using simple smooth functions. In general, a CE does not coincide with a SE.Without using flux-splitting or other special techniques, since its inception [1] the unstructured-mesh compatible CE/SE method has been used to obtain numerous accurate 1D, 2D and 3D steady and unsteady flow solutions with Mach numbers ranging from 0.0028 to 10 [42]. The phyical phenomena modeled include traveling and interacting shocks, acoustic waves, shedding vortices, viscous flows, detonation waves, cavitation, flows in fluid film bearings, heat conduction with melting and/or freezing, electrodynamics, MHD vortex, hydraulic jump, crystal growth, and chromatographic problems . In particular, the rather unique capability of the CE/SE method to resolve both strong shocks and small disturbances (e.g., acoustic waves) simultaneously [11,13,14] makes it an effective tool for attacking computational aeroacoustics (CAA) problems. Note that the fact that second-order CE/SE schemes can solve CAA problems accurately is an exception to the commonly-held belief that a second-order scheme is not adequate for solving CAA problems. Also note that, while numerical dissipation is needed for shock capturing, it may also result in annihilation of small disturbances. Thus a solver that can handle both strong shocks and small disturbances simultaneously must be able to overcome this difficulty.In spite of its past successes, there is still room for improving the CE/SE method. An example is the fact that, in a CE/SE simulation with a fixed t...

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mentioning

confidence: 69%

Section: The a Schemementioning

confidence: 99%

Section: The A-scheme and The C Schemementioning

confidence: 99%

“…To serve as a preliminary for future development, here we shall briefly review an extension of the c scheme which was introduced as a remedy for this deficiency [3,35].…”

Section: The W-α Scheme-a Special Wiggle-suppressing Schemementioning

confidence: 99%

See 2 more Smart Citations

Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Chang¹

2005

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

Self Cite

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“…[4][5]8,[11][12][13] The MOL converts the distributed dynamic system into a large system of ODEs or DAEs, which often requires a long computational time and may give rise to substantial discretization error. 14 A conservation element and solution element method, [14][15][16][17] CE/SE method for short, has been proposed to accurately and effectively solve the distributed dynamic system (or PDEs). The CE/SE method enforces both local and global flux conservation in space and time by using the Gauss's divergence theorem, and uses a simple stencil structure (two points at the previous time level and one point at the present time level) that leads to an explicit time-marching scheme.…”

Section: Introductionmentioning

confidence: 99%

Optimization of a Six-Zone Simulated-Moving-Bed Chromatographic Process

Lim

Jørgensen

2007

Ind. Eng. Chem. Res.

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Using strong cation exchange simulated moving bed (SMB) chromatography, a nitrogen-phosphate-potassium (NPK) fertilizer is produced in a cost-effective manner. The SMB process operated in a non-traditional way is divided into production and regeneration sections for exclusion of undesirable ions, and composed of six zones including two wash-water zones. This paper addresses modeling, simulation and optimization studies on this ionexchange SMB process, based upon experimental data obtained both from a pilot plant and an industrial plant. This study aims to optimize the SMB process in terms of i) maximization of productivity in the production section and ii) minimization of wash-water consumption, thereby resulting in i) increasing the profit and ii) reducing the overall operating cost in the downstream processing, respectively. The two objectives are sequentially treated within the framework of a multi-level optimization procedure (MLOP) which includes two pre-optimization levels, a productivity maximization level and a desorbent (or wash-water) consumption minimization level. In this optimization study, it is demonstrated that wash-water consumption can be reduced by 5 % at a 5% higher productivity.

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Convergence and error bound analysis for the space-time CESE method

Yang

Zhao

2001

Numer. Methods Partial Differential Eq.

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The Method of Space-Time Conservation Element and Solution Element—A New Approach for Solving the Navier-Stokes and Euler Equations

Cited by 468 publications

References 4 publications

Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Optimization of a Six-Zone Simulated-Moving-Bed Chromatographic Process

Convergence and error bound analysis for the space-time CESE method

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