M-S. Liou and C.J. Steffen. A new flux splitting scheme. Journal of Computational Physics, 107:23--29, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by increasing the order of the scheme, it is from the free transport dynamics. Although the FVS schemes are robust, the accuracy of the schemes is limited. The large artificial dissipation in the FVS scheme could easily deteriorate the physical viscous term in the Navier Stokes flow calculations [13, 27, 11]. 3. Flux Di#erence Splitting Scheme. The well known FDS schemes include the Godunov, Roe and Osher s methods. The analysis of the Godunov method in this section can also be applied to the Roe and Osher schemes in the explanation of the shock instability. 3.1. The Godunov Gas Evolution Model. ....

.... and accuracy, a recent trend in the development of upwind schemes has centered around the construction of hybrid flux splitting formulations which combine the accuracy of the FDS approach in the resolution of shear layer and the robustness of the FVS method in the capturing of strong discontinuity [3, 4, 11, 13, 15, 2, 9]. For example, starting from the FVS scheme, in order to reduce the free penetration mechanism, the correlation and collision between left and right moving waves have to be introduced. The constructions of the values M 1 2 ,U 1 2 at a cell interface in AUSM type schemes are largely based on this ....

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M.S. Liou and C.J Steffen, A New Flux Splitting Scheme,J.Comput.Phys.,107 (1993), pp. 23.

.... the arithmetic averages u = 1 2 (u j 1 u j ) w = 1 2 (w j 1 w j ) Thus these schemes are closely related to schemes which introduce separate splittings of the convective and pressure terms, such as the wave particle scheme [158, 12] the advection upwind splitting method (AUSM) [114, 197], and the convective upwind and split pressure (CUSP) schemes [85] In order to examine the shock capturing properties of these various schemes, consider the general case of a first order di#usive flux of the form 1 w j ) 34) where the matrix B j determines the properties of the ....

M.-S. Liou and C. Steffen, A new flux splitting scheme, J. Comp. Phys., 107 (1993), pp. 23--39.

.... Now the split is redefined as and the diffusive flux can be expressed as f = uwn fp, fp= p dj = cc wn 3n u Then ( and are defined as before by equations (19) and (20) using the modified eigenvalues ) defined equation (21) This splitting corresponds to the Liou Steffen splitting [12, 20]. The splitting in which the convective terms contain pE corresponds to the wave particle splitting [14, 2] As in the case of characteristic splitting, the two variations can conveniently be distinguished as the E CUSP and H CUSP schemes. 6 Implementation of limiters In the case of a scalar ....

M-S. Liou and C.J. Steffen. A new flux splitting scheme. J. Comp. Phys., 107:23-39, 1993.

....flow, and which may be used as the basis for constructing higher order schemes. Upwinding of the pressure requires the introduction of terms which may be related to the wave particle scheme of Deshpande, Rao and Balakrishnan [35, 4] and flux splitting recently proposed by Liou and Steffen [29]. This simple convective upwind and split pressure (CUSP) scheme produces discrete normal shock waves which contain two or three interior points in transonic flow, and become sharper at very high Mach numbers. Section 4 presents results of various test calculations for one , two and ....

....needed to produce perfect upwinding in the supersonic zone. The scheme retains the property of the original scheme that it is compatible with constant stagnation enthalpy in steady flow. If one derives the viscosity corresponding to the flux splitting recently proposed by Liou and Steffen [29], following equation (33) one finds that their scheme produces first order diffusion with a similar general form, and the present scheme may thus be regarded as a construction of artificial viscosity approximately equivalent to Liou Steffen splitting. 3.3 Multi dimensional systems Schemes for ....

M-S. Liou and C.J. Steffen. A new flux splitting scheme. J. Comp. Phys., 107:23-39, 1993.

.... quasi 1D inviscid Laval nozzle flow at low Mach numbers [97] Since the explicit convection pressure splitting led to stability problems when applied in the transverse direction of a 2D nozzle, the modifications introduced with similar but more general flux vector splittings by Liou and Steffen [112] and by Jameson [113] seem to be essential. Perturbation techniques have been employed for steady and unsteady flow calculations. For very low Mach numbers of O(10 Gamma2 ) to O(10 Gamma5 ) Merkle and Choi use asymptotic expansions of the Euler equations in terms of flM 2 1 and solve the ....

M.-S. Liou and C. Steffen. A new flux splitting scheme. Journal of Computational Physics, 107:23--39, 1993.

....(2.6) is equivalent to using them directly on the flux vectors F in (2.9) at the zero limit of #. RECONSTRUCTION OF UPWIND FLUXES 243 3. FLUX SPLITTING AND ENTROPY FIX Many different flux splittings have been developed in the past decade, and their development is still an active research area [17]. We have considered four representative flux splittings to implement the finite volume method via the reconstruction of the upwind fluxes of a flux splitting using BAP. They are Steger Warming splitting [27] van Leer splitting [29] kinetic flux vector splitting [20] byMandel and Deshpande, and ....

M.-S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107, 23 (1993).

.... as follows 3 (23) where the superscripts 3 and indicate ( # ) 24) Likewise, the pressure is evaluated as per Liou [13] 3 3 43 (25) 7 3 # 3F 3 ( # 3 ( 26) The dissipation term evaluation is also M dependent, although the construct is ....

....[4] 5.54 0.00 34800 34800 fill uvwT stencil2 dir2 [5] 5.24 0.00 352800 352800 gradient[6] 4.50 0.00 17400 17400 fill uvwT stencil dir1 [7] 3.95 0.00 36000 36000 fill metrik stencil dir3 [8] 2.93 0.00 34800 34800 fill metrik stencil dir2 [10] 2.48 0. 00 17400 17400 fill metrik stencil dir1 [13] 2.04 0.00 441000 882000 add2flux differences [8] 0.94 0.00 264600 705600 get 1dline 3dfield [12] 3] 19.9 2.25 7.76 20 ausm plus [3] 2.92 0.00 88200 88200 ausm plus flux [11] 2.04 0.00 441000 882000 add2flux differences [8] 1.56 0.00 441000 705600 get 1dline 3dfield [12] 0.92 0.00 ....

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M.-S.L. Liou and Ch.J. Steffen Jr. A new flux splitting scheme. Journal of Computational Physics, 107:23--39, 1993.

....V = 0:025. 2.3 CUSP Scheme In the previous sections we have described the use of an artificial viscosity based on either a scalar or matrix coefficient. Inspired by earlier work on flux vector splitting [34] Liou and coworkers designed a scheme called Advection Upstream Splitting Method (AUSM) [10, 11, 35]. This method was later refined for large scale 3 D viscous computations in [17] AUSM is based on a splitting of the flux function into convective and pressure contributions. In some sense, the pressure terms contribute to the acoustic waves while the velocity terms contribute to convective ....

M.-S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107,23(1993).

....of which the uni dimen q t R = R 1 A F x mid ( DS edges = 37 sional schemes appear to be the most robust. In this thesis, three flux formulations are used; Roe s flux difference splitting[68] van Leer s flux vector splitting[80] and Liou s Advection Upstream Splitting Method [48]. Computationally, it is a simple matter to change flux functions by replacing a single routine that computes the flux given the left and right interface states. 2.4.3.1 Roe s Flux Difference Splitting: FDS Roe s flux difference splitting solves an approximate Riemann problem at the cell ....

M.S. Liou and C.J.Steffen, Jr. A New Flux Splitting Scheme. Journal of Computational Physics, 107:23--39, 1993.

....hand, the FVS schemes do not generate these spurious solutions, since they are intrinsically solving viscous equations rather than the inviscid Euler equations. So, an optimum choice to get a better scheme is to combine both the FVS and FDS methodology, to develop the so called hybrid schemes [3, 4, 10, 13, 16, 2]. The detail explanation will be presented in this paper. In this paper, section 2 is about the physical analysis of the FVS scheme, and section 3 is related to the FDS scheme. Section 4 gives reasons for the necessity of developing hybrid schemes, which combines both the FVS and FDS methods. The ....

....the FVS scheme, in order to reduce the free penetration of different waves or particles, the correlation or collisions between left and right moving waves have to be introduced. The construction of the common M 1=2 ; U 1=2 terms in AUSM type schemes is largely based on this physical consideration[10]. Similarly, Moschetta and Pullin [16] tried to use Osher s numerical flux to replace linearly degenerate wave in the KFVS scheme. From the FDS scheme, additional dissipations are usually introduced to stabilize discontinuous solutions [27] The main dynamical effects of using two waves instead of ....

M.S. Liou and C.J Steffen (1993), "A New Flux Splitting Scheme", J. Comput. Phys., Vol.107, 23.

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M-S. Liou and C.J. Steffen. A new flux splitting scheme. Journal of Computational Physics, 107:23--29, 1993.

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M. S. Liou and Ch. J. Steffen. A new flux splitting scheme. Journal of Computational Physics, 107:23--39, 1993.

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M.S. Liou and C.J. Steffen, A New Flux Splitting Scheme,J.Comput.Phys.107 (1993), pp. 23.

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M.-S. Liou and C. J. Steffan, A new flux splitting scheme, J. Comp. Phys., 107 (1993), pp. 23--39.

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M.-S. Liou, C.J. Steffen. A New Flux Splitting Scheme. NASA Technical Memorandum 104404, Cleveland, Ohio, May 1991.

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Liou, M.-S., and Christopher J. Steffen, J., "A New Flux Splitting Scheme," NASA TM 104404, May 1991.

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