V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions,J.Comp. Phys., 118 (1995), pp. 120--130.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ( 1 ( 4 1 2 1 kk kk ff 1 2 1 ) 1 ( 1 ( 4 1 D = i i R V V kk kk ff (4.14) where V is the primitive flow variables. and D are backward and forward difference operations, respectively. Third order interpolation is obtained from k=1 3. The limiter function f in [11] and [12] is given as 2 2 2 3 ) 2 3 ) i i i i i i i V V V V V V V ee ee ff D D D = 4.15) In this paper, e i 2 is given by ( 12 3 2 10 , 0 . 24 max = i i xx ee (4.16) 4.4.3.4 Turbulence Model Nondimensional viscous coefficient is given by a sum of laminar and turbulent ....

Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady State Solutions," AIAA Paper 93-0880, Jan. 1993.

.... 0.28 0.28 0.28 0.28 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.28 0.28 0. 28 0 1 0 1 (d) Divergence of face centered velocities resulting from application of the Venkat limiter [197] for rL in (31) Figure 1: Face centered velocities constructed from the solenoidal cell centered velocity field in (a) using equations (31) and (32) are not necessarily solenoidal, as shown in (c) and (d) for two different types of slope limiters. 0 1 0 1 X Y 0.006 0.006 0.004 ....

V. Venkatakrishnan. On the accuracy of limiters and convergence to steady state solutions. Technical Report 93--0880, AIAA, 1993. Presented at the 31st Aerospace Sciences Meeting.

....Even though the limiter is frozen , the values of density and pressure are still checked to be sure that they do not dip below zero. If they do, a more restrictive limiter is used for the cell for that one time step. A new alternative to limiter freezing was recently proposed by Venkatakrishnan [63]. The limiter in Equation 3.12 is modified to be more like the Van Albada limiter [53] This new limiter prevents the limiter from acting in smooth regions of the flow, thus allowing the solution to converge. This limiter is, however, a little more diffusive, allowing some smearing of the solution ....

V. Venkatakrishnan, "On the accuracy of limiters and convergence to steady state solutions," AIAA Paper 93-0880, 1993.

....pre computation of certain purely geometric quantities used in the reconstruction, with a corresponding improvement in efficiency. Accuracy near discontinuities is poor, however, and limiting is required to prevent overshoots of O (1) Limiters that retain good convergence properties (e.g. [2]) are often computationally expensive. By design, the ENO reconstruction schemes of Harten et al. [3, 4, 5] conserve the mean, are uniformly accurate at all points for which a smooth neighborhood exists, and guarantee that overshoots will be no larger than the order of the truncation error of the ....

....function approximation or scientific computation, where such overshoots can easily produce aphysical values. This problem has typically been addressed by performing a reconstruction with geometric weights and preventing overshoots by heuristically limiting, or reducing, the derivatives (e.g. [10, 2]) While this approach is not unsuccessful, it provides only a mechanical solution to an underlying theoretical problem. Specifically, the stencil for a control volume i near a discontinuity will include control volumes j which lie across the discontinuity. Because the function is not smooth, ....

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Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady-State Solutions." AIAA paper 93-0880, Jan. 1993.

....in situations where a limiter cycle has developed. This does imply that the solution can no longer be guaranteed to be non positive. In practice, though, this restriction can give reasonable results. Although it is not guaranteed to eliminate the problem, introduction of a smoother limiter, as in [84] or [1] could help with this phenomenon. 2.4.3 Numerical Flux Construction When solving any integral conservation law using a finite volume technique, it is necessary to approximate the flux through the boundaries of the conservation volumes. By first principles, a conservation law relates the ....

V. Venkatakrishnan. On the Accuracy of Limiters and Convergence to Steady State Solutions. AIAA Paper AIAA-93-0880, 1993.

....Variable Dependencies In general, the MUSCL extrapolation scheme described here is only applicable to meshes consisting of hexahedra. For more generalized meshes the extrapolation can be based on properly limited cell gradients. For example, the limiters of Barth [53] or Venkatakrishnan [54] may be applied. 66 5.2.4 Linear System Solution The solution of the Newton iteration step given by equation (5.38) requires solving a linear system of equations of the form Ax = b; 5.56) where A = L 0 (Q n 1;p ) x = Q n 1;p 1 Gamma Q n 1;p ) b = GammaL(Q n 1;p ) The matrix, ....

V. Venkatakrishnan, "On the accuracy of limiters and convergence to steady state solutions," tech. rep., AIAA, 1993. AIAA 93-0880.

....of triangular cells. The conserved variables are reconstructed locally in each control volume. In the present work, a least squares linear reconstruction is used [12, 13] The reconstructed gradients must be limited in order to ensure monotonicity of the solution. Venkatakrishnan s limiter [14] is used because of its superior convergence properties. This limiter reduces the reconstructed gradient Preprint of AIAA J article 3 rOE at the vertex V i locally by a factor of Psi i = max neighbors j 8 : 1 Delta 2 ( Delta 2 1;max ffl 2 ) Delta 2 ....

Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady-State Solutions." AIAA paper 93-0880, Jan. 1993.

....precomputation of certain purely geometric quantities used in the reconstruction, with a corresponding improvement in efficiency. Accuracy near discontinuities is poor, however, and limiting is required to prevent overshoots of O (1) Limiters that retain good convergence properties (e.g. [2]) are often computationally expensive. By design, the ENO reconstruction schemes of Harten et al. 3, 4, 5] conserve the mean, are uniformly accurate at all points for which a smooth neighborhood exists, and guarantee that overshoots will be no larger than the order of the truncation error of the ....

....The first test case is a high angle of attack flow around a NACA 0012 airfoil, at M = 0:302 and ff = 9:86 o . Figure 6 shows the fine mesh used for this case, which contains 3323 vertices. The solution was computed by using unlimited DI L 2 , limited DI L 2 (Venkatakrishnan s limiter [2]) and unlimited DD L 2 . The solutions are virtually identical except near the suction peak on the upper surface. A detail of the surface pressure coefficient in this region is shown in Figure 7; the figure also includes a solution from INS2D to which the KarmanTsien pressure correction has been ....

Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady-State Solutions." AIAA paper 93-0880, Jan. 1993.

....based on smooth data when this is possible. They have the same convergence difficulties as structured stencil searching schemes. A more common approach to reconstruction on unstructured meshes is to use least squares reconstruction followed by some limiting procedure to eliminate overshoots [13, 14]. The reconstruction scheme described here is an extension of previous work [16] on the use of data dependent least squares reconstruction to produce ENO schemes. Previously, high order reconstruction was demonstrated in one dimension, and second order reconstruction and flow solutions were ....

....function approximation or scientific computation, where such overshoots can easily produce aphysical values. This problem has typically been addressed by performing a reconstruction with geometric weights and preventing overshoots by heuristically limiting, or reducing, the derivatives (e.g. [13, 14]) While this approach is not unsuccessful, it provides only a mechanical solution to an underlying theoretical problem. Specifically, the stencil for a control volume i near a discontinuity will include control volumes j that lie on the opposite side of the discontinuity. Because the function is ....

Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady-State Solutions. " AIAA paper 93-0880, Jan. 1993.

....precomputation of certain purely geometric quantities used in the reconstruction, with a corresponding improvement in efficiency. Accuracy near discontinuities is poor, however, and limiting is required to prevent overshoots of O (1) Limiters that retain good convergence properties (e.g. [2]) are often computationally expensive. By design, the ENO reconstruction schemes of Harten et al. 3, 4, 5] conserve the mean, are uniformly accurate at all points for which a smooth neighborhood exists, and guarantee that overshoots will be no larger than the order of the truncation error of the ....

....binary switching for small changes s j . Data independent least squares reconstruction corresponds to choosing s j to be a constant depending only on the local geometry; uniform meshes give s j = 0. This approach can lead to unphysical slopes near discontinuities. If Venkatakrishnan s limiter [2] is used, it is easy to show that s j = 2 8 Gamma 10OE j 4OE 2 j ffl 2 = u j 1 Gamma u j Gamma1 ) 2 (8) for OE j 0. These three choices for s j are shown in Figure 1. The approach taken here is to design weighting schemes for a data dependent least squares reconstruction scheme that ....

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Venkatakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady-State Solutions. " AIAA paper 93-0880, Jan. 1993.

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V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions. AIAA Paper 93-0880, Jan. 1993; to appear in J. Comp. Physics.

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V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions,J.Comp. Phys., 118 (1995), pp. 120--130.

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Venktakrishnan, V., "On the Accuracy of Limiters and Convergence to Steady State Solutions," AIAA paper 93-0880, 1993.

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Venkatakrishnan, V., #On the Accuracy of Limiters and Convergence to Steady State Solutions," AIAA 93-0880, 1993.

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