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Journal of Fluids Engineering | Volume 130 | Issue 7 | Technical Brief
Technical Briefs

Understanding the Boundary Stencil Effects on the Adjacent Field Resolution in Compact Finite Differences

[+] Author and Article Information
Stephen A. Jordan

 Naval Undersea Warfare Center, Newport, RI 02842jordansa@npt.nuwc.navy.mil

J. Fluids Eng 130(7), 074502 (Jun 25, 2008) (7 pages) doi:10.1115/1.2948366 History: Received May 31, 2007; Revised April 05, 2008; Published June 25, 2008
Article
References
Figures

When establishing the spatial resolution character of a composite compact finite differencing template for high-order field solutions, the stencils selected at nonperiodic boundaries are commonly treated independent of the interior scheme. This position quantifies a false influence of the boundary scheme on the resultant interior dispersive and dissipative consequences of the compound template. Of the three ingredients inherent in the composite template, only its numerical accuracy and global stability have been properly treated in a coupled fashion. Herein, we present a companion means for quantifying the resultant spatial resolution properties that lead to improved predictions of the salient problem physics. Compact boundary stencils with free parameters to minimize the field dispersion (or phase error) and dissipation are included in the procedure. Application of the coupled templates for resolving the viscous Burgers wave and two-dimensional acoustic scattering reveal significant differences in the predictive error.
FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
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References

1
Lele, S. K., 1992, “Compact Finite Difference Schemes With Spectral-Like Resolution,” J. Comput. Phys.  [CrossRef], 103 , pp. 16–42.
2
Carpenter, M. H., Gottlieb, D., and Abarbanel, S., 1993, “The Stability of Numerical Boundary Treatments for Compact High-Order Finite-Difference Schemes,” J. Comput. Phys.  [CrossRef], 108 , pp. 272–295.
3
Cook, A. W., and Riley, J. J., 1996, “Direct Numerical Simulation of a Turbulent Reactive Plume on a Parallel Computer,” J. Comput. Phys., 129 (2), pp. 263–283.
4
Fedioun, I., Lardjane, N., and Gökalp, I., 2001, “Revisiting Numerical Errors in Direct and Large Eddy Simulations of Turbulence: Physical and Spectral Spaces Analysis,” J. Comput. Phys.  [CrossRef], 174 , pp. 816–851.
5
Tam, C. K. W., and Hardin, C., 1997, "Second Computational Aeroacoustics (CAA) Workshop on Benchmark Problems", Hampton, VA, NASA CP-3352.
6
Hixon, R., and Turkel, E., 1998, “High-Accuracy Compact MacCormack-Type Schemes for Computational Aeroacoustics,” Report NASA∕CR-1998-208672.

Figures

Grahic Jump Location
+Spatial resolution errors of the explicit (4e and 5e) and several compact finite difference schemes; 4–8 denotes formal order
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Spatial resolution errors of the explicit (4e and 5e) and several compact finite difference schemes; 4–8 denotes formal order
Figure 1

Spatial resolution errors of the explicit (4e and 5e) and several compact finite difference schemes; 4–8 denotes formal order

Grahic Jump Location
+Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (3‐4‐3)(1); (3) third-order-accurate on boundary, (4) fourth-order accurate in field
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Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (3‐4‐3)(1); (3) third-order-accurate on boundary, (4) fourth-order accurate in field
Figure 2

Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (3‐4‐3)(1); (3) third-order-accurate on boundary, (4) fourth-order accurate in field

Grahic Jump Location
+Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (5‐5‐6‐5‐5)(3); (5) fifth-order-accurate on boundary and first interior point, (6) sixth-order Pade-type in remaining field
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Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (5‐5‐6‐5‐5)(3); (5) fifth-order-accurate on boundary and first interior point, (6) sixth-order Pade-type in remaining field
Figure 3

Real (dispersion) and imaginary (dissipation) distributions of modified wave numbers for composite compact finite difference template (5‐5‐6‐5‐5)(3); (5) fifth-order-accurate on boundary and first interior point, (6) sixth-order Pade-type in remaining field

Grahic Jump Location
+Dispersive distributions of composite compact finite difference template (22‐4‐22)(1); (22) two-parameters (a=4, η=−1∕2) second-order-accurate on boundary, (4) fourth-order Pade-type in interior
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Dispersive distributions of composite compact finite difference template (22‐4‐22)(1); (22) two-parameters (a=4, η=−1∕2) second-order-accurate on boundary, (4) fourth-order Pade-type in interior
Figure 4

Dispersive distributions of composite compact finite difference template (22‐4‐22)(1); (22) two-parameters (a=4, η=−1∕2) second-order-accurate on boundary, (4) fourth-order Pade-type in interior

Grahic Jump Location
+Real (dispersion) and imaginary (dissipation) distributions of modified wavenumbers for composite compact finite difference template (43‐4‐43)(2); (43) explicit fourth-order three-parameter family defined on boundary
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Real (dispersion) and imaginary (dissipation) distributions of modified wavenumbers for composite compact finite difference template (43‐4‐43)(2); (43) explicit fourth-order three-parameter family defined on boundary
Figure 5

Real (dispersion) and imaginary (dissipation) distributions of modified wavenumbers for composite compact finite difference template (43‐4‐43)(2); (43) explicit fourth-order three-parameter family defined on boundary

Grahic Jump Location
+Energy loss in solutions of Burgers equation using composite template (3-4-3)
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Energy loss in solutions of Burgers equation using composite template (3-4-3)
Figure 6

Energy loss in solutions of Burgers equation using composite template (3-4-3)

Grahic Jump Location
+Solutions of Burgers equation using composite template (3-4-3)
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Solutions of Burgers equation using composite template (3-4-3)
Figure 7

Solutions of Burgers equation using composite template (3-4-3)

Grahic Jump Location
+Predictive errors (L2 norm) at the fictitious boundary of composite templates as applied to the viscous Burgers equation; ‖L2‖=[(1∕N)Σ∣ue′−ufd′∣2]1∕2; optimized composite templates are (22‐4‐22)(1), (43‐4‐43)(2), and (5‐5‐6‐5‐5)(3)
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Predictive errors (L2 norm) at the fictitious boundary of composite templates as applied to the viscous Burgers equation; ‖L2‖=[(1∕N)Σ∣ue′−ufd′∣2]1∕2; optimized composite templates are (22‐4‐22)(1), (43‐4‐43)(2), and (5‐5‐6‐5‐5)(3)
Figure 8

Predictive errors (L2 norm) at the fictitious boundary of composite templates as applied to the viscous Burgers equation; ‖L2‖=[(1∕N)Σ∣ue′−ufd′∣2]1∕2; optimized composite templates are (22‐4‐22)(1), (43‐4‐43)(2), and (5‐5‐6‐5‐5)(3)

Grahic Jump Location
+Radial pressure distribution of acoustic scatter problem (45deg, t=4); optimized composite templates are (22‐4‐22)(1) and (43‐4‐43)(2)
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Radial pressure distribution of acoustic scatter problem (45deg, t=4); optimized composite templates are (22‐4‐22)(1) and (43‐4‐43)(2)
Figure 9

Radial pressure distribution of acoustic scatter problem (45deg, t=4); optimized composite templates are (22‐4‐22)(1) and (43‐4‐43)(2)

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