Practical aspects of higher-order numerical schemes for wave propagation phenomena.

*(English)*Zbl 0959.65103Summary: This paper examines practical issues related to the use of compact-difference-based fourth- and sixth-order schemes for wave propagation phenomena with focus on Maxwell’s equations of electromagnetics. An outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches: transformed plane method and physical space differencing.

In the first technique, the truncation error expansion for the sixth-order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space-differencing formulas are derived for the five-point stencil by expressing the coefficients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes.

To ensure stability for a broad range of problems, Fourier analysis is employed to develop a single-parameter family up to tenth-order tridiagonal-based spatial filters. The implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1-D cavity where they are employed to suppress a late-time instability. The paper concludes after demonstrating the application of the scheme to several 3-D canonical problems utilizing Cartesian as well as curvilinear meshes.

In the first technique, the truncation error expansion for the sixth-order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available. Physical space-differencing formulas are derived for the five-point stencil by expressing the coefficients in terms of local spacing ratios. The order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes.

To ensure stability for a broad range of problems, Fourier analysis is employed to develop a single-parameter family up to tenth-order tridiagonal-based spatial filters. The implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1-D cavity where they are employed to suppress a late-time instability. The paper concludes after demonstrating the application of the scheme to several 3-D canonical problems utilizing Cartesian as well as curvilinear meshes.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

35L60 | First-order nonlinear hyperbolic equations |

35Q60 | PDEs in connection with optics and electromagnetic theory |

##### Keywords:

finite difference method; convergence; error bounds; nonlinear first-order hyperbolic equations; wave propagation; Maxwell’s equations; mesh refinement; numerical experiment; stability
PDF
BibTeX
XML
Cite

\textit{D. V. Gaitonde} et al., Int. J. Numer. Methods Eng. 45, No. 12, 1849--1869 (1999; Zbl 0959.65103)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Shang, AIAA Journal 33 pp 491– (1995) |

[2] | Taflove, Computing Systems in Engineering 3 pp 1– (1992) |

[3] | Lele, Journal of Computational Physics 103 pp 16– (1992) |

[4] | Tam, Journal of Computational Physics 107 pp 262– (1993) |

[5] | Hirsh, Journal of Computational Physics 19 pp 90– (1975) |

[6] | Young, IEEE AP-S International Symposium 3 pp 1992– (1997) |

[7] | Young, IEEE Transactions on Antennas and Propagation 45 pp 1573– (1997) |

[8] | Gaitonde, Journal of Computational Physics 138 pp 617– (1997) |

[9] | Gustafsson, Mathematics of Computation 26 pp 649– (1972) |

[10] | Carpenter, Journal of Computational Physics 108 pp 272– (1993) |

[11] | Kreiss, Applied Numerical Mathematics 12 pp 213– (1993) |

[12] | Vichnevetsky, Mathematics and Computers in Simulation XXIII pp 344– (1981) |

[13] | Gustafsson, Journal of Computational Physics 117 pp 300– (1995) |

[14] | Carpenter, Journal of Computational Physics 111 pp 220– (1994) |

[15] | Implementation of a high-accuracy finite-difference scheme for linear wave phenomena. Proceedings of the International Conference on Spectral and High-Order Methods, June 1995. |

[16] | Harten, Journal of Computational Physics 71 pp 231– (1987) |

[17] | Elementary Numerical Analysis - An Algorithmic Approach. McGraw-Hill Book Company: New York, 1980. |

[18] | Fourier analysis of numerical approximations of hyperbolic equations. SIAM Studies in Applied Mathematics 1982. |

[19] | Time-Harmonic Electromagnetic Fields. McGraw-Hill Book Company: New York, 1961. |

[20] | Numerical Grid Generation. North-Holland: New York, 1985. |

[21] | Casper, Journal of Computational Physics 106 pp 62– (1993) |

[22] | Fung, AIAA Journal 34 pp 2029– (1996) |

[23] | Network Synthesis, vol. 1. John Wiley and Sons: New York, 1958. |

[24] | Numerical filtering for partial differential equations. Numerical Applications Memorandum. Rutgers University, NAM 156, November 1974. |

[25] | Kennedy, Applied Numerical Mathematics 14 pp 397– (1994) |

[26] | High-order finite-volume schemes in wave propagation phenomena. AIAA Paper 96-2335, June 1996. |

[27] | High-order schemes for Navier-Stokes equations: algorithm and implementation into FDL3DI. Technical Report AFRL-VA-WP-TR-1998-3060. Air Force Research Laboratory, Wright-Patterson AFB, 1998. |

[28] | Yee, IEEE Transactions on Antenna and Propagation 14 pp 302– (1966) |

[29] | Shlager, IEEE Transactions on Antenna and Propagation 41 pp 1732– (1993) |

[30] | Optimized schemes for time-domain electromagnetics. IEEE AP-S International Symposium, Montreal, Quebec, July 1997. |

[31] | Radiation Boundary Conditions. Ph.D. Thesis, Northwestern University, Evanston, IL, 1988. |

[32] | Gustafsson, Mathematics of Computation 29 pp 396– (1975) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.