High Order Numerical Methods for the Investigation of the Two Dimensional Richtmyer-Meshkov Instability
Abstract
For flows that contain significant structure, high order schemes offer large advantages over low order schemes. Fundamentally, the reason comes from the truncation error of the differencing operators. If one examines carefully the expression for the truncation error, one will see that for a fixed computational cost that the error can be made much smaller by increasing the numerical order than by increasing the number of grid points. One can readily derive the following expression which holds for systems dominated by hyperbolic effects and advanced explicitly in time: flops = const * p{sup 2} * k{sup (d+1)(p+1)/p}/E{sup (d+1)/p} where flops denotes floating point operations, p denotes numerical order, d denotes spatial dimension, where E denotes the truncation error of the difference operator, and where k denotes the Fourier wavenumber. For flows that contain structure, such as turbulent flows or any calculation where, say, vortices are present, there will be significant energy in the high values of k. Thus, one can see that the rate of growth of the flops is very different for different values of p. Further, the constant in front of the expression is also very different. With a low order scheme, one quickly reaches the limit ofmore »
- Authors:
- Publication Date:
- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- US Department of Energy (US)
- OSTI Identifier:
- 15002232
- Report Number(s):
- UCRL-ID-147252
TRN: US200408%%188
- DOE Contract Number:
- W-7405-ENG-48
- Resource Type:
- Technical Report
- Resource Relation:
- Other Information: PBD: 26 Nov 2001
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; COMPUTERS; INSTABILITY; NUMERICAL SOLUTION; TURBULENT FLOW; VORTICES
Citation Formats
Don, W-S, Gotllieb, D, Shu, C-W, and Jameson, L. High Order Numerical Methods for the Investigation of the Two Dimensional Richtmyer-Meshkov Instability. United States: N. p., 2001.
Web. doi:10.2172/15002232.
Don, W-S, Gotllieb, D, Shu, C-W, & Jameson, L. High Order Numerical Methods for the Investigation of the Two Dimensional Richtmyer-Meshkov Instability. United States. https://doi.org/10.2172/15002232
Don, W-S, Gotllieb, D, Shu, C-W, and Jameson, L. 2001.
"High Order Numerical Methods for the Investigation of the Two Dimensional Richtmyer-Meshkov Instability". United States. https://doi.org/10.2172/15002232. https://www.osti.gov/servlets/purl/15002232.
@article{osti_15002232,
title = {High Order Numerical Methods for the Investigation of the Two Dimensional Richtmyer-Meshkov Instability},
author = {Don, W-S and Gotllieb, D and Shu, C-W and Jameson, L},
abstractNote = {For flows that contain significant structure, high order schemes offer large advantages over low order schemes. Fundamentally, the reason comes from the truncation error of the differencing operators. If one examines carefully the expression for the truncation error, one will see that for a fixed computational cost that the error can be made much smaller by increasing the numerical order than by increasing the number of grid points. One can readily derive the following expression which holds for systems dominated by hyperbolic effects and advanced explicitly in time: flops = const * p{sup 2} * k{sup (d+1)(p+1)/p}/E{sup (d+1)/p} where flops denotes floating point operations, p denotes numerical order, d denotes spatial dimension, where E denotes the truncation error of the difference operator, and where k denotes the Fourier wavenumber. For flows that contain structure, such as turbulent flows or any calculation where, say, vortices are present, there will be significant energy in the high values of k. Thus, one can see that the rate of growth of the flops is very different for different values of p. Further, the constant in front of the expression is also very different. With a low order scheme, one quickly reaches the limit of the computer. With the high order scheme, one can obtain far more modes before the limit of the computer is reached. Here we examine the application of spectral methods and the Weighted Essentially Non-Oscillatory (WENO) scheme to the Richtmyer-Meshkov Instability. We show the intricate structure that these high order schemes can calculate and we show that the two methods, though very different, converge to the same numerical solution indicating that the numerical solution is very likely physically correct.},
doi = {10.2172/15002232},
url = {https://www.osti.gov/biblio/15002232},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Mon Nov 26 00:00:00 EST 2001},
month = {Mon Nov 26 00:00:00 EST 2001}
}