Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T10:52:16.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  31 October 2017

Elliott S. Wise ,
Ben T. Cox  and
Bradley E. Treeby
Elliott S. Wise*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
Ben T. Cox*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
Bradley E. Treeby*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
*
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
Get access

Abstract

Moving mesh methods provide an efficient way of solving partial differential equations for which large, localised variations in the solution necessitate locally dense spatial meshes. In one-dimension, meshes are typically specified using the arclength mesh density function. This choice is well-justified for piecewise polynomial interpolants, but it is only justified for spectral methods when model solutions include localised steep gradients. In this paper, one-dimensional mesh density functions are presented which are based on a spatially localised measure of the bandwidth of the approximated model solution. In considering bandwidth, these mesh density functions are well-justified for spectral methods, but are not strictly tied to the error properties of any particular spatial interpolant, and are hence widely applicable. The bandwidth mesh density functions are illustrated in two ways. First, by applying them to Chebyshev polynomial approximation of two test functions, and second, through use in periodic spectral and finite-difference moving mesh methods applied to a number of model problems in acoustics. These problems include a heterogeneous advection equation, the viscous Burgers’ equation, and the Korteweg-de Vries equation. Simulation results demonstrate solution convergence rates that are up to an order of magnitude faster using the bandwidth mesh density functions than uniform meshes, and around three times faster than those using the arclength mesh density function.

Keywords

Moving mesh methodpseudospectral methodbandwidth

MSC classification

Secondary: 65M06: Finite difference methods 65M50: Mesh generation and refinement 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1286 - 1308
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baltensperger, R.. Some results on linear rational trigonometric interpolation. Computers & Mathematics with Applications, 43(6-7):737–746, 2002.Google Scholar
[2] Chen, Zhong and Zhou, YongFang. An efficient algorithm for solving hilbert type singular integral equations of the second kind. Computers & Mathematics with Applications, 58(4):632–640, 2009.Google Scholar
[3] Clark, James J., Palmer, Matthew R., and Lawrence, Peter D.. A transformationmethod for the reconstruction of functions from nonuniformly spaced samples. IEEE Trans. Acoust., Speech, and Signal Process., 33(5):1151–1165,Oct 1985.Google Scholar
[4] Cohen, L.. Instantaneous ‘anything’. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 4, pages 105–108, Washington, DC, USA, 1993. IEEE Computer Society.Google Scholar
[5] Cohen, Leon. Local values in quantum mechanics. Phys. Lett. A, 212(6):315–319, 1996.Google Scholar
[6] Davidson, Keith L. and Loughlin, Patrick J.. Instantaneous spectralmoments. J. Franklin Inst., 337(4):421–436, 2000.Google Scholar
[7] Driscoll, T. A, Hale, N., and Trefethen, L. N.. Chebfun Guide. Pafnuty Publications, 2014.Google Scholar
[8] Feng, W. M., Yu, P., Hu, S. Y., and Liu, Z. K.. A fourier spectral moving mesh method for the cahn-hilliard equation with elasticity. Commun. Comput. Phys., 5(2-4):582–599, 2009.Google Scholar
[9] Feng, W. M., Yu, P., Hu, S. Y., Liu, Z. K., Du, Q., and Chen, L. Q.. Spectral implementation of an adaptive moving mesh method for phase-field equations. J. Comput. Phys., 220(1):498–510, Dec 2006.Google Scholar
[10] Guillard, Herv and Peyret, Roger. On the use of spectral methods for the numerical-solution of stiff problems. Computer Methods in AppliedMechanics and Engineering, 66(1):17–43, 1988.CrossRefGoogle Scholar
[11] Hale, Nicholas. On The Use Of Conformal Maps To Speed Up Numerical Computations. PhD thesis, University of Oxford, 2009.Google Scholar
[12] Hale, Nicholas and Tee, T.Wynn. Conformalmaps to multiply slit domains and applications. SIAM J. Sci. Comput., 31(4):3195–3215, 2009.Google Scholar
[13] Huang, Weizhang, Ren, Yuhe, and Russell, Robert D.. Moving mesh partial differential equations (mmpdes) based on the equidistribution principle. SIAMJ. Numer. Anal., 31(3):709–730, 1994.CrossRefGoogle Scholar
[14] Huang, Weizhang and Russell, Robert D.. Analysis of moving mesh partial differential equations with spatial smoothing. SIAM J. Numer. Anal., 34(3):1106–1126, 1997.Google Scholar
[15] Huang, Weizhang and Russell, Robert D.. Adaptive moving mesh methods, volume 174 of Applied Mathematical Sciences. Springer Science+Business Media, 223 Spring Street, New York, NY 10013, USA, 2011.Google Scholar
[16] Jaros, Jiri, Rendell, Alistair. P., and Treeby, Bradley E.. Full-wave nonlinear ultrasound simulation on distributed clusters with applications in high-intensity focused ultrasound. Int. J. High Perform. Comput. Appl., 30(2):137–155, 2016.Google Scholar
[17] Loughlin, Patrick J. and Tacer, Berkant. On the amplitude- and frequency-modulation decomposition of signals. J. Acoust. Soc. Am., 100(3):1594–1601, 1996.Google Scholar
[18] Mulholland, L. S., Huang, W. Z., and Sloan, D. M.. Pseudospectral solution of near-singular problems using numerical coordinate transformations based on adaptivity. SIAM J. Sci. Comput., 19(4):1261–1289, 1998.Google Scholar
[19] Shampine, L. F.. Solving 0 = f (t, y (t), y(t)) in matlab. J. Numer. Math., 10(4):291–310, 2002.Google Scholar
[20] Shen, Jie and Yang, Xiaofeng. An efficient moving mesh spectral method for the phase-field model of two-phase flows. J. Comput. Phys., 228(8):2978–2992, 2009.Google Scholar
[21] Subich, Christopher J.. A robust moving mesh method for spectral collocation solutions of time-dependent partial differential equations. J. Comput. Phys., 294:297–311, 2015.Google Scholar
[22] Tapia, Juan J. and Gilberto López, P.. Adaptive pseudospectral solution of a diffuse interface model. J. Comput. Appl. Math., 224(1):101–117, 2009.Google Scholar
[23] Wynn Tee, T.. An adaptive rational spectral method for differential equations with rapidly varying solutions. PhD thesis, University of Oxford, 2006.Google Scholar
[24] Wynn Tee, T. and Trefethen, Lloyd N.. A rational spectral collocation method with adaptively transformed chebyshev grid points. SIAM J. Sci. Comput., 28(5):1798–1811, 2006.Google Scholar
[25] Wise, Elliott S., Cox, Ben T., and Treeby, Bradley E.. A monitor function for spectral moving mesh methods applied to nonlinear acoustics. In Recent Developments in Nonlinear Acoustics: 20th International Symposium on Nonlinear Acoustics including the 2nd International Sonic Boom Forum, volume 1685, page 070009. AIP Publishing, 2015.Google Scholar
[26] Zabusky, N. J. and Kruskal, M. D.. Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15(6):240–243, 1965.Google Scholar