Skip to main content

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

  • Elliott S. Wise (a1), Ben T. Cox (a1) and Bradley E. Treeby (a1)

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.

Corresponding author
*Corresponding author. Email addresses: S. Wise), T. Cox), E. Treeby)
Hide All
[1] Baltensperger, R.. Some results on linear rational trigonometric interpolation. Computers & Mathematics with Applications, 43(6-7):737746, 2002.
[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):632640, 2009.
[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):11511165,Oct 1985.
[4] Cohen, L.. Instantaneous ‘anything’. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 4, pages 105108, Washington, DC, USA, 1993. IEEE Computer Society.
[5] Cohen, Leon. Local values in quantum mechanics. Phys. Lett. A, 212(6):315319, 1996.
[6] Davidson, Keith L. and Loughlin, Patrick J.. Instantaneous spectralmoments. J. Franklin Inst., 337(4):421436, 2000.
[7] Driscoll, T. A, Hale, N., and Trefethen, L. N.. Chebfun Guide. Pafnuty Publications, 2014.
[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):582599, 2009.
[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):498510, Dec 2006.
[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):1743, 1988.
[11] Hale, Nicholas. On The Use Of Conformal Maps To Speed Up Numerical Computations. PhD thesis, University of Oxford, 2009.
[12] Hale, Nicholas and Tee, T.Wynn. Conformalmaps to multiply slit domains and applications. SIAM J. Sci. Comput., 31(4):31953215, 2009.
[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):709730, 1994.
[14] Huang, Weizhang and Russell, Robert D.. Analysis of moving mesh partial differential equations with spatial smoothing. SIAM J. Numer. Anal., 34(3):11061126, 1997.
[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.
[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):137155, 2016.
[17] Loughlin, Patrick J. and Tacer, Berkant. On the amplitude- and frequency-modulation decomposition of signals. J. Acoust. Soc. Am., 100(3):15941601, 1996.
[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):12611289, 1998.
[19] Shampine, L. F.. Solving 0 = f (t, y (t), y(t)) in matlab. J. Numer. Math., 10(4):291310, 2002.
[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):29782992, 2009.
[21] Subich, Christopher J.. A robust moving mesh method for spectral collocation solutions of time-dependent partial differential equations. J. Comput. Phys., 294:297311, 2015.
[22] Tapia, Juan J. and Gilberto López, P.. Adaptive pseudospectral solution of a diffuse interface model. J. Comput. Appl. Math., 224(1):101117, 2009.
[23] Wynn Tee, T.. An adaptive rational spectral method for differential equations with rapidly varying solutions. PhD thesis, University of Oxford, 2006.
[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):17981811, 2006.
[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.
[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):240243, 1965.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 25 *
Loading metrics...

Abstract views

Total abstract views: 130 *
Loading metrics...

* Views captured on Cambridge Core between 31st October 2017 - 20th March 2018. This data will be updated every 24 hours.