Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T12:35:12.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Resolution Near t = 0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

Published online by Cambridge University Press:  20 August 2015

Qingshan Chen ,
Zhen Qin  and
Roger Temam
Qingshan Chen*
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA
Zhen Qin*
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA
Roger Temam*
Affiliation:
Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

The incompatibilities between the initial and boundary data will cause singularities at the time-space corners, which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions. We study the corner singularity issue for nonlinear evolution equations in 1D, and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use. Applications of the remedy procedures to the 1D viscous Burgers equation, and to the 1D nonlinear reaction-diffusion equation are presented. The remedy procedures are applicable to other nonlinear diffusion equations as well.

Keywords

35B6535K5535K5765M2065M60Compatibility conditionscorner singularitiesviscous Burgers equationnonlinear convection diffusion equationfinite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 568 - 586
Copyright
Copyright © Global Science Press Limited 2011

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]Akella, S. and Navon, I. M., A comparative study of the performance of high resolution advection schemes in the context of data assimilation, Int. J. Numer. Meth. Fluids., 51(7) (2000), 719–748.CrossRefGoogle Scholar
[2]Bieniasz, L. K., A singularity correction procedure for digital simulation of potential-step chronoamperometric transients in one-dimensional homogeneous reaction-diffusion systems, Electrochim. Acta., 50 (2005), 3253–3261.Google Scholar
[3]Boyd, J. P. and Flyer, N., Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods, Comput. Methods. Appl. Mech. Engrg., 175 (1999), 281–309.Google Scholar
[4]Chen, Q. S., Qin, Z. and Temam, R., Treatment of incompatible initial and boundary data for parabolic equations in higher dimensions, Math. Comput., to appear.Google Scholar
[5]Choi, H., Temam, R., Moin, P. and Kim, J., Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid. Mech., 253 (1993), 509–543.Google Scholar
[6]Flyer, N. and Fornberg, B., Accurate numerical resolution of transients in initial-boundary value problems for the heat equation, J. Comput. Phys., 184 (2003), 526–539.CrossRefGoogle Scholar
[7]Flyer, N. and Fornberg, B., On the nature of initial-boundary value solutions for dispersive equations, SIAM J. Appl. Math., 64 (2003), 546–564 (electronic).Google Scholar
[8]Flyer, N. and Swarztrauber, P. N., The convergence of spectral and finite difference methods for initial-boundary value problems, SIAM J. Sci. Comput., 23 (2002), 1731–1751 (electronic).CrossRefGoogle Scholar
[9]Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall Inc, Englewood Cliffs, NJ, 1964.Google Scholar
[10]Ladyženskaja, O. A., Solonnikov, V. A. and Uralceva, N. N., Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by Smith, S., Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.Google Scholar
[11]Ladyženskaya, O., On the convergence of Fourier series defining a solution of a mixed problem for hyperbolic equations, Doklady Akad. Nauk SSSR (N.S.), 85 (1952), 481–484 (Russian).Google Scholar
[12]Ladyženskaya, O. A., On solvability of the fundamental boundary problems for equations of parabolic and hyperbolic type, Dokl. Akad. Nauk SSSR (N.S.), 97 (1954), 395–398.Google Scholar
[13]Rauch, J. B. and Massey, F. J. III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303–318.Google Scholar
[14]Smale, S., Smooth solutions of the heat and wave equations, Comment. Math. Helv., 55 (1980), 1–12.Google Scholar
[15]Temam, R., Behaviour at time t=0 of the solutions of semilinear evolution equations, J. Differ. Eqs., 43 (1982), 73–92.Google Scholar
[16]Temam, R., Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Theory and Numerical Analysis, Reprint of the 1984 edition.Google Scholar
[17]Temam, R., Suitable initial conditions, J. Comput. Phys., 218 (2006), 443–450.CrossRefGoogle Scholar
[18]Zhang, D. S., Wei, G. W. and Kouri, D. J., Burgers equation with high reynolds number, Phys. Fluids., 9 (1997), 1853–1855.CrossRefGoogle Scholar