Commun. Comput. Phys., 9 (2011), pp. 568-586.


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

Qingshan Chen 1, Zhen Qin 2, Roger Temam 2*

1 Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA.
2 Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA.

Received 11 September 2009; Accepted (in revised version) 16 March 2010
Available online 17 September 2010
doi:10.4208/cicp.110909.160310s

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.

AMS subject classifications: 35B65, 35K55, 35K57, 65M20, 65M60

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Key words: Compatibility conditions, corner singularities, viscous Burgers equation, nonlinear convection diffusion equation, finite element methods.

*Corresponding author.
Email: qchen3@fsu.edu (Q. Chen), qinz@indiana.edu (Z. Qin), temam@indiana.edu (R. Temam)
 

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