Ninth MSU-UAB Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 20 (2013), pp. 53-63.

Title: Total variation stability and second-order accuracy at extrema

Author: Ritesh Kumar Dubey (SRM Univ., Tamilnadu, India)
	  
Abstract: 
 It is well known that high order total variation diminishing (TVD)
 schemes for hyperbolic conservation laws degenerate to first-order accuracy,
 even at smooth extrema; hence they suffer from clipping error.
 In this work, TVD bounds on representative three-point second-order
 accurate schemes are given for the scalar case, which show that it is possible
 to obtain second order TVD approximation at points of extrema as well as in
 steep gradient regions. These bounds can be used to improve existing
 high order TVD schemes and to reduce clipping error. In a 1D scalar test
 cases, an existing limiters based high order TVD scheme is applied,
 along with these second-order schemes using their TVD bounds to show
 improvement in the numerical results at extrema and steep gradient regions.

Published October 31, 2013.
Math Subject Classifications: 35L65, 65M06, 65M12.
Key Words: Second order accurate schemes; total variationdiminishing schemes;
           smoothness parameter; hyperbolic equations.