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June  2012, 5(3): 641-656. doi: 10.3934/dcdss.2012.5.641

## An explicit stable numerical scheme for the $1D$ transport equation

 1 Commissariat à l’Énergie Atomique (CEA), DEN/DANS/DM2S/SFME/LETR, 91191 Gif-sur-Yvette, France

Received  August 2010 Revised  October 2010 Published  October 2011

We derive in this paper a numerical scheme in order to calculate solutions of $1D$ transport equations. This $2nd$-order scheme is based on the method of characteristics and consists of two steps: the first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two $2nd$-order interpolation schemes so as to preserve the maximum principle. The resulting method is designed for classical solutions and is unconditionally stable.
Citation: Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
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