A system of conservation laws including a stiff relaxation term; the 2D case

We analyze a system of conservation laws in two space dimensions with a stiff relaxation term. A semi-implicit finite difference method approximating the system is studied and an error bound of order <img src="/fulltext-image.asp?format=htmlnonpaginated&src=G7W23214JU0X8U22_html\10543_2005_Article_BF01733792_TeX2GIFIE1.gif" border="0" alt=" $$\mathcal{O}(\sqrt {\Delta t} )$$ " /> measured inL¹ is derived. This error bound is independent of the relaxation time d > 0. Furthermore, it is proved that the solutions of the system converge towards the solution of an equilibrium model as the relaxation time d tends to zero, and that the rate of convergence measured inL¹ is of order <img src="/fulltext-image.asp?format=htmlnonpaginated&src=G7W23214JU0X8U22_html\10543_2005_Article_BF01733792_TeX2GIFIE2.gif" border="0" alt=" $$\mathcal{O}(\delta ^{1/3} )$$ " />. Finally, we present some numerical illustrations.