% \input /afs/eos.ncsu.edu/users/z/zhilin/TEX/psfig.tex
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%
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%
%
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\def\ddx{{\partial\over\partial x}}

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\def\ujmnp{U_{j-1}^{n+1}}
\def\ujpnp{U_{j+1}^{n+1}}
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\def\vnp{V^{n+1}}
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\def\wjn{W_j^n}
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\def\ejn{E_j^n}
%\def\cjmh{C_{j-1/2}}
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%\def\djmh{D_{j-1/2}}
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\def\xjph{{x_{j+1/2}}}
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\def\xiph{{x_{i+1/2}}}
\def\yjmh{{y_{j-1/2}}}
\def\yjph{{y_{j+1/2}}}
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\def\uipjmn{U_{i+1,j-1}^n}
\def\uipjpn{U_{i+1,j+1}^n}
\def\uimjpn{U_{i-1,j+1}^n}
\def\uimjmn{U_{i-1,j-1}^n}
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\def\luijnp{u_{ij}^{n+1}}
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\def\gijn{G_{ij}^n}
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\def\dobar{{\cal D}_0(\bar x,\bar t)}
\def\askgo{\quad\mbox{as}~~k\goto 0}
\def\sjn{\sigma_j^n}
\def\sig{\sigma}
\def\sjmn{\sigma_{j-1}^n}
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\def\apj{\alpha_{pj}}
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\def\sxk{S_k^x}
\def\syk{S_k^y}
\def\ext{e^{-ta\partial_x}}
\def\eyt{e^{-tb\partial_y}}
\def\exyt{e^{-t(a\partial_x + b\partial_y)}}
\def\eA{e^{-tA\partial_x}}
\def\ehA{e^{-\half tA\partial_x}}
\def\eB{e^{-tB\partial_y}}
\def\eAB{e^{-t(A\partial_x + B\partial_y)}}
\def\ekhA{e^{-\half kA\partial_x}}
\def\ekB{e^{-kB\partial_y}}
\def\ekAB{e^{-k(A\partial_x + B\partial_y)}}
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\def\dy{\partial_y}
\def\dt{\partial_t}
\def\hhx{{\cal H}_{k/2}^x}
\def\hy{{\cal H}_{k}^y}
\def\hx{{\cal H}_{k}^x}
\def\uss{u^{**}}
\def\wset{{\cal W}}
\def\dist{\mbox{dist}}
\def\kcpt{{\cal K}}
\def\LT{L_{1,{\scriptscriptstyle T}}}
\def\FG{F_{\scriptscriptstyle G}}
\def\uD{u^{\scriptscriptstyle D}}
\def\llw{L_{\scriptscriptstyle LW}}
\def\supp{\mbox{Supp}}
\def\tvt{\mbox{TV}_{\scriptscriptstyle T}}
\def\hun{\hat u^n}
\def\uhs{\hat u^*}
\def\hau{\hat A(u_l,u_r)}
\def\rh{\rho^{1/2}}
\def\FH{F_{\scriptscriptstyle H}}
\def\FL{F_{\scriptscriptstyle L}}
\def\vpj{V_{pj}}
\def\vpjp{V_{pj_p}}
\def\vpjpm{V_{p,j_p-1}}
\def\bpj{\beta_{pj}}
\def\bpjp{\beta_{pj_p}}
\def\bpjpm{\beta_{p,j_p-1}}
\def\sigpj{\sig_{pj}}
\def\sigpjp{\sig_{pj_p}}
\def\sigpjpm{\sig_{p,j_p-1}}
\def\lpj{\hat \lam_{pj}}
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