Loading [MathJax]/extensions/TeX/boldsymbol.js
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
Boundary and initial conditions
We need one boundary condition at each point on \partial\Omega :
- u is prescribed ( u=0 or known incoming wave)
- \partial u/\partial n = \normalvec\cdot\nabla u prescribed
( =0 : reflecting boundary)
- open boundary (radiation) condition: u_t + \boldsymbol{c}\cdot\nabla u =0
(let waves travel undisturbed out of the domain)
PDEs with second-order time derivative need two initial conditions:
- u=I ,
- u_t = V .
