Study guide: Finite difference methods for wave motion

Processing math: 100%

$\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\halfi}{{1/2}} \newcommand{\xpoint}{\boldsymbol{x}} \newcommand{\normalvec}{\boldsymbol{n}} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\It}{\mathcal{I}_t} \newcommand{\setb}[1]{#1^0} % set begin \newcommand{\sete}[1]{#1^{-1}} % set end \newcommand{\setl}[1]{#1^-} \newcommand{\setr}[1]{#1^+} \newcommand{\seti}[1]{#1^i} \newcommand{\Real}{\mathbb{R}}$

Step 3: Algebraic version of the initial conditions

Need to replace the derivative in the initial condition $u_t(x,0)=0$ by a finite difference approximation

The differences for $u_{tt}$ and $u_{xx}$ have second-order accuracy

Use a centered difference for $u_t(x,0)$

$[D_{2t} u]^n_i = 0,\quad n=0\quad\Rightarrow\quad u^{n-1}_i=u^{n+1}_i,\quad i=0,\ldots,N_x$

The other initial condition $u(x,0)=I(x)$ can be computed by $u_i^0 = I(x_i),\quad i=0,\ldots,N_x$