Loading [MathJax]/extensions/TeX/boldsymbol.js
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
Exact solutions of the finite difference equations
How does the truncation error depend on \uex in finite differences?
- One-sided differences: \uex''\Delta t (lowest order)
- Centered differences: \uex'''\Delta t^2 (lowest order)
- Only harmonic and geometric mean involve \uex' or \uex
Consequence:
- \uex(t)=ct+d will very often give exact solution
of the discrete equations ( R=0 )!
- Ideal for verification
- Centered schemes allow quadratic \uex
Problem: harmonic and geometric mean (error depends on \uex' and \uex )