$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Testing a manufactured solution
- Introduce common mesh parameter: \( h=\Delta t \), \( \Delta x =ch/C \)
- This \( h \) keeps \( C \) and \( \Delta t/\Delta x \) constant
- Select coarse mesh \( h \): \( h_0 \)
- Run experiments with \( h_i=2^{-i}h_0 \) (halving the cell size), \( i=0,\ldots,m \)
- Record the error \( E_i \) and \( h_i \) in each experiment
- Compute pariwise convergence rates \( r_i=
\ln E_{i+1}/E_{i}/\ln h_{i+1}/h_{i} \)
- Verification: \( r_i\rightarrow 2 \) as \( i \) increases
