Design and analysis of ADER-type schemes for model advection–diffusion–reaction equations

We construct, analyze and assess various schemes of second order of accuracy in space and time for model advection–diffusion–reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial problems and are derived following two related approaches, namely ADER and MUSCL-Hancock. Detailed analysis of linear stability and local truncation error are carried out. In addition, the schemes are implemented and assessed for various test problems. Empirical convergence rate studies confirm the theoretically expected accuracy in both space and time.