Direct-expansion forms of ADER schemes for conservation laws and their verification

To seek general-purpose numerical schemes for hyperbolic problems, the ADER approach has been reviewed on the state-series expansion forms and the direct expansion forms in the viewpoint of the numerical procedure and the accuracy, and the advantages and disadvantages of the latter forms have been discussed. As ADER direct expansion schemes, ADER-D (standard ones with Godunov states/fluxes) and ADER-waf (ones with WAF states/fluxes) are adopted. Then, verification has been carried out on the scalar conservation laws with a linear flux, nonlinear convex fluxes, and various types of nonlinear non-convex fluxes. Convergence studies have shown that all the ADER schemes achieve the designed order of accuracy up to small cell sizes, yield small errors even in large cell sizes, and have computational efficiency with keeping the CFL number close to unity. Capturability of discontinuity and rarefaction has been investigated. As results, the ADER schemes have worked well for the problem of long-time propagation in the linear cases and for the problems of complicated wave formation and interaction in the nonlinear cases corresponding to various types of convex and non-convex fluxes. It is remarkable that ADER-waf schemes have shown sharper resolvability than the other ADER schemes, but have less robustness.