Novel Adaptive Methods for Hyperbolic Conservation Laws Based on New Quasi-Linear Seventh- and Ninth-Order Schemes

We develop new adaptive numerical schemes for one- and two-dimensional hyperbolic systems of conservation laws.

The methodology relies on the use of a smoothness indicator to automatically partition the computational domain into smooth and nonsmooth (``rough``) regions.

We then follow the scheme adaption strategy recently introduced in [S.

Chu, P.

Feng, V.

A.

Kolotilov, A.

Kurganov, and V.

V.

Ostapenko, Commun.

Comput.

Phys., accepted], but instead of the quasi-linear (QL) fifth-order finite-difference scheme used there, we employ the new QL seventh- and ninth-order schemes in the smooth regions.

A series of numerical experiments for the Euler equations of gas dynamics demonstrates that the new adaptive schemes contain a smaller amount of numerical dissipation and achieve higher resolution compared with their counterpart that uses the QL fifth-order scheme in the smooth areas.