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

arXiv (Cornell University)

Abstract

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.

Authors 5

1. Shaoshuai Chu Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics , RWTH Aachen University , 52056 Aachen , Germany;

2. Pingyao Feng

   North Carolina State University

   Affiliation as printed

   Department of Mathematics , North Carolina State University , 27695 Raleigh , USA;

3. Vadim A. Kolotilov

   Lavrentyev Institute of Hydrodynamics

   Affiliation as printed

   Lavrentyev Institute of Hydrodynamics Siberian Branch of the Russian Academy of Sciences , Novosibirsk , 630090 Russia and

4. Alexander Kurganov

   Southern University of Science and Technology

   Affiliation as printed

   Department of Mathematics and

   Shenzhen International Center for Mathematics , Southern University of Science and Technology , Shenzhen , 518055 , China;

5. Владимир Викторович Остапенко

   Lavrentyev Institute of Hydrodynamics · Keldysh Institute of Applied Mathematics

   Affiliation as printed

   Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences , 125047 Moscow , Russia;

   Lavrentyev Institute of Hydrodynamics Siberian Branch of the Russian Academy of Sciences , Novosibirsk , 630090 Russia and