Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Jet schemes for advection problems

Pages: 1229 - 1259, Volume 17, Issue 4, June 2012      doi:10.3934/dcdsb.2012.17.1229

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Benjamin Seibold - Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122, United States (email)
Rodolfo R. Rosales - Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States (email)
Jean-Christophe Nave - Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montreal, QC, H3A 2K6, Canada (email)

Abstract: We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through the tracking of characteristics and the use of suitable Hermite interpolations, high order is achieved in an optimally local fashion, i.e. the update for the data at any grid point uses information from a single grid cell only. We show that jet schemes can be interpreted as advect-and-project processes in function spaces, where the projection step minimizes a stability functional. Furthermore, this function space framework makes it possible to systematically inherit update rules for the higher derivatives from the ODE solver for the characteristics. Jet schemes of orders up to five are applied in numerical benchmark tests, and systematically compared with classical WENO finite difference schemes. It is observed that jet schemes tend to possess a higher accuracy than WENO schemes of the same order.

Keywords:  Jet schemes, gradient-augmented, advection, cubic, quintic, high-order, superconsistency.
Mathematics Subject Classification:  Primary: 65M25, 65M12; Secondary: 35L04.

Received: January 2011;      Revised: November 2011;      Available Online: February 2012.