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.
Highlights
In this paper we consider a class of approaches for linear advection problems that evolve parts of the jet of the solution in time
As a test for both the accuracy and the performance of jet schemes, we consider a version of the classical “vortex in a box” flow [3, 17], adapted as follows
The jet schemes introduced in this paper form a new class of numerical methods for the linear advection equation
Summary
In this paper we consider a class of approaches for linear advection problems that evolve parts of the jet of the solution in time. Like many other semi-Lagrangian approaches, jet schemes treat boundary conditions naturally (the distinction between an ingoing and an outgoing characteristic is built into the method), and they do not possess a Courant-Friedrichs-Lewy (CFL) condition that restricts stability This latter property may be of relevance if the advection equation (1) is one step in a more complex problem that exhibits a separation of time scales. There we show how, given suitable parts of the jet of a smooth function, a cell-based Hermite interpolant can be used to obtain a high order accurate approximation This interpolant gives rise to a projection operator in function spaces, defined by evaluating the jet of a function at grid points, and constructing the piecewise Hermite interpolant.
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