ADER: Arbitrary-Order Non-Oscillatory Advection Schemes

Abstract

Here we present a new approach, called ADER, for constructing non-oscillatory advection schemes of arbitrary order of accuracy in space and time. The main ideas are presented in terms of the one-dimensional linear advection equation. As is well-known from Godunov’s theorem [8], the two requirements of (i) high accuracy and (ii) absence of spurious oscillations near discontinuities are contradictory. Solutions of hyperbolic conservation laws are piece-wise smooth, which means that in general they contain smooth parts and discontinuities. Note also that smooth portions of the solution may contain large gradients which, numerically, may be as demanding as discontinuities. A classical way of circumventing Godunov’s theorem is to construct non-linear schemes, even when applied to linear problems. A successful class of non-linear schemes are the so-called total Variation Diminishing Methods (or TVD methods) [9], [17] developed over the last two decades or so. Such schemes provide today the basis of a mature numerical technology suitable for industrial and scientific applications [14], [22], [23]. These methods are second-order accurate, almost everywhere, and reduce locally to first-order of accuracy. TVD methods however, are known to be unsuitable for special application areas such as acoustics, compressible turbulence and problems involving long-time evolution wave propagation. Extrema are clipped and numerical dissipation may become dominant The non-oscillatory methods we are attempting to construct here will be of much higher order of accuracy in both space and time and will not impose TVD-like constraints.