Fast high order ADER schemes for linear hyperbolic equations

doi:10.1016/j.jcp.2003.12.007

Journal of Computational Physics

Volume 197, Issue 2, 1 July 2004, Pages 532-539

https://doi.org/10.1016/j.jcp.2003.12.007 Get rights and content

Abstract

A reformulation of the ADER approach (Arbitrary high order schemes using DERivatives) for linear hyperbolic PDE's is presented. This reformulation leads to a drastic decrease of the computational effort. A formula for the construction of ADER schemes that are arbitrary high order accurate in space and time is given. The accuracy for some selected schemes is shown numerically for the two-dimensional linearized Euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics.

Introduction

Finite volume schemes have become very popular in computational fluid dynamics due to their robustness and flexibility. They consist of two steps: the reconstruction and the flux calculation. The discrete values are approximations of cell averages. By the reconstruction step, local values are interpolated from the average values to calculate the numerical flux between the grid cells. The construction of high order schemes concerning spatial discretization has been introduced with the idea of ENO and WENO interpolation [4], [5]. But the time integration usually becomes a limiting factor for accuracy. All these schemes are generally discretized in time with Runge–Kutta (RK) schemes. These time integration schemes become inefficient for orders of accuracy higher than four. It can be proven, that all explicit RK time integration schemes of order higher than four need more integration stages than their order. This is the so-called Butcher barrier [2].

The idea of the ADER approach of Toro (see, e.g., [7], [9], [11]) is to circumvent this efficiency barrier for the time discretization by considering finite space–time volumes, where the temporal evolution of the fluxes over the borders of the finite volumes is estimated by a Taylor series in time. The time derivatives are then replaced by space derivatives using the Lax–Wendroff procedure. With this approach it is (at least theoretically) possible to construct schemes of arbitrary high order in space and time with Δt∼Δx close to the stability limit. However, the Lax–Wendroff procedure becomes cumbersome already in the case of linear systems of PDE's for very high order ADER schemes. If the ADER approach is applied without any special treatment, it is not competitive with e. g. finite difference schemes for the solution of linear PDE's with respect to the computational effort. In this paper we develop a particular formulation of the ADER approach, which makes use of simplifications that can be taken into account for linear PDE's on structured meshes. A typical field of application would be computational aeroacoustics to simulate noise propagation in the time domain or electromagnetic wave propagation.

The scope of the paper is as follows. In Section 2 we first give the formulas which are essential for the reformulation. We derive then a finite difference-like formulation of the ADER approach which significantly enhances efficiency with respect to computational effort. In Section 3 we show the convergence rates obtained with the reformulated ADER schemes in numerical experiments for the two-dimensional linearized Euler equations up to 16th order of accuracy in space and time.

Section snippets

The fast-ADER approach

The ADER approach is a scheme developed to calculate numerical solutions of systems of hyperbolic PDE's up to an, at least theoretically, arbitrary order. Toro and Millington [11], [13] first developed the idea in one space dimension. Here, the 1st order scheme reduces to the Godunov scheme and the 2nd order scheme is equivalent to the MGRP approach by Toro [12], [13], which is a simplified GRP scheme of Ben-Artzi and Falcovitz [1]. The extension to multi-dimensions is straightforward and can

Numerical results

In Computational AeroAcoustics (CAA) good wave propagation properties are a crucial point for numerical schemes. The dispersion and dissipation errors must be as low as possible in order to provide accurate wave propagation over long distances on reasonably coarse grids. In [7], [9] we plotted the amplitude- and phase errors for the ADER approach. As nothing is neglected, they apply without any change to the fast-ADER approach. An $O 4$ ADER scheme has approximately the same phase error as a sixth

Conclusions

If the following assumptions hold: The mesh sizes Δx and Δy is constant and the PDE is linear with constant coefficients, then the fast formulation of the ADER scheme leads to a single-step scheme in time of arbitrary order of accuracy which is dramatically faster than the original formulation. For the 20th order scheme the speed-up is about 2000. We implemented the scheme in such a way, that the order of accuracy in space and time becomes only a parameter to be specified, so a really arbitrary

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Fast high order ADER schemes for linear hyperbolic equations

Abstract

Introduction

Section snippets

The fast-ADER approach

Numerical results

Conclusions

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws

Numerical Computation of Internal and External Flows Vol II: Computational Methods for Inviscid and Viscous Flow