ADER Schemes Research Articles

Solvers for the high-order Riemann problem for hyperbolic balance laws

We study three methods for solving the Cauchy problem for a system of non-linear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin, a high-order Riemann problem. Two of the methods are new; one of the them results from a re-interpretation of the high-order numerical methods proposed by Harten et al. [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231–303] and the other is a modification of the solver in [E.F. Toro, V.A. Titarev, Solution of the generalised Riemann problem for advection-reaction equations, Proc. Roy. Soc. London A 458 (2002) 271–281]. A systematic assessment of all three solvers is carried out and their relative merits are discussed. We also implement the solvers, locally, in the context of high-order finite volume numerical methods of the ADER type, on unstructured meshes. Schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations and the shallow water equations with source terms are constructed. Empirically obtained convergence rates are studied systematically and, for the tests considered, these correspond to the theoretically expected orders of accuracy. We also address the question of balance between flux gradients and source terms, for steady flow. We find that the ADER schemes may be termed asymptotically well-balanced, in the sense that the well-balanced property is attained as the order of the method increases, and this without introducing any ad-hoc fixes to the schemes or the equations.

ADER schemes for the shallow water equations in channel with irregular bottom elevation

This paper deals with the construction of high-order ADER numerical schemes for solving the one-dimensional shallow water equations with variable bed elevation. The non-linear version of the schemes is based on ENO reconstructions. The governing equations are expressed in terms of total water height, instead of total water depth, and discharge. The ENO polynomial interpolation procedure is also applied to represent the variable bottom elevation. ADER schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed, with particular attention to their convergence rates. Non-oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases. The resulting schemes can be applied to solve realistic problems characterized by non-uniform bottom geometries.

Analysis of ADER and ADER-WAF schemes

We study stability properties and truncation errors of the finite-volume ADER schemes on structured meshes as applied to the linear advection equation with constant coefficients in one-, two- and three-spatial dimensions. Stability of linear ADER schemes is analysed by means of the von Neumann method. For nonlinear schemes, we deduce the stability region from numerical experiments. The truncation error analysis is carried out for linear ADER schemes in one-, two- and three-space dimensions and for nonlinear ADER schemes in one-space dimension.

Direct-expansion forms of ADER schemes for conservation laws and their verification

To seek general-purpose numerical schemes for hyperbolic problems, the ADER approach has been reviewed on the state-series expansion forms and the direct expansion forms in the viewpoint of the numerical procedure and the accuracy, and the advantages and disadvantages of the latter forms have been discussed. As ADER direct expansion schemes, ADER-D (standard ones with Godunov states/fluxes) and ADER-waf (ones with WAF states/fluxes) are adopted. Then, verification has been carried out on the scalar conservation laws with a linear flux, nonlinear convex fluxes, and various types of nonlinear non-convex fluxes. Convergence studies have shown that all the ADER schemes achieve the designed order of accuracy up to small cell sizes, yield small errors even in large cell sizes, and have computational efficiency with keeping the CFL number close to unity. Capturability of discontinuity and rarefaction has been investigated. As results, the ADER schemes have worked well for the problem of long-time propagation in the linear cases and for the problems of complicated wave formation and interaction in the nonlinear cases corresponding to various types of convex and non-convex fluxes. It is remarkable that ADER-waf schemes have shown sharper resolvability than the other ADER schemes, but have less robustness.

TVD Fluxes for the High-Order ADER Schemes

In this paper we propose to use a TVD flux, instead of a first-order monotone flux, as the building block for designing very high-order methods; we implement the idea in the context of ADER schemes via a new flux expansion. Systematic assessment of the new schemes shows substantial gains in accuracy; these are particularly evident for problems involving long time evolution

ADER schemes on adaptive triangular meshes for scalar conservation laws

ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfaces. Currently available nonlinear ADER schemes are restricted to Cartesian meshes. This paper proposes an adaptive nonlinear finite volume ADER method on unstructured triangular meshes for scalar conservation laws, which works with WENO reconstruction. To this end, a customized stencil selection scheme is developed, and the flux evaluation of previous ADER schemes is extended to triangular meshes. Moreover, an a posteriori error indicator is used to design the required adaption rules for the dynamic modification of the triangular mesh during the simulation. The expected convergence orders of the proposed ADER method are confirmed by numerical experiments for linear and nonlinear scalar conservation laws. Finally, the good performance of the adaptive ADER method, in particular its robustness and its enhanced flexibility, is further supported by numerical results concerning Burgers equation.

ADER schemes for three-dimensional non-linear hyperbolic systems

In this paper, we carry out the extension of the ADER approach to multidimensional non-linear systems of conservation laws. We implement non-linear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order of accuracy and non-oscillatory properties of the new schemes. Compared to the state-of-art finite-volume WENO schemes the ADER schemes are faster, more accurate, need less computer memory and have no theoretical accuracy barrier.

ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions

In this paper we develop non-linear ADER schemes for time-dependent scalar linear and non-linear conservation laws in one-, two- and three-space dimensions. Numerical results of schemes of up to fifth order of accuracy in both time and space illustrate that the designed order of accuracy is achieved in all space dimensions for a fixed Courant number and essentially non-oscillatory results are obtained for solutions with discontinuities. We also present preliminary results for two-dimensional non-linear systems.

Fast high order ADER schemes for linear hyperbolic equations

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.

ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D

The ADER scheme for solving systems of linear, hyperbolic partial differential equations in two-dimensions is presented in this paper. It is a finite-volume scheme of high order in space and time. The scheme is explicit, fully discrete and advances the solution in one single step. Several numerical tests have been performed. In the first test case the dissipation and dispersion behaviour of the schemes are studied in one space dimension. Dispersion as well as dissipation effects strongly influence the discrete wave propagation over long distances and are very important for, e.g., aeroacoustical calculations. The next test, the so-called co-rotating vortex pair, is a demonstration of the ideas of the two-dimensional ADER approach. The linearised Euler equations are used for the simulation of the sound emitted by a co-rotating vortex pair.