Investigations of a New Scheme for Wave Propagation

Fan, Doreen; Roe, Philip L.

doi:10.2514/6.2015-2449

Cited by 17 publications

(9 citation statements)

References 6 publications

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“…For the scalar wave equation itself, a related method was developed independently by Hagstrom [3,15] and co-workers. The extension presented here to nonlinear acoustics in the presence of vorticity was given by Fan and Roe [12], where it was shown to give far more isotropic results than a DG method of the same formal accuracy. Nonlinear effects were included by careful choice of local values for the coefficients in the equations.…”

Section: What Else Could There Be?mentioning

confidence: 97%

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Is Discontinuous Reconstruction Really a Good Idea?

Roe

2017

J Sci Comput

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It has been almost automatically assumed for a quarter century that the numerical solution of hyperbolic conservation laws is best accomplished by making a reconstruction of the initial data that is only piecewise continuous. The effect of the discontinuities is taken into account by means of Riemann solvers. This strategy has enjoyed great practical success but introduces only one-dimensional physics as a guide to the discretization of multidimensional problems. This article points out some of the resulting defects and proposes an alternative viewpoint. The chief novelty of the new "Active Flux" method, apart from the elimination of discontinuities, is the division into advective and acoustic disturbances, with acoustics being handled by exploiting classical solutions to the scalar wave equation. Results for a standard test problem for the compressible Euler equations indicate that an order of magnitude increase in cost effectiveness may be possible by following this approach, compared to the traditional one.

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Section: What Else Could There Be?mentioning

confidence: 97%

“…This simple first-order correction to the wavespeed carries over to the nonlinear wave equation [12] and forms the basis for our treatment of the Euler equations. Figure 6 shows a typical solution to Burgers' equation using this method.…”

Section: Burgers' Equationmentioning

confidence: 99%

Is Discontinuous Reconstruction Really a Good Idea?

Roe

2017

J Sci Comput

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“…[6,10,16,23,25,28], along with "vorticity-preserving" algorithms, [13,20,15,21]. Here, our aim is to follow the insight of [8], that is, to reformulate Kirchhoff's expression into a semi-discrete (in space) Lax-Wendroff type time-marching scheme [14]. On a uniform 2D Cartesian grid, it turns out that the algorithm resulting from discretizing all the remaining terms matches the one written in [20, eqn.…”

Section: Aims and Plan Of The Papermentioning

confidence: 99%

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

Franck

Gosse

2017

Ann Univ Ferrara

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By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a timemarching, Lax-Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the H s norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former "shape functions" and "symmetric potential schemes" are highlighted.

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“…The scalar solution given by Poisson [34] is to be found in many PDE textbooks where M R u( , t) is the mean value of a function u( , t) taken at time t over [11] the surface of a sphere with center and radius R. Note that this is a fully discrete formula, giving the exact solution at a later time. As with van Leer's method, and with Godunov's method, the approximation is made to the data, but not to the evolution.…”

mentioning

confidence: 99%

“…As with van Leer's method, and with Godunov's method, the approximation is made to the data, but not to the evolution. Some manipulations [11] allow it to be applied to the wave equation in first-order form. In two dimensions, the integral is over a disc rather than a sphere.…”

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confidence: 99%