Designing an efficient solution strategy for fluid flows. 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations
Designing an efficient solution strategy for fluid flows. 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations
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DOI：
10.1006/jcph.1996.0248
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发表时间：
1996-12
影响因子：
4.1
通讯作者：
M. Gerritsen;P. Olsson
中科院分区：
文献类型：
--
作者：
M. Gerritsen;P. Olsson
Abstract We derive high-order finite difference schemes for the compressible Euler (and Navier–Stokes equations) that satisfy a semidiscrete energy estimate and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semidiscrete energy estimate is based on symmetrization of the equations, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration-by-parts procedure used in the continuous energy estimate. For the Euler equations, the symmetrization is designed such as to preserve the homogeneity of the flux vectors. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding artificial viscosity. The positioning of the subgrids and computation of the viscosity are aided by a detection algorithm which is based on a multiscale wavelet analysis of the pressure grid function. The wavelet theory provides easy-to-implement mathematical criteria to detect discontinuities, sharp gradients, and spurious oscillations quickly and efficiently. As the detection algorithm does not depend on the numerical method used, it is of general interest. The numerical method described and the detection algorithm are part of a general solution strategy for fluid flows, which is currently being developed by the authors and collaborators.
