Comparison of Several Difference Schemes for the Euler Equations in 1D and 2D
Representative ten finite difference schemes are applied on a suite of one-dimensional and two-dimensional test problems for the Euler equations. Sample results are presented in this paper while the full results are available on the web at http://www-troj
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Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Brehova 7, 115 19 Prague 1, Czech Republic liska@siduri·fjfi·cvut.cz Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA [email protected]
Summary. Representative ten finite difference schemes are applied on a suite of one-dimensional and two-dimensional test problems for the Euler equations. Sample results are presented in this paper while the full results are available on the web at http://www-troja .fjfi.cvut.cz/-liska/CompareEuler/compare.
1 Introduction Hyperbolic conservation laws, and the Euler equations of compressible fluid dynamics in particular, have been the subject of intensive research for at least the past five decades, and with good reason. The applications are many - aircraft design, stellar formation, weather prediction to name only a few. There are some theoretical results however even if the theory were perfect the applications would not be possible without methods for obtaining approximate solutions. The unfortunate situation here is that rigorous error estimates for supposed approximate solutions are almost entirely nonexistent. So, it is universally recognized that tests of methods on difficult problems are essential. Our concern here is to compare the behavior of some Eulerian methods to each other on problems that seem to us to be sufficiently difficult and representative to enable the reader to draw some conclusions about the applicability of these methods. The now classic work of this nature is the paper by Gary Sod [1] or Woodward Colella review [2]. Several difficult tests are discussed in [3] .
2 Finite Difference Schemes We have chosen ten methods that we feel are representative of the different basic finite difference approaches to solving hyperbolic conservation laws: A composite scheme (CFLF) called LWLFn consists of a cycle of n - 1 time steps of some version of the Lax-Wendroff (LW) scheme followed by one step with Lax-Friedrichs (LF). The LF step acts as a consistent (with the differential equations) filter to reduce the oscillations of LW. Here we use the 2D version called CFLF [4].
T.Y. Hou et al.(eds.), Hyperbolic Problems: Theory, Numerics, Applications © Springer-Verlag Berlin Heidelberg 2003
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Richard Liska and Burton Wendroff
A CFLF hybrid scheme (CFLFh) is also as the previous one a combination of second order LW and first order LF type schemes. The numerical flux is given by a weighted average of LF diffusive and LW oscillatory fluxes. The weight is chosen in such a way that the scheme is second order on smooth solutions, but becomes sufficiently dissipative in shocks. We use the Harten weight [5] (for overview of weights see [6]) with the CF and LF numerical fluxes [4]. Centered scheme with limiter (JT) uses discontinuous limited piecewise linear reconstruction from cell averages to get fluxes at cell edges. It uses neither dimensional splitting nor eigenvector decomposition nor any overt Riemann solver. We use the 2D version of this non-oscillatory central
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