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Center for Nonlinear Studies |
The decision to capture rather than fit shockwaves probably compromises some aspects of a computation, although exactly how has never been completely clarified. For example, in Lagrangian calculations, for which the conserved quantities are mass, velocity, and internal energy, entropy is conserved within path tubes. The good news is that physical changes to the entropy are conserved; the bad news is that errors in the calculated entropy are also conserved. Propagating entropy errors of this kind were called entropy traces by Rodzhestvenskii and Yanenko [1].
A canonical example is the Noh problem [2], a seemingly trivial Riemann problem with initial data corresponding to the collision of two equal shocks, or equivalently the reflection of a single shock from a solid wall. Virtually all shock-capturing methods of Eulerian or Lagrangian type provide quite good solutions for pressure and velocity, but predict too small a density in a small region at the origin. In consequence the temperature there is too high, so that this and related phenomena have been called wall heating. In the more than 20 years since Noh proposed the problem, no satisfactory solution has been exhibited that would carry over to other settings, nor is there even any generally accepted explanation of the mechanism. It seems common to regard the Noh problem as an isolated curiosity having only marginal relevance to practical calculations, but we will argue from the Eulerian frame that the Noh problem is merely the simplest instance of a pervasive difficulty and attempt to explain the mechanism(s) behind it.
1. Rodzhestvenskii, B.I., Yanenko, N. N., Systems of quasilinear equations, AMS Translation, 1983. 2. Noh, W. F., Errors for calculations of strong shocks using an artificialviscosity and an artificial heat flux, J. Comput. Phys., 72, p. 78, 1987.
Host: Mikhail Shashkov