Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks

1. Entropy consistency Systems of conservation laws that originate in physics frequently possess an additional conservation law for an auxiliary quantity called entropy, which is conserved in smooth solutions but increases (or decreases according to the sign convention adopted) if shock waves appear. Numerical methods for conservation laws can be expected to reveal this behavior in a general way, but only for certain methods will the correspondence be precise. A method is said to be entropy-conservative if the local changes of entropy are exactly the same as predicted by the entropy conservation law. It is said to be entropy-stable if it produces more entropy than an entropy-conservative scheme. In a previous paper [16], a new entropy-conservative scheme was proposed that is relatively inexpensive, and therefore a good candidate to be the foundation for controlling entropy production. This scheme also has the property of exactly preserving stationary contact discontinuities. An entropy-stable extension was also shown in [16], following [2]. The present paper is focused on guaranteeing that the amount of entropy produced is also correct (entropy-consistent) and to investigate the effects of entropy consistency on shock stability, which may be strongly connected to the carbuncle phenomenon [12]. 1.1. Entropy conservation and entropy-stability The physical conservation laws take the differential form in one dimension @tuþ @xf 1⁄4 0 ð1aÞ . All rights reserved. il), philroe@umich.edu (P.L. Roe). F. Ismail, P.L. Roe / Journal of Computational Physics 228 (2009) 5410–5436 5411 with the integral form I ðudx f dtÞ 1⁄4 0 ð1bÞ The additional conservation law has the differential form in one dimension @tU þ @xF 6 0 ð2aÞ with the integral form I ðU dx F dtÞ 6 0 ð2bÞ Equality prevails in regions of smooth flow. Inequality may hold if the flow contains discontinuities. The sign is a matter of convention; for ideal gas dynamics these equations are correct if one takes U 1⁄4 qgðsÞ; F 1⁄4 qugðsÞ with s 1⁄4 ln p c ln p being the physical entropy. The sign convention is common in the mathematical and computational literature but creates some linguistic tension. Where inequality holds we will speak of entropy production which seems natural from a physical viewpoint, but we need to remember that U is being reduced. The integral _ U 1⁄4 Z Z X ð@tU þ @xFÞdxdt 1⁄4 I @X ðU dx F dtÞ ð3Þ can be identified with the entropy production in a domain X with boundary @X. In all cases where entropy is produced, the physical mechanism responsible is on that is not represented in (1b). Typically it is a dissipative or dispersive process, represented mathematically by higher-order derivatives multiplied by some small parameter. In a discrete version of the governing equations, it is usually accepted that the conservation laws (1b) must be observed, and some motivation to try and enforce equality in (2b) also, in smooth regions. The question then is how to enforce inequality in (2b) at discontinuities, not only with the proper sign (entropy-stable), but in the right amount. We now make some observations concerning the significance of the amount. 1.2. Entropy production and shock resolution Consider a steady, one dimensional, discrete representation of a shockwave. Assume that this has been produced by some stable, consistent, conservative numerical method, with boundary conditions at inflow and at the outflow derived from the Rankine–Hugoniot jump conditions. Whether or not the scheme makes explicit reference to entropy, there will be some entropy flux _ mSi at the inflow (where _ m is the mass flow rate) and some entropy flux _ mSo at outflow. The nature of the scheme guarantees that these are correct, because they can be derived from conserved variables, which are certainly correct. The difference of the two entropy fluxes is accounted for by entropy production within the domain. It follows that entropy production within the domain is correct under the stated assumptions, once a steady state has been reached. Of course, if no entropy is produced, the scheme could not be stable. However, entropy production during the transient phase of the calculation affects the quality of the shockwave that is eventually produced. This statement will now be illustrated in the case of a scalar conservation law, @tuþ @xf 1⁄4 @tuþ f 0ðuÞ@xu 1⁄4 0 ð4Þ for which we choose an entropy function UðuÞ 1⁄4 u2. To conform to the physicists view that entropy is naturally increasing with time, and brings disorder, we might define the ‘‘physical entropy” as SðuÞ 1⁄4 u2. Consider a discrete shock representation uðxjÞ 1⁄4 uj ðj 1⁄4 1:::JÞ, that has not yet reached equilibrium, and suppose that during one time step, as it advances toward equilibrium, the solution changes to uðxjÞ 1⁄4 unþ1 j . Because of conservation, the mean value u will not be changed. The ‘‘physical entropy” contained in the domain is (after dropping the superscripts) X 1⁄4 X