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A finite-difference scheme for the Navier-Stokes equations of one- dimensional, isentropic, compressible flow. (English) Zbl 0727.76094

A finite difference approximation to discontinuous solutions of the Navier-Stokes equations for one-dimensional isentropic compressible flow is studied. The scheme is comparable easy to implement, avoids all problems of correct entropy production by retaining the correct physical viscosity (without any form of artificial viscosity), and is amenable to a completely rigorous convergence and error bound analysis. This analysis (rather involved technically) starts with a number of a priori energy- type estimates for the solution of the scheme which are fully discrete versions of coresponding estimates for continuous solutions developed by the second author [Trans. Amer. Math. Soc. 303, 169-181 (1987; Zbl 0656.76064)]. One of their key features is the derivation of the exponential decay of the jumps in the pressure and in the velocity gradient across the particle paths. The scheme is solvable at each timestep under certain rather technical assumptions concerning the mesh parameters, essentially equivalent to the usual conditions. The a priori estimates are applied to estimate the “weak” truncation error. A general stability result establishes the error estimates: For piecewise smooth initial data of bounded variation the error is bounded by \(0(\Delta x^{1/6})\) in a norm dominating the sup-norm of the (discontinuous!) density, and is bounded by \(0(\Delta x^{1/2})\) when the initial data are \(H^ 1\).

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
76M20 Finite difference methods applied to problems in fluid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs

Citations:

Zbl 0656.76064
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