Existence, uniqueness, and computation of solutions for mixed problems in compressible fluid flow. (English) Zbl 0777.35063

Initial-boundary value problems are considered for the one-dimensional Navier-Stokes equations for compressible flow on a finite interval. Convergence of finite difference schemes are proved for each of three different cases of initial and boundary data. For BV discontinuous initial data that is piecewise smooth the density is shown to remain discontinuous and a bound is given for the error of an approximate solution in a norm that dominates the sup-norm of the density. For \(H^ 1\) initial data a different error bound is given in the same norm.


35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Chen, G.-Q, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics, III, Acta math. sci., 6, No. 1, 75-120, (1986) · Zbl 0643.76086
[2] Glimm, J, Solutions in the large for nonlinear hyperbolic systems of conservation laws, Comm. pure appl. math., 18, 95-105, (1965)
[3] Hoff, D, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, (), 301-315 · Zbl 0635.35074
[4] Hoff, D, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. amer. math. soc., 303, No. 1, 169-181, (1987) · Zbl 0656.76064
[5] Nishida, T; Smoller, J, Mixed problems for nonlinear conservation laws, J. differential equations, 23, 244-269, (1977) · Zbl 0303.35052
[6] Smoller, J, Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
[7] Zarnowski, R; Hoff, D, A finite difference scheme for the Navier-Stokes equations of one-dimensional, isentropic, compressible flow, SIAM J. numer. anal., 28, No. 1, 78-112, (1991) · Zbl 0727.76094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.