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134
Numerical Solutions of the Euler Equations by Finite Volume Methods Using RungeKutta TimeStepping Schemes
, 1981
"... A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to deter ..."
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Cited by 517 (78 self)
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A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
Stability of largeamplitude shock waves of compressible NavierStokes equations
, 2003
"... We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be ..."
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Cited by 61 (37 self)
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We summarize recent progress on one and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the largeamplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multidimensions by a codimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently smallamplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently largeamplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory.
Viscous And Inviscid Stability Of Multidimensional Planar Shock Fronts
 Indiana Univ. Math. J
, 1999
"... . We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock profile. Our main result, generalizing earlier onedimensional calculations, is that the Evans function reduces in the longwave limit t ..."
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Cited by 61 (29 self)
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. We explore the relation between viscous and inviscid stability of multidimensional shock fronts, by studying the Evans function associated with the viscous shock profile. Our main result, generalizing earlier onedimensional calculations, is that the Evans function reduces in the longwave limit to the KreissSakamoto Lopatinski determinant obtained by Majda in the inviscid case, multiplied by a constant measuring transversality of the shock connection in the underlying (viscous) traveling wave ODE. Remarkably, this result holds independently of the nature of the viscous regularization, or the type of the shock connection. Indeed, the analysis is more general still: in the overcompressive case, we obtain a simple longwave stability criterion even in the absence of a sensible inviscid problem. An immediate consequence is that inviscid stability is necessary (but not sufficient) for viscous stability; this yields a number of interesting results on viscous instability through the in...
Wellposedness in smooth function spaces for the movingboundary 3D compressible Euler equations in physical vacuum
, 2010
"... The freeboundary compressible onedimensional Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws that are both characteristic and degenerate. Thephysical vacuum singularity (or rate of degeneracy) requires the sound speed c2 D 1 to scale as the square ..."
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Cited by 35 (7 self)
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The freeboundary compressible onedimensional Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws that are both characteristic and degenerate. Thephysical vacuum singularity (or rate of degeneracy) requires the sound speed c2 D 1 to scale as the square root of the distance to the vacuum boundary and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higherorder, Hardytype inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions. © 2010 Wiley Periodicals, Inc. 1
The block structure condition for symmetric hyperbolic systems
"... In the analysis of hyperbolic boundary value problems, the construction of Kreiss’ symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constan ..."
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Cited by 34 (8 self)
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In the analysis of hyperbolic boundary value problems, the construction of Kreiss’ symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constant multiplicity. In [2], H.O.Kreiss proved a maximal L2 energy estimate for the solutions of mixed boundaryinitial value problems for strictly hyperbolic systems and boundary conditions which satisfy the uniform Lopatinski condition (see also [8] for systems with complex coefficients). The proof is based on the construction of a symmetrizer. Thanks to the pseudodifferential calculus, the proof is reduced to the construction of an algebraic symmetrizer for the symbol of the equation (see e.g. [1]). The result extends to the case where the coefficients have finite smoothness ([3]) and, using the paradifierential calculus of J.M.BonyY.Meyer, to Lipschitzean coefficients ([7],[6]). Kreiss’ analysis is extended to a class of characteristic boundary value problems in [5].
The Stability of Compressible Vortex Sheets in Two Space Dimensions
, 2003
"... We study the linear stability of planar compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a ..."
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Cited by 32 (8 self)
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We study the linear stability of planar compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. Furthermore, the failure of the uniform KreissLopatinskii condition yields a loss of derivatives in the energy estimate. 1
NavierStokes regularization of multidimensional Euler shocks
, 2006
"... We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with “real”, or partially parabolic, viscosity including the Navier–Stokes equations of compressible gas dynamics with standard or van der Waalstype equati ..."
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Cited by 32 (18 self)
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We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with “real”, or partially parabolic, viscosity including the Navier–Stokes equations of compressible gas dynamics with standard or van der Waalstype equation of state. More precisely, given a curved Lax shock solution u 0 of the corresponding inviscid equations for which (i) each of the associated planar shocks tangent to the shock front possesses a smooth viscous profile and (ii) each of these viscous profiles satisfies a uniform spectral stability condition expressed in terms of an Evans function, we construct nearby smooth viscous shock solutions u ɛ of the viscous equations converging to u 0 as viscosity ɛ → 0, and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i)–(ii) hold always for shock waves of sufficiently small amplitude, but in general may fail for large amplitudes. We treat the viscous shock problem considered here as a representative of a larger class of multidimensional boundary problems arising in the study of viscous fluids,
Multidimensional viscous shocks. I. Degenerate symmetrizers and long time stability
 J. Amer. Math. Soc
"... Abstract. We use energy estimates to study the long time stability of multidimensional planar viscous shocks ψ(x1) for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength a ..."
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Cited by 30 (18 self)
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Abstract. We use energy estimates to study the long time stability of multidimensional planar viscous shocks ψ(x1) for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength are allowed, subject to an appropriate Evans function condition. The main tools are a conjugation argument that allows us to replace the eigenvalue equation by a problem in which the x1 dependence of the coefficients is removed, and degenerate Kreisstype symmetrizers designed to cope with the vanishing of the Evans function for zero frequency.