Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

[1]  J. Oliger,et al.  Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics , 1976 .

[2]  I. Gladwell,et al.  Analysis of explicit difference methods for a diffusion‐convection equation , 1978 .

[3]  R. F. Warming,et al.  An Implicit Factored Scheme for the Compressible Navier-Stokes Equations , 1977 .

[4]  Andrew J. Majda,et al.  Initial‐boundary value problems for hyperbolic equations with uniformly characteristic boundary , 1975 .

[5]  R. F. Warming,et al.  Alternating Direction Implicit Methods for Parabolic Equations with a Mixed Derivative , 1980 .

[6]  Gino Moretti,et al.  Importance of Boundary Conditions in the Numerical Treatment of Hyperbolic Equations , 1969 .

[7]  R. F. Warming,et al.  An extension ofA-stability to alternating direction implicit methods , 1979 .

[8]  John C. Strikwerda,et al.  Initial boundary value problems for the method of lines , 1980 .

[9]  S. Osher,et al.  One-sided difference approximations for nonlinear conservation laws , 1981 .

[10]  On boundary conditions for the numerical solution of hyperbolic differential equations , 1980 .

[11]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[12]  M. Gunzburger,et al.  Downstream boundary conditions for viscous flow problems , 1977 .

[13]  Bertil Gustafsson,et al.  Finite-Difference Methods for Computing the Steady Flow about Blunt Bodies , 1980 .

[14]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[15]  Bertil Gustafsson,et al.  The convergence rate for difference approximations to general mixed initial boundary value problems , 1981 .

[16]  Eli Turkel,et al.  Boundary conditions for multistep finite-difference methods for time-dependent equations☆ , 1978 .

[17]  John C. Strikwerda,et al.  Initial boundary value problems for incompletely parabolic systems , 1976 .

[18]  Hybrid difference methods for the initial boundary-value problem for hyperbolic equations. , 1975 .

[19]  Stability of Difference Approximations to the Mixed Initial Boundary Value Problems for Parabolic Systems , 1971 .

[20]  T. H. Pulliam,et al.  An implicit finite-difference code for inviscid and viscous cascade flow , 1980 .

[21]  E. Tadmor,et al.  Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II , 1978 .

[22]  David Gottlieb,et al.  Stability of two-dimensional initial-boundary value problems using leap-frog type schemes , 1979 .

[23]  G. W. Hedstrom,et al.  Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems , 1979 .

[24]  Heinz-Otto Kreiss,et al.  Methods for the approximate solution of time dependent problems , 1973 .

[25]  Constructing Stable Difference Methods for Hyperbolic Equations , 1979 .

[26]  Difference approximations for initial boundary-value problems , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  J. Oliger,et al.  Fourth Order Difference Methods for the Initial Boundary-Value Problem for Hyperbolic Equations , 1974 .

[28]  Bertil Gustafsson,et al.  Incompletely parabolic problems in fluid dynamics , 1978 .

[29]  W. M. Coughran.  On the approximate solution of hyperbolic initial-boundary value problems , 1980 .

[30]  C. K. Chu,et al.  Boundary conditions in finite difference fluid dynamic codes , 1974 .

[31]  Germund Dahlquist,et al.  Positive Functions and some Applications to Stability Questions for Numerical Methods , 1978 .

[32]  H. Kreiss,et al.  Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems. II , 1972 .

[33]  J. Chen.  Numerical boundary conditions and computational modes , 1973 .

[34]  R. Porter,et al.  Use of characteristics for boundaries in time dependent finite difference analysis of multidimensional gas dynamics , 1972 .

[35]  H. Kreiss,et al.  On Difference Approximations with Wrong Boundary Values , 1968 .

[36]  J. Trapp,et al.  A simple heuristic method for analyzing the effect of boundary conditions on numerical stability , 1976 .

[37]  Eli Turkel,et al.  Outflow Boundary Conditions for Fluid Dynamics , 1982 .

[38]  B. Gustafsson,et al.  Numerical solution of a PDE system describing a catalytic converter , 1978 .

[39]  Paul Garabedian,et al.  Partial Differential Equations , 1964 .

[40]  J. Steger,et al.  Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods , 1981 .

[41]  Melvyn Ciment.  Stable Matching of Difference Schemes , 1972 .

[42]  Boundary conditions for implicit solutions to the compressible Navier-Stokes equations in finite computational domains , 1979 .

[43]  D. Koster.  Stability of a Finite Difference Scheme with “Wrong” Boundary Conditions , 1973 .

[44]  M. Crandall,et al.  Monotone difference approximations for scalar conservation laws , 1979 .

[45]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[46]  M. B. Carver,et al.  Pseudo characteristic method of lines solution of the conservation equations , 1980 .

[47]  Moshe Goldberg.  On a boundary extrapolation theorem by Kreiss , 1977 .

[48]  J. Gary,et al.  On boundary conditions for hyperbolic difference schemes , 1978 .

[49]  Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems , 1979 .

[50]  David H. Rudy,et al.  A nonreflecting outflow boundary condition for subsonic navier-stokes calculations , 1980 .

[51]  Heinz-Otto Kreiss,et al.  Boundary Conditions for Time Dependent Problems with an Artificial Boundary , 1979 .

[52]  J. M. Hyman,et al.  Method of lines approach to the numerical solution of conservation laws , 1979 .

[53]  D. M. Sloan,et al.  A comparison of boundary methods for the numerical solution of hyperbolic systems of equations , 1977 .