Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations. [in gas dynamics

Abstract The application of a new implicit unconditionally stable high-resolution TVD scheme to steady-state calculations is examined. It is a member of a one-parameter family of explicit and implicit second-order accurate schemes developed by Harten for the computation of weak solutions of one-dimensional hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a fairly rapid convergence rate, but also generates a highly resolved approximation to the steady-state solution. A detailed implementation of the implicit scheme for the one- and two-dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one- and two-dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

[1]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[2]  H. C. Yee,et al.  On the application and extension of Harten's high resolution scheme , 1982 .

[3]  A. Harten,et al.  The artificial compression method for computation of shocks and contact discontinuities. I - Single conservation laws , 1977 .

[4]  H. C. Yee,et al.  A high-resolution numerical technique for inviscid gas-dynamic problems with weak solutions , 1982 .

[5]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[6]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[7]  H. C. Yee,et al.  Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates , 1987 .

[8]  A. L. Roux.  A numerical conception of entropy for quasi-linear equations , 1977 .

[9]  A. Harten,et al.  The artificial compression method for computation of shocks and contact discontinuities: III. Self , 1978 .

[10]  P. Lax,et al.  Decay of solutions of systems of nonlinear hyperbolic conservation laws , 1970 .

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

[12]  Improving the convergence rate of parabolic ADI methods , 1983 .

[13]  Sukumar Chakravarthy,et al.  High Resolution Schemes and the Entropy Condition , 1984 .

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

[15]  M. J. Baines,et al.  On convergence of Roe's scheme for the general non-linear scalar wave equation , 1984 .

[16]  G. R. Shubin,et al.  Steady shock tracking and Newton's method applied to one-dimensional duct flow , 1981 .

[17]  R. F. Warming,et al.  On the construction and application of implicit factored schemes for conservation laws. [in computational fluid dynamics] , 1978 .

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