The Time-Line Interpolation Method for Large-Time-Step Godunov-Type Schemes

Abstract This paper describes the use of the time-line interpolation procedure for the design of large-time-step, Godunov-type schemes for systems of hyperbolic conservation laws in one dimension. These schemes are based on a specific procedure to characterize the left and right states of the Riemann problems at the cell interfaces when the Courant number associated with the waves exceeds unity. To do so, the time-line interpolation technique is used. Constant and linear reconstruction techniques are presented. Sonic or critical points are seen to be a source of difficulty in the algorithm and an appropriate treatment is proposed. The algorithms are applied to the linear advection equation, to the inviscid Burgers equation, and to the set of hyperbolic conservation laws that describe shallow water flow in one dimension. These simulations show the superiority of the linear time reconstruction over the use of a constant time reconstruction. When the linear reconstruction technique is used, the modulus of the amplification factor of the scheme is equal to unity for all wave numbers, inducing oscillations in the computed profile owing to the phase error. The introduction of a slope limiter allows these oscillations to be eliminated, but yields numerical diffusion, thus restricting the range of applications of the scheme.

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

[2]  Elemér E Rosinger.  Nonlinear equivalence, reduction of PDEs to ODEs and fast convergent numerical methods , 1982 .

[3]  The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation , 2001 .

[4]  Lévêque,et al.  High resolution finite volume methods on arbitrary grids via wave propagation. Final report , 1988 .

[5]  P. Colella,et al.  An Implicit-Explicit Eulerian Godunov Scheme for Compressible Flow , 1995 .

[6]  V. Guinot.  An unconditionally stable, explicit Godunov scheme for systems of conservation laws , 2002 .

[7]  Sin-Chung Chang,et al.  Regular Article: The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws , 1999 .

[8]  A Second-Order Iterative Implicit-Explicit Hybrid Scheme for Hyperbolic Systems of Conservation Laws , 1996 .

[9]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[10]  B. P. Leonard.  Note on the von Neumann stability of explicit one-dimensional advection schemes , 1994 .

[11]  An Examination of Alternative Extrapolations to Find the Departure Point Position in a “Two-Time-Level” Semi-Lagrangian Integration , 1999 .

[12]  F. M. Holly,et al.  Dambreak flood waves computed by modified Godunov method , 1993 .

[13]  B. P. Leonard,et al.  The nirvana scheme applied to one‐dimensional advection , 1995 .

[14]  Randall J. LeVeque,et al.  Large time step shock-capturing techniques for scalar conservation laws , 1981 .

[15]  Jean-Marc Moschetta,et al.  ON THE PATHOLOGICAL BEHAVIOR OF UPWIND SCHEMES , 1998 .

[16]  Matania Ben-Artzi,et al.  Application of the “generalized Riemann problem” method to 1-D compressible flows with material interfaces , 1986 .

[17]  P. Roache.  A flux-based modified method of characteristics , 1992 .

[18]  G. Whitham.  Linear and non linear waves , 1974 .

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

[20]  Sin-Chung Chang.  The Method of Space-Time Conservation Element and Solution Element-A New Approach for Solving the Navier-Stokes and Euler Equations , 1995 .

[21]  E. Wylie,et al.  Characteristics Method Using Time‐Line Interpolations , 1983 .

[22]  Randall J. LeVeque,et al.  Convergence of a large time step generalization of Godunov's method for conservation laws , 1984 .

[23]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[24]  B. P. Leonard,et al.  Conservative Explicit Unrestricted-Time-Step Multidimensional Constancy-Preserving Advection Schemes , 1996 .

[25]  Elemer E Rosinger,et al.  Propagation of round-off errors and the role of stability in numerical methods for linear and nonlinear PDEs , 1985 .

[26]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[27]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[28]  Phillip Colella,et al.  An implicit-explicit hybrid method for Lagrangian hydrodynamics , 1986 .