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

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.