A numerical advection scheme with small phase speed errors

Abstract

SUMMARYIt is demonstrated that, for linear advection in one space dimension, a simple modification to the wellknown Lax-Wendroff integration scheme leads to a substantial reduction in phase speed errors. The modification is designed in such a way that no additional restriction is placed on the timestep used in an integration. The improved performance is comparable or superior to that of previously proposed advection schemes, yet it is achieved without recourse to expensive or cumbersome computations.The new scheme is readily extended to two space dimensions and has been used in numerical weather prediction and sea wave modelling in the Meteorological office. Anexample is given of an improved numerical forecast of the 500 mb height field which was obtained by applying the new scheme in the operational 10-level numerical weather prediction model.

Some applications of the concept of minimized integrated exponential error for low dispersion and low dissipation

“…A new advection scheme was proposed by Gadd on choosing the limiting form for a, namely, a = 3 4 (1−r 2 ) [16]. We consider Equation (12) with a replaced by 3 4 (1−r 2 ) and compute the optimal cfl of the resulting numerical scheme.…”

“…The numerical scheme approximating the 1D linear advection equation, proposed by Gadd [16], is given by…”

“…In Section 3, we use MISDE and MIEELDLD to optimize two parameters in the Takacs' family of third-order schemes. In Section 4, we optimize the scheme devised by Gadd [16]. Next, the optimal cfl for some multi-level schemes, namely, 6P50, 8P70-1, 8P70-2 and 10P90 [17] are computed using MIEELDLD in Section 5.…”

“…The numerical scheme proposed by Gadd in 1978 [16] is a simple modification of the LW algorithm which leads to a substantial reduction in phase speed errors. Gadd introduced a free parameter and four grid points for use in the spatial first derivative of the LW scheme which eliminated much of the inherent phase lag.…”

Analysis of the Turkel-Zwas scheme for the shallow-water equations

“…Ax Ay (3) and p > 1 is an integer which defines the relationship between the coarse mesh on which terms associated with the inertia-gravity waves are treated and the fine mesh on which terms associated with the slow Rossby waves are discretized. It has been proven (see Navon and de Villers, [6]) that the stability criterion for this scheme is (4) and, since for typical atmospheric conditions, .jih~(lul + lvl), (5) one can use time steps nearly p times larger than allowed by the CFL condition for the usual explicit leapfrog scheme.…”