Monotone bicompact schemes for a linear advection equation

Abstract

approximated by fourthorder compact differences, and an extended system is used. In contrast to (9), the original equation for the unknown function was sup� plemented not with an equation for its spatial deriva� tives but rather with an equation for its primitive. An advantage of this approach is obvious in the case of shockcapturing difference schemes as applied to the computation of discontinuous solutions, since the smoothness of a function's primitive is higher by one than that of the function itself. In (10) bicompact dif� ference schemes for a linear advection equation were constructed using various timestepping schemes as applied to evolution systems of differential equations obtained by the method of lines. However, the proper� ties of these schemes were examined only for smooth solutions. First, we show that the bicompact homogeneous scheme of (10), which is firstorder accurate in time and fourthorder accurate in space, as applied to a lin� ear advection equation is monotone. Second, this baseline scheme is used to construct a monotonized highorder accurate scheme in time. Both schemes were solved explicitly using the running calculation method. The numerical results produced by these schemes were compared with wellknown highorder accurate essentially nonoscillatory schemes (11, 12). The comparison showed the superiority of the schemes proposed in this paper.