A fourth-order Crowley-type advection scheme based on the multistep Warming-Kutler-Lomax (WKL) scheme is proposed in this study. This scheme utilizes a free parameter to minimize dispersion and dissipation and can be used to represent the advection of positive-definite scalars (such as moisture).
Linear Fourier component analyses indicate that the fourth-order Crowley-type scheme can reproduce the features of other modified Crowley-type schemes of third order, such as the scheme of Schlesinger and the quadratic upstream interpolation. Using the free parameter, the scheme may illustrate the limitation of the Crowley-type schemes for which diffusion is required for numerical stability of advective quantity. For these schemes, formulations that preserve amplitude are inevitably associated with smaller time steps. Adding the first cross-space term into these schemes could eliminate marginal instability or overshooting in linear advection.
Linear and nonlinear advection tests show that the performance of the proposed scheme is comparable to the fourth-order leapfrog scheme (which requires more computer memory) and the cubic upstream spline (which requires more computer time). This two-time-level advection scheme can thus be used in a numerical model to save computer resources.