A Class of Semi-Lagrangian Approximations for Fluids

This paper discusses a class of finite-difference approximations to the evolution equations of fluid dynamics. These approximations derive from elementary properties of differential forms. Values of a fluid variable ψ at any two points of a space-time continuum are related through the integral of the space-time gradient of ψ along an arbitrary contour connecting these two points (Stokes' theorem). Noting that spatial and temporal components of the gradient are related through the fluid equations, and selecting the contour composed of a parcel trajectory and an appropriate residual, leads to the integral form of the fluid equations, which is particularly convenient for finite-difference approximations. In these equations, the inertial and forcing terms are separated such that forces are integrated along a parcel trajectory (the Lagrangian aspect), whereas advection of the variable is evaluated along the residual contour (the Eulerian aspect). The virtue of this method is an extreme simplicity of the resulting solver; the entire model for a fluid may be essentially built upon a single one-dimensional Eulerian advection scheme while retaining the formal accuracy of its constant-coefficient limit. The Lagrangian aspect of the approach allows for large-Courant-number (>1) computations in a broad spectrum of dynamic applications. Theoretical considerations are illustrated with examples of applications to selected classical problems of atmospheric fluid dynamics. Since the theoretical arguments adopted in this paper assume differentiability of fluid variables, fluid systems admitting truly discontinuous solutions (e.g., shock waves, hydraulic jumps) are formally excluded from our considerations.