Parallel algorithms for semi-lagrangian advection

Abstract

Numerical time step limitations associated with the explicit treatment of advection‐dominated problems in computational fluid dynamics are often relaxed by employing Eulerian–Lagrangian methods. These are also known as semi‐Lagrangian methods in the atmospheric sciences. Such methods involve backward time integration of a characteristic equation to find the departure point of a fluid particle arriving at a Eulerian grid point. The value of the advected field at the departure point is obtained by interpolation. Both the trajectory integration and repeated interpolation influence accuracy. We compare the accuracy and performance of interpolation schemes based on piecewise cubic polynomials and cubic B‐splines in the context of a distributed memory, parallel computing environment. The computational cost and interprocessor communication requirements for both methods are reported. Spline interpolation has better conservation properties but requires the solution of a global linear system, initially appearing to hinder a distributed memory implementation. The proposed parallel algorithm for multidimensional spline interpolation has almost the same communication overhead as local piecewise polynomial interpolation. We also compare various techniques for tracking trajectories given different values for the Courant number. Large Courant numbers require a high‐order ODE solver involving multiple interpolations of the velocity field. © 1997 John Wiley & Sons, Ltd.

A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations

“…Following Malevsky and Thomas [21] we can write the time-step as a function of the grid spacing as follows:…”

Efficient parallelization of a regional ocean model for the western Mediterranean Sea

“…This has an influence on the width of the halo region because it has to contain the backtracked elements {A" ' }; see Malevsky and Thomas (1997). In fact, one may have two halo regions, namely, one halo for the calculation of the integrals…”

New numerical methods for Burgers' equation based on semi-Lagrangian and modified equation approaches

“…In addition, we require that the coefficients in the difference scheme (15) are non-negative, i.e., d m 0 and c k 0 (17) for each m and k. When the condition (17) is satisfied, the scheme (15) is monotone [6] and generally can better conserve physical properties (such as mass and entropy) in practical applications. In particular, monotonicity is essential to avoid spurious oscillations when computing solutions with steep gradients [9,22].…”

“…The authors in [13] and [17] applied the fourth-order Runge-Kutta method into their semi-Lagrangian formulation, but their results did not show improvement over those secondorder schemes. On the other hand, many practical problems, e.g., those in weather forecast, are demanding higher-order time discretizations to meet the increasing needs of accuracy and efficiency [27].…”