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Analysis of semi‐Lagrangian trajectory computations
Author(s) -
Staniforth Andrew,
White Andy,
Wood Nigel
Publication year - 2003
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1256/qj.02.115
Subject(s) - discretization , mathematics , trajectory , kinematics , angular momentum , equations of motion , stability (learning theory) , equivalence (formal languages) , context (archaeology) , classical mechanics , mathematical analysis , computer science , physics , paleontology , discrete mathematics , astronomy , machine learning , biology
An important aspect of semi‐Lagrangian models is the determination of departure points. This involves the calculation of particle displacements and is done by integrating a discrete form of the trajectory, or kinematic, equation. The way this equation is discretized affects the stability and accuracy of the equation set. Additionally, recent work in the context of a time‐centred , semi‐implicit, semi‐Lagrangian scheme has shown that the form of the discrete kinematic equation reflects the preservation or violation of a dynamical equivalence between momentum and angular momentum formulations. Here that work is extended to the case of off‐centred semi‐implicit time weighting as used in most semi‐implicit, semi‐Lagrangian models. Two types of discretization of the trajectory equation are identified which preserve dynamical equivalence: some existing discretizations are found to be approximations to them. It is also shown how to incorporate physical forcings and/or predictor–corrector dynamics into the formulation. Two simple model problems (for solid‐body rotation and wave motion) are used to provide some further insight into the accuracy and stability properties of various trajectory schemes, including: dynamically‐equivalent schemes, approximations to these, and several existing schemes. Copyright © Crown copyright, 2003. Royal Meteorological Society