Ordinary Differential Equations
Although the fundamental equations governing the evolution of geophysical fluids are partial differential equations, ordinary differential equations arise in several contexts. The trajectories of individual fluid parcels in an inviscid flow are governed by simple ordinary differential equations, and systems of ordinary differential equations may describe chemical reactions or highly idealized dynamical systems. Since basic methods for the numerical integration of ordinary differential equations are simpler than those for partial differential equations, and since the time-differencing formulae used in the numerical solution of partial differential equations are closely related to those used for ordinary differential equations, this chapter is devoted to the analysis of methods for the approximate solution of ordinary differential equations (ODE solvers). Nevertheless some approaches to the solution of partial differential equations, such as Lax–Wendroff and finite-volume methods, arise from fully discretized approximations in both space and time that cannot be correctly analyzed by considering the spatial and temporal differencing in isolation. These fully discretized approaches will be discussed in subsequent chapters.
KeywordsTruncation Error Absolute Stability Multistep Method Computational Mode Amplitude Error
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