Time integrators, solvers, and other mathematical specs#
Common specs for all solvers and time integrators, used in PKs.
There are three commonly used broad classes of time integration strategies.
“Update” methods are the simplest – they use no formal mathematical definition of a differential equation, but instead implicitly use a process by which variables at the new time are directly calculated. Typically there is an implied ODE or PDE here, but it is not stated as such and time integration routines are not used. Examples of these are common in biogeochemistry and vegetation models.
“Explicit” time methods are the next simplest. These include a variety of options from forward Euler to higher order Runge-Kutta schemes. These only require evaluating forward models where we have existing of the dependencies. If they work, these are great thanks to their deterministic nature and lack of expensive, memory-bandwith limited solvers. But they only work on some types of problems. Examples of of these include transport, where we use high order time integration schemes to preserve fronts.
“Implicit” and semi-implicit methods instead require the evaluation of a residual equation – the solution is guessed at, and the residual is calculated, which measures how far the equation is from being satisfied. This measure is then inverted, finding a correction to the guess which hopefully reduces the residual. As the residual goes to zero, the error, a measure of the difference between the guess and the true solution, also goes to zero. To do this inversion, we lean on Newton and Newton-like methods, which attempt to somehow linearize, or approximately linearize, the residual function near the guess in order to calculate an update. In this case, the time integration scheme requires both a nonlinear solver (to drive the residual to zero) and a linear solver or approximate solver (to calculate the correction).