I'm a scientific programmer, and the relevant work was
with numerical solutions of PDEs and in particular the Navier Stokes equations.; and in signal processing.
Their results are in excellent agreement
with numerical solutions in 2 and 3 dimensions.
Not exact matches
The
numerical solution of the time dependent Schrödinger equation for the investigated atoms in a strong laser field provided excellent agreement of the theory
with the experimental data in both regimes.
What is surprising in this method is that the mathematics of shape, in the form of what differential - geometers call the curvature flow, is coupled
with techniques that originated in the
numerical solution of partial differential equations to deblur and denoise an image.
He began
with fluid dynamics equations, and then used
numerical calculations to arrive at approximate
solutions for specifics combinations of source flow and spread rates, and crosswind speed.
A fourth grade teacher starts each class period by presenting a problem... along
with its precise
numerical solution.
This is a Christmas - themed idea I came up
with where students answer the questions (the
solutions are 1 to 24) and each answer links to a word; put the words in
numerical order and an entire joke is revealed!
Experience
with solution algorithms, data assimilation methods and tools, coupling of components and processes, nonlinear and linear solvers, limiters, and / or other
numerical issues common
with complex codes within earth system models of varying complexity
I am only suggesting this point - of - view because David Young's views of
numerical stability are likely very valid, but perhaps misdirected when we consider how far we can go
with an alternative statistical mechanical
solution.
From what I read here there is a great deal of understanding, in depth, of the complexity of both the physical phenomena and processes, the mathematical problems associated
with both the continuous and discrete equations,
numerical solution methods for the latter, and designing, developing, building, testing, and applying computer software that has all this stuff in it.
These include running idealized cases and comparing
with an analytical
solution, comparing
with a very high resolution
numerical solution, and testing the integral constraints (e.g. make sure the model isn't losing mass).
But the mathematical analysis of the frequencies, the demonstrations of the problems
with the continuum systems using convergent
numerical solutions on the Exponential thread, and the dynamical core manuscript itself should make the issues very clear.
I appreciate your statement «From what I read here there is a great deal of understanding, in depth, of the complexity of both the physical phenomena and processes, the mathematical problems associated
with both the continuous and discrete equations,
numerical solution methods for the latter, and designing, developing, building, testing, and applying computer software that has all this stuff in it.»