GS98 fig. 2.4-1

[gdmcbain]

The reference is:

• Gresho, P. M., Sani, R. L. (2000). Incompressible Flow and the Finite Element Method. Volume One: Advection--Diffusion. Chichester, West Sussex: Wiley

I think the problem is (see p. 92)

• ∂T/∂t = ∂²T/∂x² on 0 < x < 1
• T (1) = 0, but an ‘OBC’ (‘open boundary condition’) T' (0) = −1

The figure plots the exact solution at x = 0, viz. T (0, t) = (4t/π)^½ for 0 < t < 0.012 and superimposes the numerical solution for a grid x_i = (i/N)^1.2 and N = 20 (which is not consistent with x₁ = 0/

[gdmcbain]

So dT/dt = T'', <v, dT/dt> = <T'', v>, d <v, T> / dt = [T', v] - <T', v'>

[gdmcbain]

The time-integration is supposed to be sufficiently accurate that only the spatial discretization, particularly of the OBC, is in question. That was done in the book using LSODE but we might just use TR (trapezoidal rule) with very small timesteps. Or can we adapt scipy.integrate.LSODA?

[gdmcbain]

M dT/dt = f - lap T

where M = mass matrix, lap = <u', v'>, laplacian matrix, and f includes the forcing heat flux at the origin.

Then write as an ODE, thinking about LSODA.

dT/dt = M**-1 (f - lap T).