Transient geotherms
Numerical solutions
•Expanding (2) and re-arranging to express temperature at time n+1 as a function of the temperatura at time n we get that:
(3)
remplacing...
we get...
further replacement... and
•Rearranging terms, we obtain the following set of implicit equations for the temperature at the n+1 time step in terms of the temperature at the n-th time step:
(4)
•This formulation results in N equations and N+4 unknowns, the four additional unknowns being u0n and uN+1n, u0n+1,uN+1 n+1. However, these four additional unknowns lie outside the computational grid (time, space). Also, from the boundary conditions the following conditions must be satisfied at the n-th and n+1 time steps:
•This gives the following set of N equations for the temperature at the n+1 time step :
(6)
(5) with
These latter conditions ensure that the boundary conditions are always satisfied.