I.5 Temporal discretisation

This problem is a parabolic one : we have a set of values representing the solution at one time step, and we will solve the equations by computing successive timesteps. This timestepping represents a discretisation of time as well as space into M timesteps q^j. The time derivative can thus be expressed as

dq qj+1 - qj ---= --------- dt dt

We can insert this into (I.5). This raises the question : at what timestep are the values on the r.h.s. evaluated? If we assume they are taken as the values at timestep j, then we have an explicit scheme. If we use the values at timestep j + 1 then we have an implicit scheme.

For an explicit scheme we can write this as

qji+1 - qji = - udt-(qi+1 - qi-1)j + Gdt-(qi+1 - 2qi + qi- 1)j 2dx dx2

We can write this in terms of 2 factors

udt- Gdt- C = dx , D = dx2

C is the Courant number for the problem, and is a significant parameter in determining the stability of the scheme. D is a similar parameter relating to the diffusion. Writing this out as an algorithm :

( ) ( ) qj+1 = C-+ D qj + (1 - 2D) qj + C-- D qj i 2 i- 1 i 2 i+1

(I.6)

This is a rule for advancing the values of q through one timestep, which could be written into a spreadsheet. In the case of the implicit scheme we would write the scheme in terms of a vector of unknown values (q_i^{j + 1}), as a matrix equation :

( ) ( . ) ( . ) ... .. .. D U 0 qj+1 qj ij-+11 i-j1 L D U qij+1 = qji 0 L D qi+1 qi+1 ... ... ...

where D, U and L are constant values related to C and D. Inverting this matrix provides the solution without the stability problems mentioned above (although the Courant number is still worth calculating if there are problems).

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