QUESTION 2

Background

The transient heat equation in cylindrical coordinates (r, θ, z) may be described using: 

Q2a Write the BTCS (implicit) scheme for the 2D axisymmetric heat equation in Eqn 3 at node (j,k) for time “n+1”. The index j represents location in the r-direction, while the index k represents location in the z-direction. You must move any unknowns (at time “n+1”) to the LHS and all known values (at time “n”) to the RHS.

Q2b Undertake a von Neumann stability analysis of the difference equation for the BTCS scheme in Q2a and derive an expression for the magnitude of the ratio of errors at time n+1 to those at time n. You should write the error as

where R and S are in the r- and z-directions, respectively. You can treat R and S in the same way we treat K (=2pk/L) in the 1D heat equation. Is the BTCS scheme is the scheme unstable, conditionally stable or unconditionally stable? Justify your answer. If it is conditionally stable, derive the timestep restriction for stability.

Q2c We can write the BTCS scheme in Q2a as a matrix problem of the form

where Q is the temperature vector, [A], [B] are matrices that depend on the identity matrix [I], the Laplacian matrix [L], the timestep Dt and the diffusion coefficient a. BC is a vector that stores values associated with the boundary conditions. Specify exactly what the entries of [L] are equal to in terms of r, δr, δz and what [A], [B] are equal to in terms of [I], [L], Δt, and a (you can ignore boundary conditions in your answer).

data.all.m