discretize fails on Robin Boundary condition applied to Laplace equation in spherical domain #75
Description
Solving a simple heat conduction problem on a sphere (1D in radial direction, assumed rotational symmetry) with symmetry boundary condition on r=0 and convective heat flux boundary condition on r=R (boundary condition of Robin type) fails with error Boundary condition Differential(r)(T(0.01, t)) ~ 100 - T(0.01, t) is not on a boundary of the domain, or is not a valid boundary condition. MethodOfLines documentation claims that Robin BCs should work.
using ModelingToolkit, MethodOfLines, OrdinaryDiffEq, DomainSets
@parameters r t
@variables T(..)
Dt = Differential(t)
Dr = Differential(r)
Drr = Differential(r)^2
t_min = 0.0
t_max = 11.5
R = 10e-3 # Sphere radius
T_ambient = 100 # ambient temperature
T0(r,t) = 20 # initial temperature field
# Laplace equation in spherical coordinates, describing heat conduction in sphere
eq = [Dt(T(r,t)) ~ 1/r^2 * Dr(r^2 * Dr(T(r,t))) ]
domains = [r ∈ Interval(0, R),
t ∈ Interval(t_min, t_max)]
# BCs
bcs = [T(r,0) ~ T0(r,0), # initial values
Dr(T(0,t)) ~ 0, # symmetry at r = 0
Dr(T(R,t)) ~ T_ambient - T(R,t)] # Robin type BC describing convective heat transport on sphere surface
@named pdesys = PDESystem(eq,bcs,domains,[r,t],[T(r,t)])
N = 32
dr = 1/N
order = 2
discretization = MOLFiniteDifference([r=>dr], t, approx_order=order, upwind_order = 1, grid_align = center_align)
# fails on third Boundary Condition (Robin one) with error:
# Boundary condition Differential(r)(T(0.01, t)) ~ 100 - T(0.01, t) is not on a boundary of the domain, or is not a valid boundary condition
prob = discretize(pdesys,discretization)