MethodOfLines.jl
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Solution interface does not work when using init() and step!()
It seems that one cannot use the solution using init() and step!() from the integrator interface.
using ModelingToolkit, MethodOfLines, OrdinaryDiffEq, DomainSets
@variables x y t
@variables u(..) v(..)
Dt = Differential(t)
Dx = Differential(x)
Dy = Differential(y)
Dxx = Differential(x)^2
Dyy = Differential(y)^2
∇²(u) = Dxx(u) + Dyy(u)
brusselator_f(x, y, t) = (((x-0.3)^2 + (y-0.6)^2) <= 0.1^2) * (t >= 1.1) * 5.
x_min = y_min = t_min = 0.0
x_max = y_max = 1.0
t_max = 11.5
α = 10.
u0(x,y,t) = 22(y*(1-y))^(3/2)
v0(x,y,t) = 27(x*(1-x))^(3/2)
eq = [Dt(u(x,y,t)) ~ 1. + v(x,y,t)*u(x,y,t)^2 - 4.4*u(x,y,t) + α*∇²(u(x,y,t)) + brusselator_f(x, y, t),
Dt(v(x,y,t)) ~ 3.4*u(x,y,t) - v(x,y,t)*u(x,y,t)^2 + α*∇²(v(x,y,t))]
domains = [x ∈ Interval(x_min, x_max),
y ∈ Interval(y_min, y_max),
t ∈ Interval(t_min, t_max)]
# Periodic BCs
bcs = [u(x,y,0) ~ u0(x,y,0),
u(0,y,t) ~ u(1,y,t),
u(x,0,t) ~ u(x,1,t),
v(x,y,0) ~ v0(x,y,0),
v(0,y,t) ~ v(1,y,t),
v(x,0,t) ~ v(x,1,t)]
@named pdesys = PDESystem(eq,bcs,domains,[x,y,t],[u(x,y,t),v(x,y,t)])
N = 32
order = 2 # This may be increased to improve accuracy of some schemes
# Integers for x and y are interpreted as number of points. Use a Float to directtly specify stepsizes dx and dy.
discretization = MOLFiniteDifference([x=>N, y=>N], t, approx_order=order)
# Convert the PDE problem into an ODE problem
println("Discretization:")
@time prob = discretize(pdesys,discretization)
state = init(prob, TRBDF2())
step!(state, 0.1)
discrete_x = state.sol[x]
discrete_y = state.sol[y]
discrete_t = state.sol[t]
solu = state.sol[u(x, y, t)]
solv = state.sol[v(x, y, t)]
The error given is
ERROR: LoadError: ArgumentError: x is neither an observed nor a state variable.
with an extraordinarily long stack trace starting at build_explicit_observed_function
.