param_to_var on PDESystem parameters doesn't add initial condition BCs

[ctessum]

Description

When param_to_var promotes a PDESystem parameter to a dependent variable during coupling, no initial condition (t=0 BC) is added for the new variable. This causes discretize to fail with:

Please provide a default (`u0`), initialization equation, or guess
for the following variables:
(E_wind(t))[2, 2], (R_0(t))[2, 2], (u_y(t))[2, 2], (B_wind(t))[2, 2], (C_wind(t))[2, 2]

Root cause

The ODE coupling path doesn't have this problem because icbc(di, statevars) generates ICs for all state variables when promoting via sys + domain. But in the PDE coupling path:

1. couple2 calls param_to_var(pdesys, :R_0) to promote a parameter to a DV
2. The connector equation wires it: pdesys.R_0 ~ other_sys.R0
3. merge_pdesystems collects DVs and BCs from all subsystems
4. The new DV R_0 has no t=0 BC because it was a parameter (not a DV) in the original PDESystem

Affected variables

Any PDESystem parameter promoted via param_to_var during coupling. In the wildfire model, this includes: R_0, C_wind, B_wind, E_wind, u_x, u_y, dzdx, dzdy, φs_coeff, β_ratio, and others.

Suggested fix

Either:

1. In param_to_var(sys::PDESystem, ...): When promoting a parameter, also add a dv(t0, spatial_ivs...) ~ default BC to the PDESystem's boundary conditions.

2. In merge_pdesystems: After merging, detect DVs that lack a t=0 BC and add dv(t0, spatial_ivs...) ~ 0.0 for algebraic variables.

Workaround

Add missing ICs manually after convert(PDESystem, cs):

let t0 = DomainSets.infimum(pde.domain[1].domain)
    has_ic = Set{Symbol}()
    for bc in pde.bcs, dv in pde.dvs
        dvname = Symbol(Symbolics.tosymbol(dv, escape=false))
        if occursin(string(dvname, "(", t0), string(bc.lhs))
            push!(has_ic, dvname)
        end
    end
    new_bcs = copy(pde.bcs)
    for dv in pde.dvs
        dvname = Symbol(Symbolics.tosymbol(dv, escape=false))
        dvname ∈ has_ic && continue
        args = Symbolics.arguments(Symbolics.unwrap(dv))
        op = Symbolics.operation(Symbolics.unwrap(dv))
        push!(new_bcs, op(t0, args[2:end]...) ~ 0.0)
    end
    if length(new_bcs) > length(pde.bcs)
        pde = PDESystem(equations(pde), new_bcs, pde.domain, pde.ivs, pde.dvs, pde.ps;
            name = nameof(pde), initial_conditions = pde.initial_conditions, checks = false)
    end
end

Related

• @constants in couple2 connector equations lost during PDE merge #198 — @constants in connector equations lost during PDE merge
• Namespaced spatial coordinates (e.g. LANDFIRE₊x) not replaced during MOL discretization #199 — Namespaced spatial coordinates not replaced during MOL discretization