Use stiff solver in DEIM test (#164)

* Use stiff solver in DEIM test

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>

* Fix DEIM test init: supply consistent u0 instead of missing-guess hack

The full-order solve returned InitialFailure on the current MTK v11 /
MethodOfLines stack, so the snapshot had a single time column and
`POD(snapshot, 3)` errored with "Number of modes should be in {1,...,1}".

Root cause is initialization, not the solver. Passing
`missing_guess_value = Constant(0.0)` left every discretized state as a
missing guess, so MethodOfLines' initialization system was structurally
singular and underdetermined (1 equation for 11 unknowns). MTK fell back
to a least-squares initialization that diverged from the all-zero guess
(residual norm ~4.9e4, MaxIters), producing a garbage u0 (values ~1e19)
and an InitialFailure return code. Switching the solver to Rodas5P did
not help because the failure happens during initialization, before any
step is taken.

The boundary conditions fix every state to 0 at t=0 (v(x,0)=w(x,0)=0 and
i₀(0)=0), so the correct initial condition is simply all zeros. Providing
it explicitly makes the initialization well-posed and the solve succeeds,
yielding a 15-column snapshot that exercises POD/DEIM as intended.

Verified on the CI-failing stack (MethodOfLines 0.11.14, MTK 11.31.1,
ModelingToolkitBase 1.51.0, PDEBase 0.1.27): GROUP=Core Pkg.test() passes
(Core/deim.jl 4/4) and Runic --check is clean.

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>

---------

Co-authored-by: ChrisRackauckas-Claude <accounts@chrisrackauckas.com>

Author: ChrisRackauckas-Claude

test/deim.jl

@@ -38,11 +38,12 @@ order = 2
 discretization = MOLFiniteDifference(dxs, t; approx_order = order)
 ode_sys, tspan = symbolic_discretize(pde_sys, discretization)
 simp_sys = mtkcompile(ode_sys) # field substitutions is non-empty
-ode_prob = ODEProblem(
-    simp_sys, nothing, tspan;
-    missing_guess_value = ModelingToolkit.MissingGuessValue.Constant(0.0)
-)
-sol = solve(ode_prob, Tsit5(), saveat = 1.0)
+# The BCs fix every state to 0 at t=0, so provide those initial values directly.
+# Leaving them as missing guesses makes MethodOfLines' initialization system
+# structurally singular under MTK v11, and the least-squares solve diverges.
+u0 = [u => 0.0 for u in ModelingToolkit.get_unknowns(simp_sys)]
+ode_prob = ODEProblem(simp_sys, u0, tspan)
+sol = solve(ode_prob, Rodas5P(), saveat = 1.0)
 
 snapshot_simpsys = Array(sol.original_sol)
 pod_dim = 3
@@ -53,7 +54,7 @@ deim_sys = @test_nowarn deim(simp_sys, snapshot_simpsys, pod_dim)
 
 deim_prob = ODEProblem(complete(deim_sys), nothing, tspan)
 
-deim_sol = solve(deim_prob, Tsit5(), saveat = 1.0)
+deim_sol = solve(deim_prob, Rodas5P(), saveat = 1.0)
 
 nₓ = length(sol[x])
 nₜ = length(sol[t])