test: added test for dae_order_lowering

test/odesystem.jl

@@ -1552,3 +1552,60 @@ end
     expected_tstops = unique!(sort!(vcat(0.0:0.075:10.0, 0.1, 0.2, 0.65, 0.35, 0.45)))
     @test all(x -> any(isapprox(x, atol = 1e-6), sol2.t), expected_tstops)
 end
+
+@testset "dae_order_lowering basic test" begin
+    @parameters a
+    @variables x(t) y(t) z(t)
+    @named dae_sys = ODESystem([
+        D(x) ~ y,
+        0 ~ x + z,
+        0 ~ x - y + z
+    ], t, [z, y, x], [])
+
+    lowered_dae_sys = dae_order_lowering(dae_sys)
+    @variables x1(t) y1(t) z1(t) 
+    expected_eqs = [
+        0 ~ x + z,
+        0 ~ x - y + z,
+        Differential(t)(x) ~ y
+    ]
+    lowered_eqs = equations(lowered_dae_sys)
+    sorted_lowered_eqs = sort(lowered_eqs, by=string)
+    sorted_expected_eqs = sort(expected_eqs, by=string)
+    @test sorted_lowered_eqs == sorted_expected_eqs
+
+    expected_vars = Set([z, y, x])
+    lowered_vars = Set(unknowns(lowered_dae_sys))
+    @test lowered_vars == expected_vars
+end
+
+@testset "dae_order_lowering test with structural_simplify" begin
+    @variables x(t) y(t) z(t)
+    @parameters M b k
+    eqs = [
+        D(D(x)) ~ -b / M * D(x) - k / M * x,
+        0 ~ y - D(x),
+        0 ~ z - x
+    ]
+    ps = [M, b, k]
+    default_u0 = [
+        D(x) => 0.0, x => 10.0, y => 0.0, z => 10.0
+    ]
+    default_p = [M => 1.0, b => 1.0, k => 1.0]
+    @named dae_sys = ODESystem(eqs, t, [x, y, z], ps; defaults = [default_u0; default_p])
+
+    simplified_dae_sys = structural_simplify(dae_sys)
+
+    lowered_dae_sys = dae_order_lowering(simplified_dae_sys)
+    lowered_dae_sys = complete(lowered_dae_sys)
+
+    tspan = (0.0, 10.0)
+    prob = ODEProblem(lowered_dae_sys, nothing, tspan)
+    sol = solve(prob, Tsit5())
+
+    @test sol.t[end] == tspan[end]
+    @test sum(abs, sol.u[end]) < 1
+
+    prob = ODEProblem{false}(lowered_dae_sys; u0_constructor = x -> SVector(x...))
+    @test prob.u0 isa SVector
+end