adding tests for pseudotransient method

test/basictests.jl

@@ -62,7 +62,7 @@ u0 = [1.0, 1.0]
 
 precs = [
     NonlinearSolve.DEFAULT_PRECS,
-    (args...) -> (Diagonal(rand!(similar(u0))), nothing)
+    (args...) -> (Diagonal(rand!(similar(u0))), nothing),
 ]
 
 for prec in precs, linsolve in (nothing, KrylovJL_GMRES())
@@ -772,3 +772,44 @@ for options in list_of_options
     sol = solve(probN, alg, abstol = 1e-10)
     @test all(abs.(f(u, p)) .< 1e-10)
 end
+
+# --- PseudoTransient tests ---
+
+function test_f!(du, u, p)
+    du[1] = 2 - 2u[1]
+    du[2] = u[1] - 4u[2]
+    return du
+end
+
+u0 = zeros(2)
+
+probN = NonlinearProblem{true}(test_f!, u0)
+
+sol = solve(probN, PseudoTransient())
+
+@test sol.retcode == ReturnCode.Success
+du = zeros(2)
+test_f!(du, sol.u, 4.0)
+@test du≈[0, 0] atol=1e-7
+
+#oop
+simple_test(u, p) = 0.9f0 .* u .- u
+probN = NonlinearProblem{false}(simple_test, [1.0f0])
+sol = solve(probN, PseudoTransient(alpha_initial = 1.0))
+
+@test sol.retcode == ReturnCode.Success
+@test abs(sol.u[1]) <= 1.0f-4
+
+#test jacobian free PseudoTransient method
+function f!(res, u, p)
+    @. res = u * u - p
+end
+u0 = [1.0, 1.0]
+p = 2.0
+probN = NonlinearProblem(f!, u0, p)
+
+linsolve = LinearSolve.KrylovJL_GMRES()
+sol = solve(probN,
+    PseudoTransient(alpha_initial = 10.0, linsolve = linsolve),
+    reltol = 1e-9)
+@test sol.retcode == ReturnCode.Success