From 1448dde67779613217b267e2fea152a656839402 Mon Sep 17 00:00:00 2001
From: Valentin Churavy <v.churavy@gmail.com>
Date: Tue, 20 May 2025 10:41:11 +0200
Subject: [PATCH] Multi-derivative implicit

---
 examples/implicit.jl | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/examples/implicit.jl b/examples/implicit.jl
index 21b8e5c..ee522ef 100644
--- a/examples/implicit.jl
+++ b/examples/implicit.jl
@@ -36,6 +36,18 @@ function G_Trapezoid!(res, uₙ, Δt, f!, du, u, p, t)
     return nothing
 end
 
+# ## Implicit Euler -- multiderivative
+#
+
+function G_Euler_MD!(res, uₙ, Δt, f!, du, u, p, t)
+    f!(du, u, p, t)
+    autodiff(Forward, f!, Duplicated(du, res), Duplicated(u, copy(du)), Const(p), Const(t))
+
+
+    res .= uₙ .+ Δt .* du .+ Δt^2 / 2 .* res .- u
+    return nothing
+end
+
 # ## Jacobian of various G
 
 function jacobian(G!, f!, uₙ, p, Δt, t)