From 2a4fed56d14a79e8152dbc18132d8e5608cd3120 Mon Sep 17 00:00:00 2001
From: Michael Kraus <michael.kraus@ipp.mpg.de>
Date: Thu, 8 Feb 2024 14:01:13 +0100
Subject: [PATCH] Fix in docs and README.

---
 README.md                    | 16 ++++++++--------
 docs/src/integrators/vprk.md |  2 +-
 docs/src/tutorial.md         |  2 +-
 3 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/README.md b/README.md
index 0d6265146..bb5043153 100644
--- a/README.md
+++ b/README.md
@@ -35,24 +35,24 @@ Before any use, we need to load `GeometricIntegrators`,
 ```julia
 using GeometricIntegrators
 ```
-Then we can create an `ODE` object for the equation ẋ(t) = x(t) with initial condition x(0)=1 by
-```julia
-ode = ODE((t, x, ẋ) -> ẋ[1] = x[1], [1.0]);
+Then we can create an ODE problem for the equation $\dot{x} (t) = x(t)$ with integration time span $(0, 1)$. a time step of $\Delta t = 0.1$, and initial condition $x(0) = 1$, 
+```@example 1
+prob = ODEProblem((ẋ, t, x, params) -> ẋ[1] = x[1], (0.0, 1.0), 0.1, [1.0])
 ```
-An integrator for this ODE, using the tableau for the explicit Euler method and a time step of Δt=0.1, is obtained by
+An integrator for this ODE, using the explicit Euler method is obtained by
 ```julia
-int = Integrator(ode, TableauExplicitEuler(), 0.1);
+int = GeometricIntegrator(prob, ExplicitEuler())
 ```
-With that, the solution for nₜ=10 time steps is simply computed by
+With that, the solution is simply computed by
 ```julia
-sol = integrate(ode, int, 10);
+sol = integrate(int)
 ```
 which returns an object holding the solution for all time steps.
 With the help of the *[Plots.jl](https://github.com/JuliaPlots/Plots.jl)* package we can visualise the result and compare with the exact solution:
 ```julia
 using Plots
 plot(xlims=[0,1], xlab="t", ylab="x(t)", legend=:bottomright)
-plot!(sol.t, sol.q[1,:], label="numeric")
+plot!(sol.t, sol.q[:,1], label="numeric")
 plot!(sol.t, exp.(sol.t), label="exact")
 ```
 
diff --git a/docs/src/integrators/vprk.md b/docs/src/integrators/vprk.md
index e150438a0..461c768ce 100644
--- a/docs/src/integrators/vprk.md
+++ b/docs/src/integrators/vprk.md
@@ -4,7 +4,7 @@ CurrentModule = GeometricIntegrators.Integrators
 
 # [Variational Partitioned Runge-Kutta Integrators](@id vprk)
 
-Variational partitioned Runge-Kutta methods solve Lagranian systems in implicit form, i.e.,
+Variational partitioned Runge-Kutta methods solve Lagrangian systems in implicit form, i.e.,
 ```math
 \begin{aligned}
 p       &= \dfrac{\partial L}{\partial \dot{q}} (q, \dot{q}) , &
diff --git a/docs/src/tutorial.md b/docs/src/tutorial.md
index d4729caf0..861bb7587 100644
--- a/docs/src/tutorial.md
+++ b/docs/src/tutorial.md
@@ -33,7 +33,7 @@ Before any use, we need to load GeometricIntegrators,
 ```@example 1
 using GeometricIntegrators
 ```
-Then we can create an ODE object for the equation $\dot{x} (t) = x(t)$ with initial condition $x(0) = 1$, integration time span $(0, 1)$ and a time step of $\Delta t = 0.1$,
+Then we can create an ODE problem for the equation $\dot{x} (t) = x(t)$ with integration time span $(0, 1)$. a time step of $\Delta t = 0.1$, and initial condition $x(0) = 1$, 
 ```@example 1
 prob = ODEProblem((ẋ, t, x, params) -> ẋ[1] = x[1], (0.0, 1.0), 0.1, [1.0])
 ```