fix some typos in the docs (#202)

Author: ranocha

docs/src/tutorials/ARK_order_conditions.md

@@ -2,7 +2,7 @@
 
 Consider an ordinary differential equation (ODE) of the form
 ```math
-u'(t) = \sum_\nu^N f^\nu(t, u(t)).
+u'(t) = \sum_{\nu=1}^N f^\nu(t, u(t)).
 ```
 
 An additive Runge-Kutta (ARK) method with ``s`` stages is given by its
@@ -21,8 +21,8 @@ is assumed, which reduces all analysis to autonomous problems.
 The step from ``u^{n}`` to ``u^{n+1}`` is given by
 ```math
 \begin{aligned}
-  y^i &= u^n + \Delta t \sum_\nu \sum_j a^\nu_{i,j} f^\nu(y^i), \\
-  u^{n+1} &= u^n + \Delta t \sum_\nu \sum_i b^\nu_{i} f^\nu(y^i),
+  y^i &= u^n + \Delta t \sum_\nu \sum_j a^\nu_{i,j} f^\nu(t^n + c_j \Delta t, y^j), \\
+  u^{n+1} &= u^n + \Delta t \sum_\nu \sum_i b^\nu_{i} f^\nu(t^n + c_i \Delta t, y^i),
 \end{aligned}
 ```
 where ``y^i`` are the stage values.
@@ -34,7 +34,7 @@ ARK methods are represented as
 
 ## Order conditions
 
-The order conditions of RK methods can be derived using colored rooted trees.
+The order conditions of ARK methods can be derived using colored rooted trees.
 In [RootedTrees.jl](https://github.com/SciML/RootedTrees.jl), this
 functionality is implemented in [`residual_order_condition`](@ref).
 Thus, an [`AdditiveRungeKuttaMethod`](@ref) is of order ``p`` if the

docs/src/tutorials/RK_order_conditions.md

@@ -29,7 +29,7 @@ is assumed, which reduces all analysis to autonomous problems.
 The step from ``u^{n}`` to ``u^{n+1}`` is given by
 ```math
 \begin{aligned}
-  y^i &= u^n + \Delta t \sum_j a_{i,j} f(t^n + c_i \Delta t, y^i), \\
+  y^i &= u^n + \Delta t \sum_j a_{i,j} f(t^n + c_j \Delta t, y^j), \\
   u^{n+1} &= u^n + \Delta t \sum_i b_{i} f(t^n + c_i \Delta t, y^i),
 \end{aligned}
 ```

docs/src/tutorials/Rosenbrock_order_conditions.md

@@ -22,7 +22,7 @@ is assumed, which reduces all analysis to autonomous problems.
 The step from ``u^{n}`` to ``u^{n+1}`` is given by
 ```math
 \begin{aligned}
-  k^i &= \Delta t f\bigl(u^n + \sum_j a_{i,j} k^j \bigr) + \Delta t J \sum_j \gamma_{ij} k_j, \\
+  k^i &= \Delta t f\bigl(u^n + \sum_j a_{i,j} k^j \bigr) + \Delta t J \sum_j \gamma_{ij} k^j, \\
   u^{n+1} &= u^n + \sum_i b_{i} k^i.
 \end{aligned}
 ```

docs/src/tutorials/basics.md

@@ -16,7 +16,7 @@ end
 
 Depending on your background, you may be more familiar with the classical
 notation used in the books of Butcher or Hairer & Wanner. You can get these
-representation via [`butcher_representation`](@ref).
+representations via [`butcher_representation`](@ref).
 
 ```@example basics
 for t in RootedTreeIterator(4)
@@ -92,7 +92,7 @@ for t in RootedTreeIterator(4)
 end
 ```
 
-![latex elemenary weights](https://private-user-images.githubusercontent.com/125130707/298310491-8a035faf-fd1a-4fc0-92be-c3387eb53177.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJnaXRodWIuY29tIiwiYXVkIjoicmF3LmdpdGh1YnVzZXJjb250ZW50LmNvbSIsImtleSI6ImtleTUiLCJleHAiOjE3MDYxNjY1ODAsIm5iZiI6MTcwNjE2NjI4MCwicGF0aCI6Ii8xMjUxMzA3MDcvMjk4MzEwNDkxLThhMDM1ZmFmLWZkMWEtNGZjMC05MmJlLWMzMzg3ZWI1MzE3Ny5wbmc_WC1BbXotQWxnb3JpdGhtPUFXUzQtSE1BQy1TSEEyNTYmWC1BbXotQ3JlZGVudGlhbD1BS0lBVkNPRFlMU0E1M1BRSzRaQSUyRjIwMjQwMTI1JTJGdXMtZWFzdC0xJTJGczMlMkZhd3M0X3JlcXVlc3QmWC1BbXotRGF0ZT0yMDI0MDEyNVQwNzA0NDBaJlgtQW16LUV4cGlyZXM9MzAwJlgtQW16LVNpZ25hdHVyZT1kYzQyYjM0MWY0MDU5YWZlYzBmODA4MjFiZGIxN2E3YjhkYTdmZDNkYTU5NmI5OTEwNWFiZjg0OGZjNDg1MzZhJlgtQW16LVNpZ25lZEhlYWRlcnM9aG9zdCZhY3Rvcl9pZD0wJmtleV9pZD0wJnJlcG9faWQ9MCJ9.GB-PigOlQqenzgruzWg19qslzM6RXeX4xWwCNreOvNY)
+![latex elementary weights](https://private-user-images.githubusercontent.com/125130707/298310491-8a035faf-fd1a-4fc0-92be-c3387eb53177.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJnaXRodWIuY29tIiwiYXVkIjoicmF3LmdpdGh1YnVzZXJjb250ZW50LmNvbSIsImtleSI6ImtleTUiLCJleHAiOjE3MDYxNjY1ODAsIm5iZiI6MTcwNjE2NjI4MCwicGF0aCI6Ii8xMjUxMzA3MDcvMjk4MzEwNDkxLThhMDM1ZmFmLWZkMWEtNGZjMC05MmJlLWMzMzg3ZWI1MzE3Ny5wbmc_WC1BbXotQWxnb3JpdGhtPUFXUzQtSE1BQy1TSEEyNTYmWC1BbXotQ3JlZGVudGlhbD1BS0lBVkNPRFlMU0E1M1BRSzRaQSUyRjIwMjQwMTI1JTJGdXMtZWFzdC0xJTJGczMlMkZhd3M0X3JlcXVlc3QmWC1BbXotRGF0ZT0yMDI0MDEyNVQwNzA0NDBaJlgtQW16LUV4cGlyZXM9MzAwJlgtQW16LVNpZ25hdHVyZT1kYzQyYjM0MWY0MDU5YWZlYzBmODA4MjFiZGIxN2E3YjhkYTdmZDNkYTU5NmI5OTEwNWFiZjg0OGZjNDg1MzZhJlgtQW16LVNpZ25lZEhlYWRlcnM9aG9zdCZhY3Rvcl9pZD0wJmtleV9pZD0wJnJlcG9faWQ9MCJ9.GB-PigOlQqenzgruzWg19qslzM6RXeX4xWwCNreOvNY)
 
 
 ## Number of trees

src/colored_trees.jl

@@ -393,7 +393,7 @@ end
 """
     BicoloredRootedTreeIterator(order::Integer)
 
-Iterator over all bi-colored rooted trees of given `order`. The returned trees
+Iterator over all bicolored rooted trees of given `order`. The returned trees
 are views to an internal tree modified during the iteration. If the returned
 trees shall be stored or modified during the iteration, a `copy` has to be made.
 """

src/time_integration_methods.jl

@@ -148,8 +148,8 @@ An additive Runge-Kutta method applied to the ODE problem
 has the form
 ```math
 \\begin{aligned}
-  y^i &= u^n + \\Delta t \\sum_\\nu \\sum_j a^\\nu_{i,j} f^\\nu(y^i), \\\\
-  u^{n+1} &= u^n + \\Delta t \\sum_\\nu \\sum_i b^\\nu_{i} f^\\nu(y^i).
+  y^i &= u^n + \\Delta t \\sum_\\nu \\sum_j a^\\nu_{i,j} f^\\nu(t^n + c_j \\Delta t, y^j), \\\\
+  u^{n+1} &= u^n + \\Delta t \\sum_\\nu \\sum_i b^\\nu_{i} f^\\nu(t^n + c_i \\Delta t, y^i).
 \\end{aligned}
 ```