1. Sec 232: The coefficient tableau 2ed order RK method Expand all f around (x,y) using Taylor expansion in 2d and then choose constants such that the equation match as many terms in Taylor expansion. Explicit Runge–Kutta methods As the order being sought increases, the algebraic conditions on the coefficients of the method become increasingly complicated. The pattern behind these conditions is known

2. Tree (graph theory) Graph a graph is an ordered pair G=(V,E) comprising a set V of vertices or nodes together with a set E of edges or lines The order of a graph is |V| (the number of vertices) The degree of a vertex is the number of edges that connect to it a tree is a graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree.

3. Tree (graph theory) a tree is a graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree. A rooted tree is a tree in which a special ("labeled") node is singled out. A leaf is a vertex of degree 1. An internal vertex is a vertex of degree at least 2. the ‘leaves’ of a tree are the vertices from which there are no outward-growing arcs; in other words, a leaf is a vertex, other than the root, which has exactly one other vertex joined to it.

4. Sec 233: The coefficient tableau 3ed order RK method Expand all f around (x,y) using Taylor expansion in 2d and then choose constants such that the equation match as many terms in Taylor expansion. 2ed order RK method For each tree t, a corresponding polynomial in the coefficients of the method can be written down. Denote this by Φ(t). Also associated with each tree t is an integer γ(t).

5. Sec 233: The coefficient tableau associate with each vertex of the tree, except the leaves, a label i, j,..., and assume that i is the label attached to the root Write down a sequence of factors of which the first is bi. For each arc of the tree, other than an arc that terminates in a leaf, write down a factor, say ajk, where j and k are the beginning and end of the arc (assuming that all directions are in the sense of movement away from the root). Having written down this sequence of factors, sum their product for all possible choices of each of the labels, in the set {1, 2,..., s}. associate a factor with each vertex of the tree For the leaves this factor is 1, and for all other vertices it is equal to the sum of the factors attached to all outward-growing neighbours, plus 1. The product of the factors, for all vertices of the tree, is the value of γ(t).

6. Sec 233: The coefficient tableau 3ed order RK method

7. Sec 233: The coefficient tableau order conditions

8. Sec 233: The coefficient tableau 4th order RK method

9. Sec 233: The coefficient tableau 4th order RK method RK41 RK42

10. Sec 236: Higher orders Higher orders Because the number of rooted trees of various orders increases rapidly for orders greater than 4, the complexity of the order conditions also increases. Above order 4, it is no longer possible to obtain order s with just s stages. For order 5, six stages are required, and for order 6, seven stages are required. Above this order, there are even sharper increases in the required numbers of stages. Explicit Runge–Kutta methods

11. Sec 237: Implicit Runge–Kutta Methods Explicit Runge–Kutta methods Implicit Runge–Kutta methods Explicit Runge–Kutta methods are generally unsuitable for the solution of stiff equations because their region of absolute stability Semi-explicit Diagonally explicit we can evaluate by simple (‘fixed point’) iteration. That is, we calculate a sequence of approximations

12. Sec 237: Implicit Runge–Kutta Methods order 3 order 2 order 4 In the implicit case, the cost of using a specific method depends not so much on the number of stages, as on the difficulty in evaluating the stages. From this point of view, (237a) is the easiest to use because only one of the stages is implicit; (237b) and (237c) each have two interconnected implicit stages (237a) (237b) (237c)

13. Sec 237: Implicit Runge–Kutta Methods Fehlberg's 4(5) method The first row of coefficients at the bottom of the table gives the fifth-order accurate method, and the second row gives the fourth-order accurate method.

14. Sec 237: Implicit Runge–Kutta Methods MATLAB Built-in Command ode23, ode45, ode113, ode15s, ode23s, ode23t, ode23tb, ode15i, Example 1 An example of a nonstiff system is the system of equations describing the motion of a rigid body without external forces. To simulate this system, create a function rigid containing the equations function dy = rigid(t,y) dy = zeros(3,1); % a column vector dy(1) = y(2) * y(3); dy(2) = -y(1) * y(3); dy(3) = -0.51 * y(1) * y(2); In this example we change the error tolerances using the odeset command and solve on a time interval [0 12] with an initial condition vector [0 1 1] at time 0. options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5]); [T,Y] = ode45(@rigid,[0 12],[0 1 1],options); Plotting the columns of the returned array Y versus T shows the solution plot(T,Y(:,1),'-',T,Y(:,2),'-.',T,Y(:,3),'.')