1. Runge Kutta schemes Taylor series method Numeric solutions of ordinary differential equations

2. Idea of Runge-Kutta methods The exact solution of the trivial differential equation is Since a very rich theory and powerful methods exist to compute integrals numerically, they can also be utilized in the numerical solution of general ODE’s This is the rationale behind Runge-Kutta methods.

3. Gaussian quadrature Replacing an integral with a finite sum is a procedure known as quadrature. Let be a non-negative function acting in the interval (a,b) such that is called the weight function.

4. We approximate as follows: where and are called the quadrature weights and nodes respectively. They are independent of the function f(x), but, in general, depend upon, a, and b.

5. How good is the approximation? Suppose that the quadrature matches the integral exactly whenever f(x) is a polynomial of degree It can then be easily proved that, for every function f(x) with p smooth derivatives, where the constant c > 0 is independent of f(x). Such a quadrature formula is said to be of order p.

6. We denote the set of all real polynomials of degree m by. Then is of order p if it is exact for every Given any distinct set of nodes, it is possible to find a unique set of weights such that the quadrature formula is of order

7. The weights can be derived explicitly: where The simplest method is to choose the quadrature nodes equispaced in [a,b] and this leads to the so-called Newton-Cotes methods. This procedure is, however, far from being optimal: by making an adroit choice of we can, in fact, double the order to 2n.

8. Each weight function determines an inner product in the interval (a,b), namely whose domain is the set of all functions f(x), g(x) such that

9. We say that is an m-th orthogonal polynomial (with respect to the weight function ) if Orthogonal polynomials are not unique since we can always multiply p m by a non-zero constant without violating the above condition. However if we consider only polynomials with the leading coefficient equal to one, we achieve the uniqueness.

10. Classical orthogonal polynomials Three families of weights give rise to classical orthogonal polynomials. Let where Such polynomials are called Jacobi polynomials

11. Legendre polynomials

13. Chebyshev polynomials

16. Laguerre polynomials Hermite polynomials

17. The classical orthogonal polynomials have been very extensively studied and occur in a very wide range of applications. They have several remarkable properties that, in a well-defined sense, make them the ‘simplest’ orthogonal’ polynomials. For example are the only orthogonal polynomials whose derivatives are also orthogonal with some weight function.

18. All m zeros of an orthogonal polynomial p m lie in the interval (a,b) and they are simple

19. where then the quadrature method based on these nodes and weights is of order 2n and no other quadrature can exceed this order. Let be the zeros of an orthogonal polynomial and

20. Explicit Runge-Kutta schemes How do we extend a quadrature formula to the ODE The obvious approach is to integrate from and to replace the integral by a quadrature.

21. The above formula might have been the method except that we do not know the values of y at the nodes We must resort to an approximation.

22. We denote We let c 1 = 0, then The idea behind explicit Runge-Kutta (ERK) methods is to express each by updating y k with a linear combination of

23. Specifically, we let

24. The matrix is called the RK matrix, while and are the RK nodes and RK weights respectively.

25. When choosing the RK matrix, RK nodes, and RK weights we expand everything into Taylor series about (t k, y k ). For n = 2, the popular choices are:

26. Some instances of third-order ERK methods: the classical RK method the Nystrom scheme

27. The best known fourth-order RK method