Diagonally implicit Runge-Kutta formulae for the numerical integration of nonlinear two-point boundary value problems (Q1058641)

This paper considers the standard implicit Runge Kutta formulae for solving a system of equations of the form (1) (d/dt)\b{y}\(=\underline f(t,\underline y)\) with (2) g(\b{y}(a),\b{y}(b))\(=0\). The fully implicit Runge Kutta formulae lead to a set of non-linear equations. These can be simplified using the diagonally implicit Runge Kutta formula which reduces the difficulty in the algebra. However non-linear equations are still produced which incorporate (2). A special example of the simplifications is considered using the implicit mid-point rule and separated boundary conditions. Numerical results are obtained for several well-known systems. Although the analysis is applicable to both separated and non-separated boundary conditions only the former appear in the numerical examples. The main purpose of this paper is the derivation of an efficient class of Runge-Kutta formulae for two point boundary value problems and estimates of the number of operations involved in the algorithms are given.