4TH ORDER RUNGE-KUTTA METHOD IN 2D FOR LINEAR BOUNDARY-VALUE-PROBLEMS

The standard formulae of the fourth order Runge-Kutta method are transformed into a finite difference equation which allow us to solve linear boundary value problems. This equation is manipulated further and extended into 2D for solving second order linear partial differential equations. The various transformations result in four equations for the two first derivatives, mixed second derivative and the function value in the depicted point of the mesh. Each of these quantities is expressed by the linear combination of values of the function given in the eight nearest nodes around the depicted node. Explicit formulation of the equations for corner, boundary, symmetry and internal nodes is given. The results of the new computer code are compared with an analytical result and precise measurements, as well as a FEM calculation. It is concluded, for the specific example investigated, that the new method is about ten or more times precise than the FEM with the same number of elements. The major merit of the method lies in the fact that it can be implemented on personal computers.