Motivated by interpolation problems arising in applications to image analysis, computer vision, shape reconstruction and signal processing, we develop an algorithm to simulate curve straightening flows under which curves in R^n of fixed length and prescribed boundary conditions to first order evolve to elasticae, i.e., to critical points of the elastic energy E given by the integral of the square of the curvature function. We also consider variations in which the length L is allowed to vary and the flows seek to minimize the scale-invariant elastic energy, or the free elastic energy. Details of the implementations, experimental results, and applications to contour interpolation problems are also discussed.