In this paper, we develop convergent numerical schemes to approximate absolutely continuous invariant measures of piecewise convex maps with countably many branches, based on a modified piecewise-linear Markov method and a novel piecewise quadratic B-spline method. The proposed schemes are effective for studying piecewise convex chaotic maps. Notably, while it ensures the same convergence order as the piecewise-linear method, the piecewise quadratic B-spline method enhances the smoothness of the approximate density. Numerical examples demonstrate the efficacy of these approaches.
Yang et al. (Thu,) studied this question.