Filtration simplification consists of simplifying a given filtration while simultaneously controlling the perturbation in the associated persistence diagrams. In this paper, we propose a filtration simplification algorithm for orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Given a lower-star filtration of the mesh, faces are added into contiguous clusters according to a "height" function and a parameter. Faces in the same cluster are merged into a single face, resulting in a lower resolution mesh and a simpler filtration. We prove that the parameter bounds the perturbation in the original persistence diagrams, and we provide experiments demonstrating the computational advantages of the simplification process.