  We present a framework for solving partial different equations on evolving
surfaces. Based on the grid-based particle method (GBPM) [18], the method can
naturally resample the surface even under large deformation from the motion
law. We introduce a new component in the local reconstruction step of the
algorithm and demonstrate numerically that the modification can improve
computational accuracy when a large curvature region is developed during
evolution. The method also incorporates a recently developed constrained
least-squares ghost sample points (CLS-GSP) formulation, which can lead to a
better-conditioned discretized matrix for computing some surface differential
operators. The proposed framework can incorporate many methods and link various
approaches to the same problem. Several numerical experiments are carried out
to show the accuracy and effectiveness of the proposed method.
