  The evolution of a closed two-dimensional surface driven by both mean
curvature flow and a reaction--diffusion process on the surface is formulated
into a system, which couples the velocity law not only to the surface partial
differential equation but also to the evolution equations for the geometric
quantities, namely the normal vector and the mean curvature on the surface. Two
algorithms are considered for the obtained system. Both methods combine surface
finite elements as a space discretisation and linearly implicit backward
difference formulae for time integration. Based on our recent results for mean
curvature flow, one of the algorithms directly admits a convergence proof for
its full discretisation in the case of finite elements of polynomial degree at
least two and backward difference formulae of orders two to five. Numerical
examples are provided to support and complement the theoretical convergence
results (demonstrating the convergence properties of the method without error
estimate), and demonstrate the effectiveness of the methods in simulating a
three-dimensional tumour growth model.
