A stable and convergent O-method for general moving hypersurfaces

We present a finite volume method for transport, diffusion and reaction problems on evolving hyper-surfaces. The surface motion is assumed to be given. The numerical scheme is built on a sequence of general polygonal hyper-surfaces approximating the continuous hyper-surface and whose no des propagate with the actual velocity field. Our approach consists of using a dual strategy to approximate the solution of our partial differential equation (PDE). First we use a suitable interpretation of the flux continuity condition on a dual mesh and a proper minimization strategy to construct an adequate operator dependent piece-wise linear interpolant around nodal points. The interpolant builds from discrete points around nodes a piece-wise linear function whose the piece-wise constant gradient satisfies an appropriate flux continuity condition on the sub-cells induced by the space discretization on the dual mesh. Next we integrate the PDE on cells using the Gauss formula and the gradients of the above introduced functions. The diffusion operators as well as the reaction operators are approximated implicitly while the advection operators are approximated explicitly using the upwind procedure and an adapted min-mode strategy. The obtained semi-implicit scheme is a cell center finite volume which is second order convergent in special L^2 norm and first order in special H^1 norm. Finally, we provide several examples to support the theory