A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems

In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of the nonlinear partial differential equation u(t)+ div(qf(u)) - Delta phi(u) = 0 in a 1D, 2D or 3D domain. The function phi is supposed to be strictly increasing, but some values s such that phi'(s) = 0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic two point boundary value problem (qf(v) + (phi(v))')' = 0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes.