Finite Difference Approximations for a Class of Non-local Parabolic Equations


In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero as $t \rr \infty$ exponentially. The numerical results of some examples are presented , which support our theoretical justifications.