Difference scheme for a parabolic problem with changing time direction (Q1907491)

The paper is devoted to construction and theoretical study of finite difference schemes for forward-backward parabolic equations of the type \(v(x)u_t = Lu + f\) in \((-1,1) \times (0,T)\), where \(L\) is a symmetric elliptic operator and \(v(x)\) is positive for \(x > 0\) and negative for \(x < 0\). The solution is subject to stable initial-boundary conditions. The stability of the constructed scheme is studied using maximum principle. The convergence for smooth solutions is proven in discrete energy norm. An iterative method for the linear system is proposed and studied.