Compact difference scheme for parabolic and Schrödinger-type equations with variable coefficients

We develop a new compact scheme for the second-order PDE (parabolic and Schrödinger type) with a variable time-independent coefficient. It has a higher order and smaller error than classic implicit scheme. The Dirichlet and Neumann boundary problems are considered. The relative finite-difference operator is almost self-adjoint. •High order compact scheme for 1D parabolic and Schrodinger-type equations with variable coefficients is constructed.•The approximation order was confirmed by various numerical experiments.•The Richardson extrapolation method improves the approximation order up to 6th.•High-order Neumann boundary conditions approximation are constructed.