Compact difference scheme for parabolic and Schr\"odinger-type equations with variable coefficients

TL;DR

This paper introduces a new compact finite-difference scheme for parabolic and Schrödinger-type PDEs with variable coefficients, achieving higher accuracy and smaller errors than traditional implicit methods, applicable to boundary value problems.

Contribution

The paper presents a novel compact scheme with higher order accuracy for variable coefficient PDEs, improving upon classic implicit methods in error reduction.

Findings

Higher order accuracy demonstrated

Reduced error compared to classic schemes

Applicable to Dirichlet and Neumann boundary problems

Abstract

We develop a new compact scheme for second-order PDE (parabolic and Schr\"odinger 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.