On a family of finite-difference schemes with discrete transparent boundary conditions for a parabolic equation on the half-axis

TL;DR

This paper develops and analyzes a broad family of finite-difference schemes with transparent boundary conditions for a 1D parabolic equation on the half-axis, ensuring stability and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a new family of finite-difference schemes with discrete transparent boundary conditions for parabolic equations, with rigorous stability analysis and numerical validation.

Findings

Schemes are stable in two norms.

Transparent boundary conditions are rigorously derived.

Numerical experiments confirm the schemes' effectiveness.

Abstract

An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averagings both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.