# Energy Stability and Convergence of SAV Block-centered Finite Difference Method for Gradient Flows

## TL;DR

This paper develops and analyzes a second-order accurate block-centered finite difference method for gradient flows, demonstrating its stability, convergence, and efficiency through theoretical proofs and numerical experiments.

## Contribution

It introduces a novel SAV/CN-BCFD scheme with rigorous stability and convergence analysis for gradient flows, enhancing accuracy and efficiency.

## Key findings

- Second-order accuracy in time and space confirmed
- Scheme is stable and convergent under various norms
- Numerical tests verify robustness and efficiency

## Abstract

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.