# Efficient and accurate exponential SAV algorithms with relaxation for dissipative system

## TL;DR

This paper introduces two linear, unconditionally energy-stable exponential SAV schemes with relaxation for dissipative systems, including Navier-Stokes equations, ensuring positivity and high-order accuracy with demonstrated effectiveness.

## Contribution

The paper develops novel R-ESAV schemes that improve positivity preservation and stability over existing methods, and extends high-order BDF schemes for complex dissipative systems.

## Key findings

- Schemes are linear and unconditionally energy stable.
- Numerical examples confirm high accuracy and effectiveness.
- Approaches preserve positivity without additional assumptions.

## Abstract

In this paper, we construct two kinds of exponential SAV approach with relaxation (R-ESAV) for dissipative system. The constructed schemes are linear and unconditionally energy stable. They can guarantee the positive property of SAV without any assumption compared with R-SAV and R-GSAV approaches, preserve all the advantages of the ESAV approach and satiesfy dissipation law with respect to a modified energy which is directly related to the original free energy. Moreover the second version of R-ESAV approach is easy to construct high-order BDF$k$ schemes. Especially for Navier-Stokes equations, we construct wo kinds of novel schemes based on the R-ESAV method. Finally, ample numerical examples are presented to exhibit that the proposed approaches are accurate and effective.