Novel linear, decoupled, and energy dissipative schemes for the Navier-Stokes-Darcy model and extension to related two-phase flow

TL;DR

This paper introduces energy-dissipative, linear, and decoupled numerical schemes for the Navier-Stokes-Darcy model and related two-phase flows, ensuring stability and energy dissipation with efficient linear solves.

Contribution

The paper develops novel energy-dissipative schemes with a relaxation technique that guarantees unconditional energy stability and requires only linear solves with constant coefficients.

Findings

Schemes are unconditionally energy dissipative.

Numerical experiments confirm accuracy and stability.

Error analysis supports theoretical results.

Abstract

We construct efficient original-energy-dissipative schemes for the Navier-Stokes-Darcy model and related two-phase flows using a prediction-correction framework. A new relaxation technique is incorporated in the correction step to guarantee dissipation of the original energy, thereby ensuring unconditional boundedness of the numerical solutions for velocity and hydraulic head in the $l^{\infty}(L^2)$ and $l^2(H^1)$ norms. At each time step, the schemes require solving only a sequence of linear equations with constant coefficients. We rigorously prove that the schemes dissipate the original energy and, as an example, carry out a rigorous error analysis of the first-order scheme for the Navier-Stokes-Darcy model. Finally, a series of benchmark numerical experiments are conducted to demonstrate the accuracy, stability, and effectiveness of the proposed methods.