Energy Stability Property of the CPR Method Based on Subcell Second-Order CNNW Limiting in Solving Conservation Laws

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Applied mathematics

Background:

• The correction procedure via reconstruction (CPR) method is crucial for solving fluid dynamics problems.
• Ensuring energy stability is vital for the reliability of numerical methods in fluid dynamics.
• Subcell limiting techniques are employed to handle discontinuities and maintain solution accuracy.

Purpose of the Study:

• To investigate the energy stability properties of the CPR method with staggered flux points.
• To analyze the impact of second-order subcell limiting on the CPR method's stability.
• To theoretically and numerically validate the stability of the proposed approach.

Main Methods:

• Utilizing a second-order subcell compact nonuniform nonlinear weighted (CNNW2) scheme for limiting in troubled cells.
• Employing a shock indicator to identify cells with potential discontinuities.
• Calculating smooth cells using the CPR method with staggered flux points.
• Proving the linear energy stability of the linear CNNW2 scheme theoretically.

Main Results:

• The CPR method with staggered flux points and subcell linear CNNW2 limiting is shown to be energy-stable.
• The CPR method with staggered flux points and subcell nonlinear CNNW2 limiting is demonstrated to be nonlinearly stable.
• Numerical experiments confirm the theoretical findings on energy stability.

Conclusions:

• The CPR method with staggered flux points, particularly when combined with second-order subcell limiting, offers a stable and reliable approach for computational fluid dynamics.
• Both linear and nonlinear CNNW2 limiting schemes contribute to the overall stability of the CPR method.
• The findings provide a theoretical and numerical foundation for the application of this method in complex flow simulations.