An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

Area of Science:

• Computational fluid dynamics
• Kinetic theory
• Numerical analysis

Background:

• Existing numerical methods for compressible fluid dynamics often face limitations in explicit time-stepping stability and computational cost.
• High-order accurate methods are crucial for resolving complex flow phenomena.

Purpose of the Study:

• To present a class of arbitrarily high-order, fully explicit kinetic numerical methods for compressible fluid dynamics.
• To extend these methods to multi-dimensional systems and ensure they are asymptotic preserving.

Main Methods:

• Development of fully explicit kinetic numerical methods in both time and space, incorporating relaxation schemes.
• Utilizing a small parameter (Knudsen number) for asymptotic preservation.
• Extension of previous one-dimensional work to multi-dimensional systems on Cartesian meshes.

Main Results:

• The proposed methods allow for CFL numbers greater than or equal to unity in multi-dimensional cases.
• The methods are proven to be asymptotic preserving with respect to the Knudsen number.
• Computational costs are comparable to standard explicit schemes.
• Robustness and high-order accuracy were demonstrated on 2D scalar and Euler equation problems.

Conclusions:

• The new kinetic numerical methods offer a robust and accurate approach for compressible fluid dynamics.
• These methods provide significant advantages in terms of stability and computational efficiency for multi-dimensional problems.