A high-order pseudo arc-length method with positivity-preserving flux limiter for compressible multi-medium flows

This paper proposes a high-order pseudo arc-length method (PALM) for multi-medium flows with strong robustness, stability, and positivity preservation for solving one- and two-dimensional compressible Euler equations. The main idea of the proposed scheme is to add an additional arc-length constraint equation to the original control equation and map it to the uniform orthogonal arc-length space. We discretize the space with high accuracy by using the high-order weighted essentially non-oscillatory (WENO) interpolation reconstruction, which overcomes the difficulty of constructing the high-order format due to the physical space deformation caused by the grid movement. The application scope of the positivity-preserving algorithm is further expanded, and the positivity-preserving limiter of the high-order WENO pseudo arc-length adaptive method in the coordinate system of the arc-length calculation is constructed and proved, solving the problem of the negative density and pressure caused by the interaction between a strong shock wave and a strong sparse wave. For grid motion after interpolation of the level set function, a third-order non-conservative interpolation scheme is offered to ensure the interface capture accuracy. Finally, combined with level set interface tracking and the real ghost fluid method interface-processing techniques, the algorithm is applied to calculate multi-medium flows. Numerical examples show that the PALM almost eliminates the mass loss near the interface and maintains the high-accuracy and high-resolution characteristics of the algorithm when dealing with extreme problems such as low density, low pressure, strong shock waves, or strong sparse waves.