High-Order Finite Difference AWENO Lagrangian Scheme for the Multidimensional Compressible Euler Equations in Curvilinear Coordinates

The Lagrangian method has attracted increasing interest in multimaterial fluid flow fields, as it allows the grid to move with the fluid velocity, and it can track the material interfaces automatically and sharply. There are many studies discussing the finite volume (FV) or discontinuous Galerkin (DG) Lagrangian schemes, but little attention has been paid to the finite difference Lagrangian schemes. In this paper, we construct a high-order conservative finite difference alternative weighted essentially nonoscillatory (AWENO) Lagrangian scheme for multidimensional compressible Euler equations in curvilinear coordinates. This is the first high-order finite difference work in pure Lagrangian framework. Specifically, the governing equations are represented in Lagrangian coordinates and are truly hyperbolic. Then we design a fifth-order finite difference AWENO scheme with third-order strong stability preserving Runge–Kutta time-marching. This scheme is conservative, and it can not only achieve arbitrary high-order accuracy in smooth regions, but also avoid numerical oscillations near discontinuities. Moreover, by appropriately discretizing the metric derivatives, the discrete compatibility conditions and free-stream preserving property are proved. More importantly, there is no mass exchange across the moving grid points. Compared to the FV or DG Lagrangian schemes, our finite difference Lagrangian scheme is simpler in coding, it easier to achieve arbitrary high-order accuracy, and, most importantly, it is easily extended and is more efficient in high dimensions. A series of numerical experiments including one-dimensional, two-dimensional, and three-dimensional examples are given to demonstrate the good performance of the scheme in terms of high-order accuracy, nonoscillation, high resolution for discontinuities, and free-stream preservation.