RBF-FD meshless simulation of compressible flows

The accurate and efficient simulation of compressible flows in complex geometries remains a central challenge in computational fluid dynamics, particularly when strong gradients and shock waves are present. Traditional mesh-based methods often require significant effort in grid generation and adaptation, especially for complex geometries and/or when repeated geometry modifications are required, as in shape optimization problems. In this work, a meshless numerical framework based on the Radial Basis Function–Finite Difference (RBF-FD) method is developed and applied to the solution of two-dimensional compressible flow in a converging–diverging nozzle. The governing compressible Euler equations are discretized in space using local RBF-FD stencils constructed on scattered nodes, eliminating the need for a structured or unstructured mesh while retaining the simplicity, locality and computational efficiency of finite difference schemes.

The proposed approach employs multiquadric RBFs augmented with polynomial terms to achieve high-order spatial accuracy, and proper hyperviscosity damping to achieve numerical stability. Boundary conditions are enforced directly at boundary nodes, demonstrating the flexibility of the meshless formulation in handling arbitrary geometries. The RBF-FD solutions are compared against reference finite volume solutions, showing good agreement in terms of pressure, density and velocity fields.
In addition, shape optimization is performed on a reference test case to demonstrate the flexibility of the proposed method which does not require remeshing. The results demonstrate that the RBF-FD method provides an accurate and robust alternative to conventional mesh-based methods, with reduced preprocessing effort and enhanced geometric flexibility, making it a promising tool for compressible flow simulations coupled with shape optimization.