A p-adaptive high-order mesh-free framework for fluid simulations in complex geometries

TL;DR

This work introduces a $p$-adaptive, high-order mesh-free framework based on Local Anisotropic Basis Functions (LABFM) to solve fluid problems in complex geometries. A Laplacian-based refinement indicator drives local changes in polynomial order $p$, enabling high accuracy where needed while reducing overall cost. The method demonstrates strong convergence behavior and stability across benchmarks (convergence tests, Burgers’ equation, Kelvin–Helmholtz instability) and applies to compressible reacting flows, achieving up to substantial speed-ups compared with fixed-high-order schemes while preserving accuracy. The results underscore the approach’s potential for efficient, accurate simulations in porous media and other complex geometries, with future work focusing on parallel load balancing and dynamic domain partitioning.

Abstract

This paper presents a novel p-adaptive, high-order mesh-free framework for the accurate and efficient simulation of fluid flows in complex geometries. High-order differential operators are constructed locally for arbitrary node distributions using linear combinations of anisotropic basis functions, formulated to ensure the exact reproduction of polynomial fields up to the specified p order. A dynamic p-refinement strategy is developed to refine (increase) or de-refine (decrease) the polynomial order used to approximate derivatives at each node. A new refinement indicator for mesh-free methods is proposed, based on local error estimates of the Laplacian operator, and is incorporated into the solution procedure at minimal added computational cost. Based on this error indicator, a refinement criterion is established to locally adjust the polynomial order p for the solution. The proposed adaptive mesh-free scheme is then applied to a range of canonical PDEs, and its potential is demonstrated in two-dimensional simulations of a compressible reacting flow in porous media. For the test cases studied, the proposed method exhibits potential to save up to 50% of computational costs while maintaining the specified level of accuracy. The results confirm that the developed p-adaptive high-order mesh-free method effectively captures highly non-linear regions where high-order approximation is necessary and reduces computational costs compared to the non-adaptive method, preserving high accuracy and solution stability.

Figures (15)

• Figure 1: An illustration of the computational stencil.
• Figure 2: Convergence behaviour of the differential operator approximation using the adaptive and non-adaptive methods, showing Laplacian approximation (left) and gradient approximation (right).
• Figure 3: Convergence test results using the adaptive method, showing error versus cost with comparison against the non-adaptive method (left) and local polynomial order $p$ (right).
• Figure 4: Numerical solution of the travelling wave case using the adaptive method [$Re = 500$, $s=1/160$, $t=0.5$], showing the value of $u$ (left) and local polynomial order $p$ (right).
• Figure 5: Numerical solution of the travelling wave case using the adaptive method [$Re = 200$, $s=1/160$, $t=0.5$], showing the value of $u$ (left) and local polynomial order $p$ (right).