Gradient Reconstruction on Adaptive Meshes

Gradient approximation from grid data is a critical component of high-order numerical schemes. However, certain problems are highly sensitive to the quality of gradient reconstruction, particularly those involving the development of hydrodynamic instabilities. In such cases, even minor perturbations, including numerical ones, can significantly alter the dynamics of the process. This study identifies scenarios where asymmetry in gradient reconstruction on adaptive meshes leads to the formation of Richtmyer–Meshkov instability in one-dimensional problem without actual perturbations. To address this issue, a modified least squares method is proposed. The new approach ensures symmetry and eliminates numerical instability, thereby improving the robustness of numerical solutions.