Modification of Multiresolution Analysis for Efficient Implicit Temporal Integration

Abstract

A modified method has been suggested to improve the computational efficiency of a multiresolution analysis for implicit temporal integrations. The fundamental idea behind multiresolution analysis is that high cost computations such as flux evaluation are performed only at the cells where the salient flow features exist, thereby reducing the computational cost. The multiresolution analysis, however, suffers from numerical inefficiency in implicit time integration because the efficiency of the multiresolution analysis method arises mainly from a reduction in the expensive flux evaluation for spatial discretization. This paper demonstrates that the implicit temporal integration of multiresolution analysis can be substantially improved by reducing the implicit operator matrix to the size of the additional residual dataset. To suppress the numerical error from the reduced implicit operation, the operator matrix is derived based on the additional residual dataset constructed by the decomposition and thresholding of the residuals, and the thresholding criterion is modified to retain the numerical accuracy at a high Courant–Friedrichs–Lewy number. The accuracy and efficiency of the present modified multiresolution analysis are verified through various application flow problems, such as a stationary vortex, a steady airfoil, an airfoil vortex interaction, and a transonic wing, ranging from two-dimensional steady/unsteady to three-dimensional steady applications. The results show that the computing speed of the modified multiresolution analysis with lower–upper symmetric Gauss–Seidel is about twice as fast as that of a baseline computational fluid dynamics solver on steady and unsteady simulations, while maintaining the accuracy of the solver.