Fast Construction of Forward Flow Maps using Eulerian Based Interpolation Schemes We propose a modification to a recently developed Eulerian interpolation scheme for constructing the flow map for autonomous, periodic and aperiodic dynamical systems. We show that the proposed methods significantly improve the computational efficiency when a large number of flow maps are needed, yet retain secondorder accuracy as in the original approach. The idea is to precompute and to store a carefully selected set of intermediate flow maps. When the initial and the final time of a required flow map are known, our proposed methods can simply load these preprocessed shorttime flow maps for flow map construction. Numerical examples are included to validate our theoretical prediction, and demonstrate the effectiveness of these proposed Eulerian interpolation schemes as a simple numerical tool for other applications such as the finitetime Lyapunov exponent in determining the Lagrangian coherent structure of dynamical systems and the geometrical optics problems.

Analysis of Second Order Time Filtered Backward Euler Method for MHD Equations The present work is devoted to introduce the backward Euler based modular time filter method for MHD flow. The proposed method improves the accuracy of the solution without a significant change in the complexity of the system. Since time filters for fluid variables are added as separate post processing steps, the method can be easily incorporated into an existing backward Euler code. We show that the time filtered backward Euler method delivers better correct energy and crosshelicity balance in comparison with the backward Euler method. Longtime stability, stability and second order convergence of the method are also proven. The influences of introduced time filter method on several numerical experiments are given, which both verify the theoretical findings and illustrate its usefulness on practical problems.