Eulerian Algorithms for Computing the Forward Finite Time Lyapunov Exponent Without Finite Difference Upon the Flow Map

In this paper,effective Eulerian algorithms are introduced for the computation of the forwardfinite time Lyapunov exponent(FTLE)of smoothflowfields.The advantages of the proposed algorithms mainly manifest in two aspects.First,previous Eulerian approaches for computing the FTLEfield are improved so that the Jacobian of theflow map can be obtained by directly solving a corresponding system of partial differential equations,rather than by implementing certainfinite difference upon theflow map,which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges.Second,in the proposed algorithms,all computations are done on thefly,that is,all required partial differential equations are solved forward in time,which is practically more natural.The new algorithms still maintain the optimal computational complexity as well as the second order accuracy.Numerical examples demonstrate the effectiveness of the proposed algorithms.