Abstract: Fourier Methods and Second Order Weak Taylor Scheme for Backward Stochastic Differential Equations in Computational Finance In this presentation we will explain how we can solve linear, semi-linear as well as nonlinear partial differential equations by the concept of backward stochastic differential equations and Fourier cosine expansions. We will discuss the highly efficient pricing of financial options in the Fourier context. We extend the Fourier method to solve so-called decoupled forward-backward stochastic differential equations (FBSDEs) with second-order accuracy. The FSDE is approximated by different Taylor schemes, such as the Euler, Milstein, and Order-2.0-weak-Taylor schemes. The conditional expectations appearing are approximated by using the characteristic function for these schemes and Fourier-cosine series expansions.