sufficiently small time step. Crank-Nicolson for nonlinear problems Because the Crank-Nicolson method is implicit, it is generally impossible to solve exactly. Instead, an iterative technique should be used to converge to the . One option is to use Newton's method to converge on the prediction, but this requires the computation of the Jacobian. For a high-dimensional system like those in computational fluid dynamics or numerical relativity, it may be infeasible to compute this Jacobian. A Jacobian-free alternative is fixed-point iteration. If is the velocity of the system, then the Crank-Nicolson prediction will be a fixed point of the map . If the map iteration does not converge, the parameterized map , with , may be better behaved. In expanded form, the update formula is , where is the current guess and is the previous time-step. Even for high dimensional systems, iteration of this map can converge surprisingly quickly. Application in financial mathematics Because a number of other phenomena can be modeled with the heat equation (often called the diffusion equation in financial mathematics), the Crank–Nicolson method has been applied to those areas as well. Particularly, the Black–Scholes option pricing model's differential equation can be transformed into the heat equation, and thus numerical solutions for option pricing can be obtained with the Crank–Nicolson method. The importance of this for finance is that option pricing problems, when extended beyond the standard assumptions (e.g. incorporating changing dividends), cannot be solved in closed form, but can be solved using this method. Note however, that for non-smooth final conditions (which happen for most financial instruments), the Crank–Nicolson method is not satisfactory as numerical oscillations are not damped. For vanilla options, this results in oscillation in the gamma value around the strike price. Therefore, special damping initialization steps are necessary (e.g., fully implicit finite difference method). See also Financial mathematics Trapezoidal rule References External links Numerical PDE Techniques for Scientists and Engineers, open access Lectures and Codes for Numerical PDEs An example of how to apply and implement the Crank-Nicolson method for the Advection equation Mathematical finance Numerical differential equations Finite differences