How to Solve nonlinear system by newton method?

How do I use the Newton-Raphson method for an implicit /explicit finite difference method with nonlinear boundary conditions?

• I've built the Crank-Nicolson 2D matrix for the heat equation, and I have discretized my nonlinear BC. I have to combine both and equal them to zero, find my jacobian and do my iterations. However I am confused by the implicit (t+1) terms in my equation.

• Answer:

  Without knowing the details of your problem it is difficult to be very explicit but in general you can use Newton's method to numerically integrate nonlinear differential equations. Once you discretize your problem in the spatial variable(s) you will presumably end up with a (high-dimensional) system of ode's so the following should be relevant. If you want to solve [math]\frac{du}{dt} = f(u(t)) [/math] where [math]u \in R^N[/math] implicitly via finite differences then you might  approximate the derivative by [math]\frac{du}{dt} \approx \frac{1}{dt} (u(t+1)-u(t)) [/math] and evaluate the right hand side at t=t+1. Or perhaps you might want to evaluate the RHS at both t and t+1. In the former case you'll have: [math] \frac{1}{dt}(u(t+1)-u(t)) = f(u(t+1))[/math] and something similar in the latter. In either case the Newton Raphson method can be used to solve for u(t+1). In the former you'd have to solve the nonlinear equation: [math] u(t+1)-u(t) -dt f(u(t+1) = 0 [/math] for the variable u(t+1). I'm not sure of the precise nature of your difficulty is but you can ignore the time index. It might clarify matters to write [math]u(t+1)=x[/math] where "x" is the unknown you are looking for and also you might as well write [math]u(t)=k[/math] where k is "known". Then the equations you need to solve are: [math]x-k + dt f(x)=0[/math] which is a bread and butter problem for Newton's method which I assume you are already familiar with.