Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems
Any set of finite difference, finite volume, or finite element equations is just a set of coupled nonlinear equations, that can be abbreviated as:
We perform a Taylor series expansion about the latest approximation to the solution, and assume that the difference between the true solution and the last approximation is very small. This means that the jth component of the vector function can be expressed as:
Back to vector notation, we get:
The biggest piece of work is coming up with the elements of the Jacobian Matrix J. There are several possible approaches:
When you are creating a solution procedure from scratch, you should implement two of the above to cross check your work. Programming errors in Jacobian elements are very difficult to detect, since they may only slow, not prevent, convergence of the Newton iteration. In practice some combination of the first and fourth options will be most efficient.