Understanding the Newton Method for Solving Nonlinear Equations

The Newton method is a powerful tool for solving nonlinear equations, particularly in complex systems where multiple variables interact. In this context, we can define a new system of equations, represented as ( f_i(D) = D_{i-1} + a_{1,i}D_i + a_{k,i}D_i^2 + a_2D_{i+1} - b_i ). Here, the variable ( D ) serves as an approximation to another variable ( C' ), and at the beginning of the iteration, these approximations align with known values of ( C ). Our goal is to adjust ( D ) so that all ( f_i ) values approach zero, indicating that we have arrived at the correct solution.

The approach begins by focusing on the boundary conditions, specifically the first and last equations in the system. For instance, in a Cottrell experiment, the first equation simplifies to ( f_1(D) = a_{1,1}D_1 + a_{k,1}D_1^2 + a_2D_2 - b_1 ), where the boundary value ( D_0 ) is set to zero. Adjustments can also be made for derivative boundary conditions using linear approximations, although multivariate derivatives complicate the situation.

For the last equation, ( f_N(D) ) involves the bulk value ( D_{N+1} ), which is known and is determined by the time step ( T + \delta T ). It is crucial to treat the two bulk values differently to avoid confusion. With the setup established, we can now implement the Newton method, which involves iterative corrections to reach the desired ( D ) values.

The Newton method relies on Taylor expansion to create a linear approximation around the current ( D ) values. This results in a set of equations organized in a vector/matrix format, leading to a linear system that can be expressed as ( J \cdot d = -F(D) ), where ( J ) is the Jacobian matrix. This tridiagonal system is then solvable using efficient algorithms such as the Thomas algorithm.

To ensure convergence, we can either monitor the residual norm or check the correction vector ( d ). The goal is to achieve a norm below a predefined threshold, such as ( 10^{-6} ). While a few iterations—typically 2 to 3—are generally sufficient, the iterative nature of this method often provides more accurate results than linearized versions, making it a valuable technique in computational analysis and simulations.