I am trying to solve a time-dependent diffusion equation with finite-difference discretizations using the Newton–Raphson method. However, I encountered some problems in convergence--the solution result changes when changing time step size. Here I want to find some coding examples to refer to. Thank you.
In general, the name diffusion equation (time dependent or not) identifies a linear PDE, whose finite difference discretization leads to a linear algebraic problem that does not require a nonlinear solver like Newton Raphson. This means that your problem is a nonlinear one, for example with nonlinear viscosity (porous media equation and such), and that an implicit method is being used for the time discretization. The correct way to approach these problems is to write the space and time discretization (there is no mention of what time discretization method is used) which leads to a (potentially large) nonlinear system to be solved
at each time step. Then at each time step a nonlinear solver is used to compute the solution of thie algebraic problem. Any of these steps might have been coded incorrectly leading to the problem you mention. One simple way to check what is going on is to compare with the results of a simple explicit method (say forward Euler) employed with a very small time step, such a method would not require NR iterations and therefore any incorrectness in the NR solver would not affect it.
The video below explains Application of Finite Differences in Newton-Raphson's Method with MATLAB code.
https://www.youtube.com/watch?v=eNPfsxZxJLM
Please study the following paper:
Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations.
Dear all: Thank you very much for your answers.
The PDE may be like:
K∂C/∂t−∇⋅(λ∇C)=q
> what time discretization method is used?
In the iteration loop, I just divide the total time into N steps each with Δt size. With the initial value C_n, the obtained C_n+1 by Newton iteration will be the initial value of the next time step.
For each time step (Δt), I calculate residual R:
R=K(∂C)−∇⋅(λ∇C)Δt−qΔt
and
Its Jacobian R' = ∂R/∂C
Then do Newton iteration and solve equations -R/R' = ΔC (= C_n+1 - C_n) to get ΔC in each iteration.
In the iteration, the linear searches here are conducted to get C_n+1 as C_n + alpha * ΔC to increase the effectiveness. The alpha decreases with a factor k (e.g., = 0.5).
Perhaps the problems are:
1. the Jacobian is not accurate.
alpha can iteratively change to a very small value. Does this mean the Jacobian is not correct?
2. the setting of linear search is not right
I had found the solution changes with the decremental factor k of alpha. (e.g., 0.5 - 0.9 ).
> compare with the results of a simple explicit method (say forward Euler) employed with a very small time step
I will check the suggestion.
Thank you for all your answers. I am still waiting for a code example with similar usage (for solving time-dependent nonlinear PDE using FDM and Newton-Raphson method ) if there are some.
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