SplitNewton

Unbounded SPLIT Newton with pseudo-transient continuation and backtracking

Good for ill-conditioned problems where there are two different sets of systems

Particular applications include

• Fast-Slow Reaction-Diffusion systems
• CFD - Pressure-Velocity coupling

What does 'split' mean?

The system is divided into two and for ease of communication, let's refer to first set of variables as "outer" and the second as "inner".

• Holding the outer variables fixed, Newton iteration is performed till convergence using the sub-Jacobian

• One Newton step is performed for the outer variables with inner held fixed (using its sub-Jacobian)

• This process is repeated till convergence criterion is met for the full system (same as in Newton)

How to install and execute?

Just run

pip install splitnewton

There is an examples folder that contains a test function and driver program

How good is this?

Consider the test problem

$\lambda_{a} = 10^{6}$, $\lambda_{b} = 10^{2}$

and the second system $\lambda_{c} = 10^{-1}$ $\lambda_{d} = 10^{-4}$

and using logspace for variation in $\lambda_{i}$

$$ F(u) = \lambda_{a} u^{4}{1} + ... + \lambda{b} u^{4}{\lfloor N/2 \rfloor} + \lambda{c} u^{4}{\lceil N/2 \rceil} + ... + \lambda{d} u^{4}_{N}$$

$$ J(u) = 3 * \begin{bmatrix} \lambda_a & \dots & 0 & 0 & \dots & 0 \newline \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \newline 0 & \dots & \lambda_b & 0 & \dots & 0 \newline 0 & \dots & 0 & \lambda_c & \dots & 0 \newline \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \newline 0 & \dots & 0 & 0 & \dots & \lambda_d \end{bmatrix} u^{2} $$

For N=5000 (with no backtracking and pseudo-transient continuation),

Method	Time	Iterations
Split Newton	9 seconds	32
Newton	not converged > 1 min	NA

Whom to contact?

Please direct your queries to gpavanb1 for any questions.