Implementation of psuedo-arclength continuation

[TokenMonkey]

Hi there,

 

I have a quick question regarding pseudo-arclength continuation. As some background, I am a chemical engineer, not a mathematician, applied or otherwise, so my knowledge of numerical methods is limited to the "standard" stuff.

 

I've been reading up as extensively as I can on pseudo-arclength continuation, but unfortunately, it's all from second-hand sources; I don't have access to Keller's original paper. Here's what I understand at this point:

 

We want to solve a problem [math]F(x,\lambda)=0[/math]. We assume that the solution is known at [math]x^0[/math] and [math]\lambda^0[/math]. To avoid the singularity of the Jacobian, and therefore the breakdown of Newton's method, at turning points, [math]x[/math] and [math]\lambda[/math] both become parameterised by arclength ([math]s[/math]), and we end up with an augmented system of equations to solve:

 

[math]F(x,\lambda)=0[/math]

[math]\left(u-u^{0}\right)\mathrm{d}u^{0}/\mathrm{d}s+\left(\lambda-\lambda^{0}\right)\mathrm{d}\lambda^{0}/\mathrm{d}s-\Delta S=0[/math]

 

While this seems simple enough, how does one obtain the derivatives w.r.t [math]s[/math]? Not a single text seems to mention this. Ideas that spring to mind are forward differences using, say, cubic splines; however, that seems horrendously inefficient to me. There must be a better way!

 

Thanks,

TM

[TokenMonkey]

I found the answer: a procedure is described in Kubicek's Algorithm 502 in ACM Trans. Math. Software. dl.acm.org/citation.cfm?id=355675