# Why are asymptotic expansions not common for solving partial and algebraic differential equations?

There are many different numerical methods for solving partial and algebraic differential equations. The common commercial and open source packages (e.g. Elmer and OpenFOAM) use discretization/meshing based methods such as finite difference or finite volume.

When I was in undergrad we used to use power series for this matter all the time. They have several advantages over other methods:

1. The result is easily differentiable/integratable
2. They seem to be computationally less expensive (the solution can be found analytically/symbolically)
3. Complicated boundary and initial conditions can be applied easily (like moving objects)
4. No meshing is required
5. solutions are reusable, only the boundary/initial conditions need to be applied

However, It is a surprise that there are not many proprietary or open source software implementing this algorithms. At least I haven't seen much. The only thing I have seen so far is the Mathematica's AsymptoticDSolveValue function, which is only for ODEs.

Now my question is why asymptotic methods are not as common for solving nonlinear differential equations. Are there any scientific studies showing they are less efficient than common methods? Maybe there are some products and I am not aware of them. If that's the case I would appreciate if you could let me know.

P.S.1. There are other series too:

• Chebyshev polynomials
• Padé rational functions approximant
• Lagrange interpolation
• Laurent series
• Fourier series

to mention some.

P.S.2. Other relevant topics: Frobenius method, WKB approximation, Spectral method, Finite element method

• I think that is a good question. I think it may be too complicated. – mick Jul 30 '18 at 9:36
• @mick you mean the answer to my question or asymptotic methods in general? – Foad Jul 30 '18 at 9:37
• Well If you keep one variable fixed , you end up with a simpler equation. But that Also implies not keeping it fixed must give a “ meta solution “. – mick Jul 30 '18 at 9:39
• @Foad I think you're confused. The phrase 'they are not as common as FEM and CFD methods' doesn't make sense at all. CFD stands for computational fluid dynamics, any numerical method that yields an accurate solution would be considered a CFD method. Finite differencing and expansion in terms of orthogonal basis functions are two different numerical approaches to computing a solution. FEM is a way of dealing with complex geometries by splitting your domain into subdomains and then solving the problem in each subdomain any way you can, including using basis functions or finite differencing. – Mattos Jul 31 '18 at 1:17
• @Foad E&M= Electricity and Magnetism. I'm thinking of Maxwell's partial differential equations. To find a high frequency solution in complex geometries, you can can try an asymptotic solution such as the program 'Xpatch' attempts. Google 'Xpatch' to find more information. – skbmoore Jul 31 '18 at 16:44

## 1 Answer

This question is very broad, and its answers subjective (how do you quantify usefulness?), but I can give some comments.

1. Asymptotic convergence is not broadly guaranteed a priori.

When solving an elliptic PDE using the finite element method, there are well-established results such as the Lax-Milgram theorem and Cea's lemma that furnish straightforward "guarantees" of solving efficiency for such a problem. This is nice because the validity of such theorems depends only on the nature of the equation, not of the solution.

Compare that with, say, trying to obtain a Taylor series representation of a PDE solution over some expansion point. Who says the function is analytic over the whole domain? And estimating the radius of convergence with something like the Cauchy-Hadamard theorem requires informed guesses as to the behavior of the coefficients of such a series, which are themselves obtained from solution attempts.

You often get the same problem in perturbation theory; who says that a matched boundary layer approach or WKB approach will yield a convergent series? Often perturbation expansions of this type rely on constructing a sequence of linear operator equations out of a single nonlinear operator equation, and hoping that your sequence of equations has a solution up to some order (as determined by the Fredholm alternative, or other such theorems). This involves getting rid of terms that are incompatible with these operators as they appear in the solution sequences, which is a fairly unpredictable process. As I understand it, there are very few theorems that guarantee convergence of such series without having some information about the solution.

2. Asymptotic methods are fundamentally local.

Series approaches involving orthogonal function sets (Fourier series, Chebyshev polynomials) use the geometry of Lebesgue spaces to make guarantees about the "goodness" of the approximation: for example, that the $$L^2$$ norm of the residual always decreases as you increase the terms in the series, that the series is the best representation of the function in a specific subspace, etc. This is due to the fact that such series are approximating the function globally, whereas asymptotic methods approach solutions locally. As a result, there are few such broad, general guarantees in asymptotic series, which can show weird divergent behaviors in specific regions of the domain as terms in the approximation increase even if it behaves perfectly well near the point of local expansion.

The strange behaviors of asymptotic series are, for me, epitomized by Carrier's rule: "Divergent series converge faster than convergent series because they don't have to converge".

3. It is difficult to predict the computational efficiency and solvability of algorithmic asymptotic methods.

Because asymptotic methods are local and have few a priori guarantees of behaving "well", it is very difficult to predict whether an algorithm employing such methods terminates or, if it does, how quickly it would do so. Consider, in perhaps a naive way, an algorithm attempting an asymptotic expansion that must match some boundary condition in a region far from the expansion point, and the algorithm terminates when the approximation is within some error margin. It may well be the case that no such expansion of the type you want exists (à la Stokes or Whitehead paradox), and as a result your algorithm keeps fruitlessly searching for the right function ad infinitum.

• Thanks for the reply. There are many terminologies that I am not familiar with. So I need more study to be able to comment, but the moment: "how do you quantify usefulness?" I think computational efficiency. What is the convergence rate versus computational resources (i.e CPU time) – Foad Jul 30 '18 at 15:00
• I'll add some information to the answer regarding computational complexity shortly. – aghostinthefigures Jul 30 '18 at 16:59
• I posted a follow up question here I would appreciate if you could take a look. – Foad Aug 2 '18 at 12:01