Talk:Numerical methods for partial differential equations

solver for axisymetric Laplace equation
Is there a numerical solver for axisymetric Laplace equation? Amir 132.66.50.172 10:30, 29 July 2007 (UTC)
 * What do you mean by axisymetric? Do you mean in cylindrical coordinates? What about Galerkin methods using cylindrical Bessel functions? This can be done also analytically, if I remember right.

some criticism about this Wiki page
I really don't want to be unfair. But this wiki site is far from being comprehensive or conclusive. To my experience, the numerical solution of PDEs works via tempo-spatial or ansatz-function discretization, and mixtures of both. And one should try to categorize the different methods accordingly. Finite differences is based on tempo-spatial discretisation at certain discrete time-space points (solution is then e.g. linearly interpolated between these points). Finite volume is also based on tempo-spatial discretization, but volume averaged. That is, the value of a solution is just a mean value inside a small volume. In the other category, there are the Galerkin-methods located. The discretization happens via discrete ansatz-functions, which often are orthogonal function bases such as the Fourier modes (which are here mentioned under "spectral methods". But the method itself is much more general, and you could also use orthogonal polynomial bases or spherical and cylindrical Bessel functions and the like. Important is that you express the solution of a PDE in terms of expansions of these ansatz-functions with some ansatz-coefficients, which have to be determined by inserting this ansatz into the original PDE, thereby disretizing it. The next step to achieve an algebraic system, which can be solved e.g. by means of computer, is to integrate out the tempo-spatial dependencies. This can be done in several ways: projection onto the used base system (also known as collocation methods; the coefficients are determined such that the such way obtained algebraic equation is fulfilled, i.e. the collocation rest is zero), or squaring and integrating the whole discretized equation (the residual) and minimizing w.r.t. the ansatz coefficients (if hitting zero, one has the exact solution). Please, check the Bronstein: Mathematics handbook.

Finite elements are a special case. The ansatz functions are "elements", meaning that they are defined on a small volume (the support of the element). That's why one obtains also sort of a spatial discretization. In my opinion, the next step is to project the whole equation onto the elements, but this is confusingly often referred to as a "variational approach" what sounds more like the minimizing approach mentioned above. So, I count FEM to the collocation methods.

Most of the other methods are in my eyes only modifications. One big exception is the use of Monte-Carlo methods, which represent an extra and separate class of numerical methods. Look, for instance, for quantum monte carlo methods, which are used to solve the Schroedinger or the Dirac equation. Or look for molecular dynamical methods which in principle solve the Boltzmann equation. Here btw. I remember also the Lattice-Boltzmann method, which also tries to solve the Boltzmann equation. Up to now, but, I have no idea how to categorize this method. It appears to be kind of a dimension-reduced finite difference method.

And quite recently, I found even people that consider solving a PDE as a kind of inference process, and try to solve the equation by means of Bayesian inference ;) Well, for uncertainty quantification by using uncertain parameters, I understand this to be useful (see polynomial chaos expansion). But throwing away the some or all of the information the PDE itself represents is in my eyes not professional and especially not in the sense of Bayesian inference, which tries to involve ALL available information!

And insofar the discussion, why a certain method is used for which problem, should be discussed in more details. Because every method can be used in every problem. But not every method is practically appropriate for each problem. In fluid mechanics, you need conservation of the fluid volume (mass). So, you use methods, which can accomplish this - in the simplest manner. If you have a complicated geometry or complex boundary conditions, you better do not use finite differences, although you can! But its difficult e.g. to adapt finite differences to curvilinear coordinates, or to fit Cartesian discrete finite difference points to round boundaries! So, I think this wiki site is really worth to think about once again. — Preceding unsigned comment added by 2001:4CA0:0:F000:79A4:71EA:3DAE:5A1B (talk) 15:14, 12 January 2017 (UTC)

Addendum: Btw. if using delta-functions or piecewise (closed areawise) constant functions in ansatz methods, this restores the finite difference and finite volume method, respectively. Under this aspect, they are all sort of Galerkin-methods. — Preceding unsigned comment added by 2001:4CA0:0:F000:A1D2:32:C928:FC4D (talk) 09:06, 18 January 2017 (UTC)