Usability of upwind finite difference schemes

Question

NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here.

Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the advection equation:

$$ \dfrac{\partial \psi}{\partial t} + \dfrac{\partial }{\partial x}(u\psi) = 0 \qquad (1) $$

and this makes sense, because the scalar $\psi$ is being transported by the wind speed $u$.

My question is, what is the extent to which these schemes could be used? Could they be used to solve all partial differential equations? Does it make sense? For example you can find in the literature attempts to solve the diffusion equation with an upwind method. And if we were to use them to solve an equation, for example, of the form,

$$ \dfrac{\partial \psi}{\partial t} + a(x,t)\dfrac{\partial \psi}{\partial x} = 0\qquad (2) $$

should we use a virtual wind speed, i.e.

$$ u=\dfrac{1}{\psi} \int a \dfrac{\partial \psi}{\partial x}dx $$

to determine the stencils?

Answer 1

There are multiple questions here, but let's start with the basics. You have written two hyperbolic PDEs; (1) is the continuity equation, which is conservative and (2) is the color equation, which is not conservative.

What are the characteristic speeds for these equations?

For (1), you have stated correctly that the characteristic speed is $u(x,t)$. For (2), you have made incorrect suggestions in both your answer and your comment. The characteristic speed for (2) is $a(x,t)$. Therefore, the "virtual wind speed" you introduce is not useful.

Can an upwind scheme be used to solve the color equation (2)?

Yes. Furthermore, and contrary to what @Jan has suggested, (2) is well-posed even for typical discontinuous functions $a$. An upwind discretization for the problem with discontinuous $a$ will be convergent, under the usual restriction on the CFL number.

It is not directly related to your question, but since @Jan brought it up I will add that integrability of $a$ is not the relevant condition. The simplest situation in which a classical solution fails to exist is when there is a point where $a$ changes sign from positive to negative. A delta function must form at that point. Exactly the same thing happens for (1) in the case that $u$ changes sign, so this is nothing special about the color equation.

Could upwind methods be used to solve all PDEs?

It depends on what exactly you mean by "upwind methods". For the diffusion equation, there is no directional bias in the transmission of information. An explicit, purely one-sided stencil cannot be convergent for diffusion (see the CFL paper). You could use a stencil that is biased toward one direction (but includes some points in both directions) for any PDE.

To learn more:

For a lengthy and excellent discussion of differences between (1) and (2) and their numerical discretization by upwind methods, see Chapter 9 of LeVeque's book on finite volume methods.

Answer 2

Let me assume that upwind refers to the upwind discretization of a advection term in Finite Volumes or Finite Differences Schemes, see e.g. the book Numerical Treatment of Partial Differential Equations by Grossmann, Roos, and Stynes.

Can upwind be used to solve all equations?

Certainly not. If in $(1)$ the wind $a$ is not integrable, then there won't be a classical solution that can be approximated by a numerical scheme.

So lets assume that $a$ is continuous, that the problem is posed on a regular domain, with reasonable data and the discretization scheme is reasonable. Then, upwind applied to $(1)$ will give useful results because it is stable and consistent.

Should we use a virtual windspeed?

Any reformulation of your problem that results in a form like $(1)$ with a smooth $a$ can thus be well tackled by an upwind discretization. You can go more general and also have a source term and a linear coercive operator like diffusion.

Can we solve equations of type $(2)$?

Probably yes. Doing the reformulation $a\partial_x\psi = \partial_x (a\psi) - \partial_x(a) \psi $ gives you the form $(1)$ plus an additional operator that hopefully does not destroy the stability of the scheme.

Anyways, the source of an equation like $(2)$ is most probably a differential form of a conservation law (like the momentum equation in the Navier-Stokes equation) that takes the form $(2)$ because $a$ is constant in space (in 1D) or divergence free (2D or 3D). In this case, in the proposed reformulation, the additional term is zero and you get Equation $(1)$.