# Difference between strong, weak and differential forms?

We recently covered fundamental laws in our Computational Continuum Mechanics course last term. There are many forms of these laws and I am confused which one to use and when.

For instance in mass continuity, we can write the integral version (I am not gonna write it here) as usual. Then we can use the Jacobian to write it like this $$\int_{V_0} \rho_0 - J \rho \ \ dV$$ which can be said to be the weak form. Then by localisation, we can have $$J(\pmb{x},t) = \frac{\rho_0(t)}{\rho(\pmb{y},t)}$$ where $\pmb{y}(\pmb{x},t)$ is the deformed configuration position and $\pmb{x}$ is the undeformed configuration position (if that make sense) which is now the strong form. So isn't this good enough already? Why don't we use this in our code, because this is also point-wise.

We can also derive the differential form of course which is $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \pmb{u})$$ where $u$ is velocity.

If I just wanted density with time, is it better to use the strong form? Would I just use the differential form if I want to include velocity or something? Also our lecturer mentioned that the Jacobian strong form is in Lagrangian form but differential is in Eulerian form. I know the meaning of both but I am not sure what it means in this context.

If your course started with an indigestible dose of multi-dimensional calculus without much discussion of how the math relates to the physics (and there are certainly textbooks on computational CFD which start that way!) the answers to these questions may become clearer when you start developing the actual numerical methods and seeing how they perform solving "real" problems.

There are two basic ways to devise a numerical solution scheme. One way is to consider the values of quantities defined at specific points in space (e.g. the grid points of a finite difference mesh). The computational method ignores what happens to the quantities "in between" the mesh points, and you hope that as you increase the number of points, the computed values will converge to the solution of the underlying differential equations. (Sometimes, you can prove the solution will converge in this sense, but for most "real world" problems involving nonlinear differential equations you can't prove it).

The other way is to consider the value of properties over regions of space - for example the mass, energy and momentum of a "unit cell" defined by the grid points that surround it, and the flows into and out of the cell across its boundaries. This has the advantage that the solution method can guarantee that it satisfies some fundamental physical laws like conservation of mass and energy. Usually, a solution method based only on values at specific points will not automatically satisfy such conservation laws, though (if it is a useful solution method!) increasing the number of solution points should reduce the errors.

But satisfying the global conservation laws doesn't necessarily mean that the grid-point values are "accurate" - for example there may be unphysical oscillations at adjacent grid points which don't affect the "average" energy, momentum, etc.

Also our lecturer mentioned that the Jacobian strong form is in Lagrangian form but differential is in Eulerian form. I know the meaning of both but I am not sure what it means in this context.

Without knowing exactly what was said (or in the lecture notes) I'm not sure what that means either. You can certainly create four versions of the mathematical formulation, i.e. weak and strong forms of both the Lagrangian or Euleran equations - though some are more intuitively "obvious" than others. In fact some numerical methods use a combination of more than one of the four options.

• Yes I definitely agree this will make a lot more sense when I start coding simple stuff and getting plots or working or some research code. I am guessing from what I understand, your second paragraph refers to a Lagrangian kind of approach and third refers to Eulerian one. I understand that when you mesh, say a shaft and you have material particles, that is more of a Lagrangian solution and if you just had coordinates and have a flow going through it, that is more Eulerian. I am just trying to make the connection between equations and reference frames. – thephysicsguy Dec 29 '16 at 17:44
• I think I just need to read into the reference frames a little bit more and definitely ask my lecturer lots of questions after the break. Also regarding the weak and strong forms, I think it will clear up once I start working on real examples as you mentioned. – thephysicsguy Dec 29 '16 at 17:45
• I'll add a simple analogy that might help. Lagrangian = you sit on a particle of the material inside a box, Eulerian = you watch a particle from outside the box. Eulerian equations have extra terms because of that. – Biswajit Banerjee Dec 29 '16 at 21:00