Conservation Vs Non-conservation Forms of conservation Equations

Question

I understand mathematically how one can obtain the conservation equations in both the conservative $${\partial\rho\over\partial t}+\nabla\cdot(\rho \textbf{u})=0$$

$${\partial\rho{\textbf{u}}\over\partial t}+\nabla\cdot(\rho \textbf{u})\textbf{u}+\nabla p=0$$

$${\partial E\over\partial t}+\nabla\cdot(\textbf{u}(E+p))=0$$

and non-conservative forms. However, I am still confused, why do we call them conservative and non-conservative forms? can any one explain from a physical and mathematical point of view?

Many off-site threads deal with this question (here and here), but none of them provides a good enough answer for me!

If any one can provide some hints, I will be very grateful.

Answer 1

What does it mean?

The reason they are conservative or non-conservative has to do with the splitting of the derivatives. Consider the conservative derivative:

$$ \frac{\partial \rho u}{\partial x} $$

When we discretize this, using a simple numerical derivative just to highlight the point, we get:

$$ \frac{\partial \rho u}{\partial x} \approx \frac{(\rho u)_i - (\rho u)_{i-1}}{\Delta x} $$

Now, in non-conservative form, the derivative is split apart as:

$$ \rho \frac{\partial u}{\partial x} + u \frac{\partial \rho}{\partial x} $$

Using the same numerical approximation, we get:

$$ \rho \frac{\partial u}{\partial x} + u \frac{\partial \rho}{\partial x} = \rho_i \frac{u_i - u_{i-1}}{\Delta x} + u_i \frac{\rho_i - \rho_{i-1}}{\Delta x} $$

So now you can see (hopefully!) there are some issues. While the original derivative is mathematically the same, the discrete form is not the same. Of particular difficulty is the choice of the terms multiplying the derivative. Here I took it at point $i$, but is $i-1$ better? Maybe at $i-1/2$? But then how do we get it at $i-1/2$? Simple average? Higher order reconstructions?

Those arguments just show that the non-conservative form is different, and in some ways harder, but why is it called non-conservative? For a derivative to be conservative, it must form a telescoping series. In other words, when you add up the terms over a grid, only the boundary terms should remain and the artificial interior points should cancel out.

So let's look at both forms to see how those do. Let's assume a 4 point grid, ranging from $i=0$ to $i=3$. The conservative form expands as:

$$ \frac{(\rho u)_1 - (\rho u)_0}{\Delta x} + \frac{(\rho u)_2 - (\rho u)_1}{\Delta x} + \frac{(\rho u)_3 - (\rho u)_2}{\Delta x} $$

You can see that when you add it all up, you end up with only the boundary terms ($i = 0$ and $i = 3$). The interior points, $i = 1$ and $i = 2$ have canceled out.

Now let's look at the non-conservative form:

$$ \rho_1 \frac{u_1 - u_0}{\Delta x} + u_1 \frac{\rho_1 - \rho_0}{\Delta x} + \rho_2 \frac{u_2 - u_1}{\Delta x} + u_2 \frac{\rho_2 - \rho_1}{\Delta x} + \rho_3 \frac{u_3 - u_2}{\Delta x} + u_3 \frac{\rho_3 - \rho_2}{\Delta x} $$

So now, you end up with no terms canceling! Every time you add a new grid point, you are adding in a new term and the number of terms in the sum grows. In other words, what comes in does not balance what goes out, so it's non-conservative.

You can repeat the analysis by playing with altering the coordinate of those terms outside the derivative, for example by trying $i-1/2$ where that is just the average of the value at $i$ and $i-1$.

How to choose which to use?

Now, more to the point, when do you want to use each scheme? If your solution is expected to be smooth, then non-conservative may work. For fluids, this is shock-free flows.

If you have shocks, or chemical reactions, or any other sharp interfaces, then you want to use the conservative form.

There are other considerations. Many real world, engineering situations actually like non-conservative schemes when solving problems with shocks. The classic example is the Murman-Cole scheme for the transonic potential equations. It contains a switch between a central and upwind scheme, but it turns out to be non-conservative.

At the time it was introduced, it got incredibly accurate results. Results that were comparable to the full Navier-Stokes results, despite using the potential equations which contain no viscosity. They discovered their error and published a new paper, but the results were much "worse" relative to the original scheme. It turns out the non-conservation introduced an artificial viscosity, making the equations behave more like the Navier-Stokes equations at a tiny fraction of the cost.

Needless to say, engineers loved this. "Better" results for significantly less cost!

Answer 2

You show the Euler Equations, reduced forms of Navier-Stokes, which are conservation laws for mass, momentum, and energy along with Stoke's hypothesis. Now, conservative/non-conservative has nothing to do with conservation laws.

In conservative form, you can directly integrate the derivatives once on a control volume and conserve the flux quantities through the control surfaces (i.e., finite volume method). That's conservative vs. non-conservative form (the flux).

In CFD algorithms the form has important implications with shock placements and propagation speeds. For compressible flows, you want to utilize the conservative forms (where you have shocks).

Answer 3

All answers are good, but those do not answer the question. The question is very basic, and reads:

Q: when is a mathematical model called conservative and when non-conservative?

A: a mathematical model is called conservative when it directly stems from conservation of mathematical quantities within a control volume, i.e. from a conservation law.

More in detail, if beside simple conservation (i.e. mass or momentum conservation) we also apply the chain rule then we obtain a non-conservative mathematical model, so called because it does not describe anymore conservation of a quantity on a control volume, since it does not directly stem from a conservation law. But aren't the two formulation equivalent? NO!!! They are equivalent only if the solution is continuous. We cannot apply the chain rule if the solution is not continuous! Therefore, the non-conservative form cannot properly describe the propagation of discontinuities (i.e. shocks). See Toro's book for more details.

Summarizing: the question is about the mathematical model, not about the numerical discretization. Clearly there is a strong link between the numerical and mathematical model, and all the other answers are great. But they don't answer the question.

Answer 4

One important reason (maybe not the only one) to the labels "conservative" and "non-conservative" forms of conservation equations is related to their numerical solution. When written in conservative form, after discretization of the differential equation by a numerical method, such as, the finite volume method, the resulting algebraic equation still holds the conservation principle, regardless of the size of the finite volume. In other words, one can still see the balance of the conserved variable (mass, momentum, species concentration) in the algebraic equation. On the other hand, when an equation in non-conservative form is discretized by, let´s say, a finite difference or finite element method, conservation is only guaranteed when the numerical mesh is refined. In this case, the balance cannot be seen in the algebraic, discretized equation. Therefore, this label only makes sense when handling conservation equations. It does not make any sense to try to write a differential equation that has no relation to a conservation principle in conservative or non-conservative form.