[granzer]

A basic question. I know governing equations (i am considering CFD or structural governing equations) in differential equation form(strong form?) can be integrated directly (with a test function?) to get a weak form (integral form) of the governing equation. But integral form can also be derived from using RTT, and not from the differential equation. Are these integral equations also in weak form?

Are all integral form (both conservative and non-conservative integral from) of governing equations in weak form?

[FMDenaro]

Originally Posted by granzer

A basic question. I know governing equations (i am considering CFD or structural governing equations) in differential equation form(strong form?) can be integrated directly (with a test function?) to get a weak form (integral form) of the governing equation. But integral form can also be derived from using RTT, and not from the differential equation. Are these integral equations also in weak form?

Are all integral form (both conservative and non-conservative integral from) of governing equations in weak form?

The conservation law in integral form, as derived from the RTT, is a particular weak form but you can derive other weak forms from the differential equation using several test functions.

Have a look to Sec.11.11 here https://www.google.com/url?sa=t&rct=...Fc3FQFbNPpgtTL

[granzer]

Originally Posted by FMDenaro

The conservation law in integral form, as derived from the RTT, is a particular weak form but you can derive other weak forms from the differential equation using several test functions.

Have a look to Sec.11.11 here https://www.google.com/url?sa=t&rct=...Fc3FQFbNPpgtTL

@FMDenaro But are all integral forms of the differential equations considered weak form?

[FMDenaro]

Originally Posted by granzer

@FMDenaro But are all integral forms of the differential equations considered weak form?

You can integrate a differential equation but to get a weak form you need to eliminate any differentiation on the variable and transfer that to the test function. That is possible if integrating by part you can do that. For example, starting from the quasi-linear form of the Euler equation and coming back to the divergence form.



If you consider the NSE you have that the diffusive terms are second order and differentiation remains.

[praveen]

Starting from the definition of weak form, you can show that integral form of conservation law holds true for any control volume.



The finite volume method is based on integral form. Lax-Wendroff theorem shows that if the numerical solutions from a finite volume method converge, then the limiting solution is a weak solution.


Due to these results, the finite volume solution can be considered as an approximation to the weak solution.

[Ford Prefect]

Slightly off topic, but I have always found it strange to start with the partial differential equations and then go backwards to the flux balance expression. It feels more natural to skip the partial differential equation all together and just work with the control volume fluxes.

[praveen]

Originally Posted by Ford Prefect

Slightly off topic, but I have always found it strange to start with the partial differential equations and then go backwards to the flux balance expression. It feels more natural to skip the partial differential equation all together and just work with the control volume fluxes.

I agree with your sentiments. If your interest is doing numerics only, then there is no need to deal with differential equations and the weak form.


But if you want to do some math theory of existence/uniqueness of solutions, and also want to do numerical analysis in terms of showing convergence of numerical approximations and theoretical error estimates, then the weak formulation is unavoidable.

[FMDenaro]

Originally Posted by Ford Prefect

Slightly off topic, but I have always found it strange to start with the partial differential equations and then go backwards to the flux balance expression. It feels more natural to skip the partial differential equation all together and just work with the control volume fluxes.

The process to define the conservation laws has the Reynolds Transport theorem as foundation. That is the physical way to express a conservation of an extensive quantity (mass, momentum, energy) is expressed by means of the integral balance. The derivation of the differential form is more a "mathematical" consequence of the assumption that the functions are regular and differentiable. In terms of a physical law there is no a real meaning in thinking about an equation valid in a pointwise sense (that is a point of vanishing measure). Again, the differential equation is valid under the continuum hypothesis, the poit is actually a small but finite volume and the pointwise variable is actually an averaged value over such volume.
I conclusion, an integral law is always implied.
There is a recent paper of Len Margolin addressing the topic.