[cfdnewb123]

Hi, I have only worked with hyperbolic PDEs and that mainly involved solving the conservation system of Euler equation explicitly. But what about problems which contain mixed type (e.g., elliptic, parabolic, hyperbolic) of PDE like Navier-Stokes equation which is parabolic? Do we consider which type of PDE is dominant before applying the specific numerical methods for that particular PDE?

For example, at high Mach number, the Navier-Stokes equation is dominated by the hyperbolic character of the PDEs and thus, should be solved explicitly. But if the Mach number is low, the Navier-Stokes equation is mainly diffusive and thus, parabolic, meaning it should be solved implicitly like the heat equation?

What about transitional flow whereby neither the hyperbolic nor parabolic aspect of the Navier Stokes equation dominate each other? Do we separate the Navier Stokes equation to solve the Euler equation explicitly while the right hand side of the Navier-Stokes equation (e.g., viscous stress) is solved implicitly?

Any resources is appreciated. Thank you.

[LuckyTran]

The short answer is yes. Of course you must always consider the nature of characteristics and choose the appropriate solver.

A slightly longer answer is we choose different discretizations and flux schemes for each term in the Navier-Stokes equation to handle the locally different characteristics. The advection term is treated one way and viscous terms treated another way.

I want to point out that most of what you say is true for 1D equations with homogeneous flow characteristics, which is very academic... Practical problems in physics (whether it be Navier-Stokes or Maxwell equations or Schrodinger's equation) often result in boundary layers where you have one region of the flow/solution that is dominated by one characteristic and another region that is dominated by another. In N-S for wall bounded flows for example, the region of flow near walls is dominated by viscous effects (i.e. parabolic ) and not-so-far from walls it is very inviscid. So you must carefully choose your methods in each region regardless of whether the outer Mach number is high or low. You must always consider the local characteristics of each individual computational cell.

Explicit versus implicit offers numerical advantages that have less to do with the PDE being solver and more to do with the hardware and scalability. Implicit solvers need lots of fast memory. Memory often comes in high capacity and slow (e.g. disks) or low capacity but fast (e.g. cache). Explicit solvers demand fast arithmetic, i.e. many super fast cpu's. You don't choose explicit versus implicit in a vacuum. You can solve heat equation with explicit methods and it will run very very fast and be super accurate, for small problems (e.g. FTCS) but it does not scale well.

Transitional flow has a special meaning in N-S and I recommend you don't use it to refer to flows that are neither viscous nor inviscid. The way you use the term transitional flow... just means any flow. I adapt a quote from one of my advisors for this purpose: Saying you are writing a book on non-linear dynamics is like saying you are writing a book about the study of non-elephant species. There are so few flows that are purely viscous or purely inviscid that calling the thingy in-between transitional flow is just weird! There's nothing transitional about it... that's how most non-elementary flows are! Most living organisms are not elephants. The term you actually meant to use is transonic (not transitional) but again I would not recommend transonic to refer to flows that are somewhere in between 0 m/s and the speed of light.

Of course your question raises valid concerns. These were the same mathematical conundrums that plagued great practicioners until Prandtl developed his boundary layer theory! I highly recommend anyone interested in physical aspects (beyond being a computer science nerd) of the topic to look into the work Prandtl, Blasius, and Nikuradse for a non-numerical, theoretical foundation with historical anecdotes.

[cfdnewb123]

Originally Posted by LuckyTran

The short answer is yes. Of course you must always consider the nature of characteristics and choose the appropriate solver.

A slightly longer answer is we choose different discretizations and flux schemes for each term in the Navier-Stokes equation to handle the locally different characteristics. The advection term is treated one way and viscous terms treated another way.

I want to point out that most of what you say is true for 1D equations with homogeneous flow characteristics, which is very academic... Practical problems in physics (whether it be Navier-Stokes or Maxwell equations or Schrodinger's equation) often result in boundary layers where you have one region of the flow/solution that is dominated by one characteristic and another region that is dominated by another. In N-S for wall bounded flows, the region of flow near walls is dominated by viscous effects (i.e. parabolic ) and far from walls it is very inviscid. So you must carefully choose your methods in each region regardless of whether the outer Mach number is high or low.

Explicit versus implicit offers numerical advantages that have less to do with the PDE being solver and more to do with the hardware and scalability. Implicit solvers need lots of fast memory. Memory often comes in high capacity and slow (e.g. disks) or low capacity but fast (e.g. cache). Explicit solvers demand fast arithmetic, i.e. many super fast cpu's. You don't choose explicit versus implicit in a vacuum. You can solve heat equation with explicit methods and it will run very very fast and be super accurate, for small problems (e.g. FTCS) but it does not scale well.

Transitional flow has a special meaning in N-S and I recommend you don't use it to refer to flows that are neither viscous nor inviscid. The way you use the term transitional flow... just means any flow. I adapt a quote from one of my advisors for this purpose: Saying you are writing a book on non-linear dynamics is like saying you are writing a book about the study of non-elephant species. There are so few flows that are purely viscous or purely inviscid that calling the thingy in-between transitional flow is just weird! There's nothing transitional about it... that's how most non-elementary flows are!

Thanks for your response. I guess my main confusion is whether there is a need to classify flows based on the type of PDEs since the reason I see is that it helps in deciding a good choice for numerical methods. This may be true for cases that are purely one type like lapace/heat/wave equation where there are different techniques to handle each type of PDEs.

However, for Navier-Stokes equation, it is mix of these different types of PDEs, usually parabolic and hyperbolic. In that case, what techniques are used to resolve these mix types of PDEs? From what I have seen so far, the problem is treated as hyperbolic nature and solved using Godunov scheme with the viscous term handled using simple finite difference. This doesn't seem right to me especially when the numerical dissipation is much larger than the actual dissipation from viscous (or even turbulence) effect.

[LuckyTran]

Well consider that advection dominated characteristics are advection dominated. You can sprinkle in some dissipation (whether physical or artificial) and it won't change the fact that the characteristic is advection dominated.

Consider that you throw a ball upwards and it travels in a projectile motion. Its trajectory is mostly governed by ballistic effects (e.g. gravity). Double or triple the air drag, it makes not much difference. Now if you fired a bullet on the other hand... That is a little different because there the speed of the bullet is limited by drag effects. That is why the characteristics are so important

[FMDenaro]

Just as simple suggestion, a more easy way to approach the matter is to start from the hypothesis you introduce in the flow formulation.

Pure incompressible flows (M=0), low Mach flows (M very small, typical of combustion), compressible flows (M>0.3).
Starting from such formulations it should be simple to deduce the mathematical character depending on the further choice between ideal or real fluid.

[cfdnewb123]

Originally Posted by FMDenaro

Just as simple suggestion, a more easy way to approach the matter is to start from the hypothesis you introduce in the flow formulation.

Pure incompressible flows (M=0), low Mach flows (M very small, typical of combustion), compressible flows (M>0.3).
Starting from such formulations it should be simple to deduce the mathematical character depending on the further choice between ideal or real fluid.

I am aware about the mathematical character of the different types of flow but knowing whether the flow is hyperbolic/parabolic/elliptic only seem useful if the flow only exhibit one mathematical character (e.g., heat/diffusion/laplace) as that is when we can select specific solvers/b.c.s/i.c.s to tackle them.

But what about problem that has mixed character (E.g., NS equation which is parabolic-hyperbolic)? Does knowing that it is parabolic-hyperbolic help me with deciding what type of solver/b.c.s/i.c.s to use?

[FMDenaro]

Originally Posted by cfdnewb123

I am aware about the mathematical character of the different types of flow but knowing whether the flow is hyperbolic/parabolic/elliptic only seem useful if the flow only exhibit one mathematical character (e.g., heat/diffusion/laplace) as that is when we can select specific solvers/b.c.s/i.c.s to tackle them.

But what about problem that has mixed character (E.g., NS equation which is parabolic-hyperbolic)? Does knowing that it is parabolic-hyperbolic help me with deciding what type of solver/b.c.s/i.c.s to use?

Let me illustrate an example, incompressible viscous flows. The continuity equation is hyperbolic and the momentum equation is parabolic. However, the continuity equation is "converted" in the elliptic equation for the pressure.
Therefore, the solver has both the algorithm for parabolic momentum equation and that for elliptic pressure equation.


As further example, compressible ideal fluids are hyperbolics, no matter about the Mach number.

[LuckyTran]

You yourself are aware of practical examples of how Navier-Stokes equations are solved. Where is the confusion?

It sounds like you are having doubts about having wasted time in your mathematical studies and worried that they have been a fruitless endeavor. Learning the basics of how to approach purely parabolic and purely hyperbolic problems sets you up for understanding how to solve problems when both effects are present. Namely, you wouldn't use a central differencing scheme for the hyperbolic part and you wouldn't use a purely marching scheme for the parabolic part