[moon_light]

In order to suppress the oscillation, we cite a limiter resulting in avoiding the appearance of new extremum points. I am wondering can we add cells in the position of new extremum points but do not use a limiter. I hope in this way, we can obtain a more accurate simulation of discontinuous solution? Is there any existing techniques which associated with this method? Thank you!

[FMDenaro]

Originally Posted by moon_light

In order to suppress the oscillation, we cite a limiter resulting in avoiding the appearance of new extremum points. I am wondering can we add cells in the position of new extremum points but do not use a limiter. I hope in this way, we can obtain a more accurate simulation of discontinuous solution? Is there any existing techniques which associated with this method? Thank you!

You can use an adaptive refinement, in any case the presence of numerical oscillations must be controlled by a suitable scheme (e.g., ENO/WENO)

[moon_light]

Originally Posted by FMDenaro

You can use an adaptive refinement, in any case the presence of numerical oscillations must be controlled by a suitable scheme (e.g., ENO/WENO)

When you say the adaptive refinement, are you referring to the moving mesh?
Generally speaking, the width of a discontinue part like a shock, is very small(less than the width of a grid), right?
I am wondering is there any algorithm which can find out the discontinue part automatically and then use low order schemes like two order scheme (linear TVD scheme) scheme but at the same time densify the grids to the extent that the scheme is TVD scheme and the accuracy requirements can be satisfied? Thank you!

[FMDenaro]

Originally Posted by moon_light

When you say the adaptive refinement, are you referring to the moving mesh?
Generally speaking, the width of a discontinue part like a shock, is very small(less than the width of a grid), right?
I am wondering is there any algorithm which can find out the discontinue part automatically and then use low order schemes like two order scheme (linear TVD scheme) scheme but at the same time densify the grids to the extent that the scheme is TVD scheme and the accuracy requirements can be satisfied? Thank you!

no, not a moving mesh but a local adaptive mesh based on the gradients threshold ...shock capturing schemes are designed for getting the discontinuity on the computational grid but, as you stated, the shock wave for NS equations can be as small as some mean free path, therefore is practically unresolvable on a grid. You must accept tha the shock layer is spreaded on some cells...
I suggest reading also the book of LeVeque on FV methods for hyperbolic systems.

[moon_light]

Originally Posted by FMDenaro

no, not a moving mesh but a local adaptive mesh based on the gradients threshold ...shock capturing schemes are designed for getting the discontinuity on the computational grid but, as you stated, the shock wave for NS equations can be as small as some mean free path, therefore is practically unresolvable on a grid. You must accept tha the shock layer is spreaded on some cells...
I suggest reading also the book of LeVeque on FV methods for hyperbolic systems.

Thank you! are you refering this book Finite Volume Methods for Hyperbolic Problems?

[FMDenaro]

Originally Posted by moon_light

Thank you! are you refering this book Finite Volume Methods for Hyperbolic Problems?

yes, this one

[praveen]

If you are using second or higher order scheme for a hyperbolic problem, and your solution has discontinuities, then limiter are absolutely necessary. Even if you adapt the mesh, oscillations cannot be eliminated.

But if your problem is parabolic (like Navier-Stokes), then solutions will be smooth though the gradients might be large in some regions. Then with enough grid adaptation, you can get non-oscillatory solutions without limiters.

[FMDenaro]

Originally Posted by praveen

If you are using second or higher order scheme for a hyperbolic problem, and your solution has discontinuities, then limiter are absolutely necessary. Even if you adapt the mesh, oscillations cannot be eliminated.

But if your problem is parabolic (like Navier-Stokes), then solutions will be smooth though the gradients might be large in some regions. Then with enough grid adaptation, you can get non-oscillatory solutions without limiters.

yes, a well know theorem states that monotone linear scheme can be only first order accurate. The key is to build a non-linear scheme (also for solving linear hyperbolic equations) and a limiter is a way to do that.. however, I suggest more modern schemes such as ENO/WENO reconstructions