Local DG predictor and polynomial reconstruction

krolm · August 4, 2021, 08:49

Hi,

I am new here and in numerical methods as well. I’m trying to write some one-dimensional finite volume code for two-phase flows. My goal was to make it of high order accuracy and this ultimately brought me to local Discontinous Galerkin predictor. I think I quite understand now finite volume method, however, Galerkin methods are something I have problems with. As I have no degree in mathematics or numerical methods it is hard for me to understand every equation used to obtain the predictor. I hope some of you could explain me this issue in more comprehensible form, as I would like to learn this.

I use this lecture notes as a basis and I will refer to them when pointing where I have problems:
http://www.olindozanotti.net/wp-cont...-Frankfurt.pdf
I have also access to some other papers, including those appearing in references in above notes.

For every problem below, assume one spatial dimension and polynomials of degree 2.

My problems start at the polynomial reconstruction step. Could you explain me, how do I get the system of linear equations from eq (1.42)? I know integration by gaussian quadrature and Einstein summation convention, but I don’t understand how this should give me different equations for each cell inside stencil.
Moreover, eqs (1.42) and (1.43) are very confusing for me. As above, I understand the summation convention, but here, with multiple indices, it’s hard to me to figure out how to calculate it.

Going to the calculations of predictor itself, there are two functions theta - basis function with index p and test function with index q. Should I assume them as the same? And how do I get the polynomial psi for time variable in eq (1.49)?

I’m sorry for these beginner’s questions, but I couldn’t find any answer yet, so eventually I decided to seek some help. Probably I will have other questions once I find solutions to those above.

Eifoehn4 · August 8, 2021, 15:30

The literature you mentioned is really bad.
First step would be to search for good literature.

Moreover, i would recommend to implement RK-DG instead of ADER-DG first.

Regards

krolm · August 10, 2021, 04:22

Thanks for your answer. Could you then recommend me some good literature? I’ve preferred finite volume methods over discontinuous Galerkin by now, as DG methodology is difficult to understand for me.

Eifoehn4 · August 15, 2021, 07:26

Don't get me wrong, the literature you mentioned is not bad in the theoretical part. However, it is missing many practical implementation details. Perhaps, the book of Kopriva will help you:

Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers

Regards

August 4, 2021, 08:49	Local DG predictor and polynomial reconstruction	#1
krolm New Member Join Date: Aug 2021 Posts: 3 Rep Power: 4	Hi, I am new here and in numerical methods as well. I’m trying to write some one-dimensional finite volume code for two-phase flows. My goal was to make it of high order accuracy and this ultimately brought me to local Discontinous Galerkin predictor. I think I quite understand now finite volume method, however, Galerkin methods are something I have problems with. As I have no degree in mathematics or numerical methods it is hard for me to understand every equation used to obtain the predictor. I hope some of you could explain me this issue in more comprehensible form, as I would like to learn this. I use this lecture notes as a basis and I will refer to them when pointing where I have problems: http://www.olindozanotti.net/wp-cont...-Frankfurt.pdf I have also access to some other papers, including those appearing in references in above notes. For every problem below, assume one spatial dimension and polynomials of degree 2. My problems start at the polynomial reconstruction step. Could you explain me, how do I get the system of linear equations from eq (1.42)? I know integration by gaussian quadrature and Einstein summation convention, but I don’t understand how this should give me different equations for each cell inside stencil. Moreover, eqs (1.42) and (1.43) are very confusing for me. As above, I understand the summation convention, but here, with multiple indices, it’s hard to me to figure out how to calculate it. Going to the calculations of predictor itself, there are two functions theta - basis function with index p and test function with index q. Should I assume them as the same? And how do I get the polynomial psi for time variable in eq (1.49)? I’m sorry for these beginner’s questions, but I couldn’t find any answer yet, so eventually I decided to seek some help. Probably I will have other questions once I find solutions to those above.

August 8, 2021, 15:30		#2
Eifoehn4 Senior Member - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 13	The literature you mentioned is really bad. First step would be to search for good literature. Moreover, i would recommend to implement RK-DG instead of ADER-DG first. Regards __________________ Check out my side project: A multiphysics discontinuous Galerkin framework: Youtube, Gitlab.

August 15, 2021, 07:26		#4
Eifoehn4 Senior Member - Join Date: Jul 2012 Location: Germany Posts: 184 Rep Power: 13	Don't get me wrong, the literature you mentioned is not bad in the theoretical part. However, it is missing many practical implementation details. Perhaps, the book of Kopriva will help you: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers Regards __________________ Check out my side project: A multiphysics discontinuous Galerkin framework: Youtube, Gitlab.

August 10, 2021, 04:22		#3
krolm New Member Join Date: Aug 2021 Posts: 3 Rep Power: 4	Thanks for your answer. Could you then recommend me some good literature? I’ve preferred finite volume methods over discontinuous Galerkin by now, as DG methodology is difficult to understand for me.