Thanks for the response. I am totally open to the idea of the algorithm not
being stable.
The term shown in the graph is from a fairly simple equation. It arises
from the need of using the Local Discontinuous Galerkin method for a second
order derivative in one of the hyperbolic equations. The equation being
solved is:
grad a - q = 0
where a is the variable that is equal to 1 everywhere and q is to be solved
for. I try and do a strong form and end up with:
\int_cell phi * q = \int_cell phi * grad a + \int_face phi*( {{a}} -
a_interior ) * normal
where {{a}} is the average of the values of a on the faces.
This though leads to the horribly broken solution of q that has positive
and negative values and jumps when it should be smooth. Is the answer to
use a limiter or filter at this point? I am new to discontinuous galerkin
but I haven't seen a filter or limiter being used in local discontinuous
galerkin before.
On Thursday, July 18, 2024 at 1:13:53 PM UTC-6 Wolfgang Bangerth wrote:
>
> Sean:
>
> >
> VectorTools::interpolate(mapping,dof_handler_DG,Functions::ConstantFunction<dim>(1.),alpha_solution);
> >
> > Obviously changing the final vector for different variables.
> >
> > Later, when the gradient of these terms are used the gradient calculated
> by
> > deal.ii is not 0. I do get very small values on the order of 10^(-14)
> and when
> > I account for the inverse jacobian factor the unit_gradient returns
> values
> > ranging between +/- .9x10^(-15) & +/- 2x10^(-15).
>
> Right, and this can not be avoided. Your alpha_solution corresponds to a
> function of the form
> \sum_j c_j phi_j(x)
> where the coefficients c_j in your case happen to all be ones. Then if you
> compute the gradient of this function, it is computed as
> \sum_j c_j [grad phi_j(x)]
> which *should* be zero, but because grad phi_j(x) is not zero, you are
> adding
> together things that are subject to round-off errors and so you have to
> expect
> that the terms in the sum do not exactly cancel but are on the order of
> round-off. That's of course exactly what you observe.
>
>
> > These seem very small but due to their values being periodic they
> quickly grow
> > and destroy stability of my simulation. Are there any known fixes for
> this?
>
> This is the real problem in your program: That cannot tolerate small
> perturbations. This will be true of all perturbations, not just those that
> come from round-off. In other words, your algorithm is not *stable*: Small
> perturbations grow without bounds. No algorithm that is not stable is
> useful.
> You need to figure out what it is that makes your algorithm instable.
>
> Best
> W.
>
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
>
>
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