I've been learning about the Discontinous Galkerin Method by reading the book by Hesthaven and Warburton and have ran into a problem with the advection equation with forcing

$u_t + u_x = g(x,t)$

Following the book I've been able to implement a method that works when $g(x,t) = 0$ in Mathematica but am having trouble to get a stable method when the forcing function is active.

Additionally I've got the domain as $[0,1]$ with 4 elements and have tried both situations below with differing order of elements and boundary and initial conditions are

$u(0,t) = sin(t)$

$u(x,0) = -sin(x)$

I assume I'm missing something very obvious but I've tried implementing it as a 'direct method'

$\textbf{u}_t =-\mathcal{M}^{-1}\mathcal{S} \textbf{u} + \mathcal{M}^{-1}[\textbf{f}^*] + \mathcal{M}^{-1} \textbf{g}_h $


$[\textbf{f}^*]$ are the DG fluxes and

$\textbf{g}_h = \int g(x,t) l(x) dx $

and $l(x)$ is the Lagrange polynomial, which appears to work so long as the forcing function goes to zero at the origin but fails when it doesn't and is particularly bad when g(x,t) = 1

as well as an 'indirect' method where

$\textbf{u}_t =-\mathcal{M}^{-1}\mathcal{S} \textbf{u} + \mathcal{M}^{-1}[\textbf{f}^*] + g(x,t) $

i.e. Following the book the mass matrix you get from the discretisation of g (which I'm not 100% clear on but that's a different question) is cancelled out by the inverse mass matrix from $u_t$ term which doens't work at all.

The questions are, am I correct in assuming that in general $g(x,t)$ can be any function, including a non-zero constant, without additional corrections e.g. artificial viscosity?

And am I missing anything in the formulation, e.g. do the inter-element fluxes need to include additional terms from $g(x,t)$?

  • $\begingroup$ Yes, in principle $g$ can be any function of $x$ and $t$. $\endgroup$ – chris Nov 2 '17 at 8:57
  • $\begingroup$ No it cannot be any function of $x$ and $t$. It must be bounded or well-posedness is lost. $\endgroup$ – ekkilop Jul 14 '19 at 13:51

Your numerical integration technique also need to be stable to $u_{t}=g(x,t)$ which may have different time step restrictions than the hyperbolic terms.


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