Howdy,
So I have the 1-d heat equation u_{t} = \alpha u_{xx} +s
I represent the solution u=\sum c_j(t)\phi_j(x), do integrations by parts on the spatial derivative, and plug in the solution to yield an ode system
A\dot{c} = Bc +d(t) i.e do a semi-discretization.
I then solve the ODE system, to produce the solution to the original PDE. When I use “CG” everything works great, and I find a convergence order of 2 when I use 1st order polyonmals (theoretically I should see convergence of 1 degree larger than the polynomial order I used). When I take this exact same code and change it to “DG”, I do not recover 2nd order convergence, and the solution is wrong.
My boundary conditions are being enforced correctly in both the CG and DG case, which makes me think that the problem lies in the time stepper. I could believe if somehow the DG makes the right hand side of the ODE not smooth that could create problems. I’m curious if maybe you have to do special time stepping for DG? To me since I did everything for arbitary basis \phi_j(x), I would expect the mathematics work out if I used either continous basis function or discontinous. Any opinions?
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