Study how to reconstruct an exponential representation of the solution, given a set of moments of the solution.
The above plot demonstrates that the approximation for g_R() base on 1st order Gauss-Lobatto quadrature is not 1st order accurate. This is because calculating involves terms, which requires more than 1st order accuracy to be accurate. At least it is a postive definite approximation.
Test out iteration with limiters
Here are the full iteration equations, for 1D passive advection, for a cell that has no flux into it:
This is a plot of on the next time step as a function of on the previous time step, for lambda = v*dt/dx = 0.05.
The linear extrapolation to the boundary flux has the problem that increases every time step and will eventually exceed the bound .
Using the cubic extrapolation to the boundary (which is a kind of Pade approximation to the exponential extrapolation), does slow down the rate per step by which increases, but it still eventually exceeds the limit.
The only way to rigorously satisfy the limit on is to limit beta to be less than 1, where beta is the mulitplier on the interior volume average term . I think the way to interpret this is that when approaches 1, then the Euler time step is breaking down, because it takes very little time for the whole solultion to advect out of the right boundary. I think it is okay to apply a limiter there, as if we were reverting to a 2cd order centered method, because the order of accuracy for the time integration only holds in smooth regions, where , and this limiter on beta preserves an important feature of the solution while retaining at least first-order accuracy in time.