Question

\frac{d}{dt}v_j = D_+v_j is a stable method-of-line scheme for $u_t = u_x$ with periodic boundary condition. but it is only first-order accurate.\newline Correct $D_+$ with $D_+D_-$ and $D_+D_+D_-$ to get a third-order method-of-line scheme. Namely, find a spatial discretization of the form \newline $Q = D_+ + \alpha D_+D_- + \beta D_+D_+D_-$ \newline so that $\frac{d}{dt}v_j = Qv_j$ is third-order accurate. \newline (Hint: show $Qe^{iwx} = (iw + O(h^3))e^{iwx}$. The result can also be used in Question 2.)

          \frac{d}{dt}v_j = D_+v_j is a stable method-of-line scheme for $u_t = u_x$ with periodic boundary condition. but it is only first-order accurate.\newline Correct $D_+$ with $D_+D_-$ and $D_+D_+D_-$ to get a third-order method-of-line scheme. Namely, find a spatial discretization of the form
\newline $Q = D_+ + \alpha D_+D_- + \beta D_+D_+D_-$
\newline so that $\frac{d}{dt}v_j = Qv_j$ is third-order accurate.
\newline (Hint: show $Qe^{iwx} = (iw + O(h^3))e^{iwx}$. The result can also be used in Question 2.)

(d)/(dt)vj = D+vj is a stable method-of-line scheme for ut = ux with periodic boundary condition. but it is only first-order accurate.Correct D+ with D+D- and D+D+D- to get a third-order method-of-line scheme. Namely, find a spatial discretization of the form
Q = D+ + α D+D- + β D+D+D-
so that (d)/(dt)vj = Qvj is third-order accurate.
(Hint: show Qe^iwx = (iw + O(h^3))e^iwx. The result can also be used in Question 2.)

Added by Elizabeth T.

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