= Solution
Let $x_m=mh$ be a uniform periodic mesh, with indices interpreted modulo the number of nodes. Write the <finite element> approximation as $u_h(x,t)=\sum_jU_j(t)\phi_j(x)$ using the <chapeau functions>. The <Galerkin method> requires
$$
\int_0^1\partial_tu_h\,\phi_m\,dx=\int_0^1\partial_xu_h\,\phi_m\,dx.
$$
The <mass matrix> has $M_{mm}=2h/3$ and $M_{m,m\pm1}=h/6$. The spatial <matrix> $C_{mj}=\int\phi_m\phi_j'$ has $C_{m,m+1}=1/2$ and $C_{m,m-1}=-1/2$, with zero diagonal. Therefore the semidiscrete equations are
$$
\boxed{\frac h6\left(\dot U_{m-1}+4\dot U_m+\dot U_{m+1}\right)=\frac12(U_{m+1}-U_{m-1}),}
$$
or equivalently $M\dot U=CU$. The coefficients come from integrating the products of the two overlapping <piecewise-linear hat functions> on each interval; periodicity supplies the wraparound entries.
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