$T=1$ and we computed the discrete maximum error on the quadrature nodes and the normalized $L^2$ error defined as $\|e_h(\cdot,t^n)\|/\|e_h(\cdot,0)\|,$ where $e_{h}=u-U$. The numerical method appears to converge with an optimal rate in space ($r=4$) but with a suboptimal rate equal to three in time.

    $N$     $M$       $L^{\infty}$ Error       Rate          $L^2$ Error          Rate
  ------- -------- ------------------------- --------- ------------------------ ---------
    $4$    $1000$   $ 0.2630\times 10^{-1}$      –      $0.4263\times 10^{-1}$      –
    $8$    $1000$   $0.2654\times 10^{-2}$    $3.309$   $0.4125\times 10^{-2}$   $3.370$
   $16$    $1000$   $ 0.1916\times 10^{-3}$   $3.793$   $0.2686\times 10^{-3}$   $3.941$
   $32$    $1000$   $0.1243\times 10^{-4}$    $3.945$   $0.1693\times 10^{-4}$   $3.988$
   $64$    $1000$   $0.7863\times 10^{-6}$    $3.983$   $0.1060\times 10^{-5}$   $3.997$
   $128$   $1000$   $ 0.5068\times 10^{-7}$   $3.956$   $0.6636\times 10^{-7}$   $3.998$

  : \[tav1\]Spatial rates of convergence (hybrid scheme)

    $N$     $M$      $L^{\infty}$ Error      Rate           $L^2$ Error          Rate
  ------- ------- ------------------------ --------- ------------------------- ---------
   $20$    $ 20$   $0.1301\times 10^{-3}$      –      $0.1249\times 10^{-3}$       –
   $40$    $40$    $0.1866\times 10^{-4}$   $2.802$   $0.1678\times 10^{-4}$    $2.896$
   $80$    $80$    $0.3888\times 10^{-5}$   $2.262$   $0.3733\times 10^{-5}$    $2.169$
   $160$   $160$   $0.5566\times 10^{-6}$   $2.804$   $0.5465\times 10^{-6}$    $2.772$
   $320$   $320$   $0.7289\times 10^{-7}$   $2.933$   $0.7101\times 10^{-7}$    $2.944$
   $640$   $640$   $0.9443\times 10^{-8}$   $2.948$   $0.8994\times 10^{-8 }$   $2.981$

  : \[tav2\]Temporal rates of convergence (hybrid scheme)

Tables \[tav1\] and \[tav2\] show the numerical spatial and temporal rates of 