\documentclass[tikz,border=5pt]{standalone}
\usepackage{amsmath}
\usepackage{tikz,pgfplots}
\usepgfplotslibrary{groupplots}
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\definecolor{darkorange25512714}{RGB}{255,127,14}
\definecolor{forestgreen4416044}{RGB}{44,160,44}
\definecolor{steelblue31119180}{RGB}{31,119,180}
\begin{document}
\begin{tikzpicture}
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\definecolor{lightgray204}{RGB}{204,204,204}
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\begin{groupplot}[group style={group size=2 by 1, horizontal sep=65pt}, yticklabel style={font=\large}, xticklabel style={font=\large}, legend style={at={(1.1,-0.55)}, anchor=south, legend columns=3,column sep=10pt}]
\nextgroupplot[log basis y={10}, tick align=outside, tick pos=left, x grid style={darkgray176}, xlabel={Time (t)}, xlabel style={yshift=-4pt}, xmin=-0.05, xmax=1.05, xtick style={color=black}, y grid style={darkgray176}, ylabel={Changes}, ylabel style={yshift=8pt}, ymin=1e-08, ymax=1e-07, ymode=log, ytick style={color=black}]\addplot[thick, lightcoral246112136, dashed] coordinates {(0,0)(0.01,6.13201098786931e-08)(0.1,6.13201096566485e-08)(0.2,6.13201092125593e-08)(0.3,6.13201096566485e-08)(0.4,6.13201096566485e-08)(0.5,6.13201094346039e-08)(0.6,6.13201094346039e-08)(0.7,6.132010876847e-08)(0.8,6.13201094346039e-08)(0.9,6.13201094346039e-08)(1,6.13201096566485e-08)};
\addlegendentry{L2 energy (N=100)}\addplot[thick, lightcoral246112136] coordinates {(0,0)(0.001,6.13201101e-08)(0.1,6.13201098e-08)(0.2,6.13201101e-08)(0.3,6.13201101e-08)(0.4,6.13201101e-08)(0.5,6.13201101e-08)(0.6,6.13201098e-08)(0.7,6.13201101e-08)(0.8,6.13201101e-08)(0.9,6.13201103e-08)(1,6.13201101e-08)};
\addlegendentry{L2 energy (N=1000)}\addplot[thick, dodgerblue59163236, dashed] coordinates {(0,0)(0.01,3.7330857960427e-08)(0.1,3.72764823453053e-08)(0.2,3.72045283469902e-08)(0.3,3.72233230905294e-08)(0.4,3.72745581067591e-08)(0.5,3.72722719355068e-08)(0.6,3.72562509731722e-08)(0.7,3.72776298718236e-08)(0.8,3.72898876221939e-08)(0.9,3.72659361147498e-08)(1,3.72664397119138e-08)};
\addlegendentry{Hamiltonian energy (N=100)}\addplot[thick, dodgerblue59163236] coordinates {(0,0)(0.001,3.73320077e-08)(0.1,3.73320041e-08)(0.2,3.73319970e-08)(0.3,3.73319988e-08)(0.4,3.73320045e-08)(0.5,3.73320041e-08)(0.6,3.73320023e-08)(0.7,3.73320045e-08)(0.8,3.73320059e-08)(0.9,3.73320041e-08)(1,3.73320041e-08)};
\addlegendentry{Hamiltonian energy (N=1000)}\foreach \y in {2,3,4,5,6,7,8,9} {\addplot[gray,dotted,forget plot] coordinates {(-0.05, \y*1e-08)(1.05, \y*1e-08)};
}\foreach \x in {0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0} {\addplot[gray,dotted,forget plot] coordinates {(\x, 1e-09)(\x, 100)};
}\foreach \p in {-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1} {\addplot[gray,dotted,forget plot] coordinates {(-0.05, 10^\p)(1.05, 10^\p)};
}\addlegendimage{thick, steelblue31119180}\addlegendentry{MHK Example \ref{ex: matrix, nonlinear, time-dependent}}\addlegendimage{thick, forestgreen4416044}\addlegendentry{MHK Example \ref{ex: matrix, nonlinear, time-independent}}\addlegendimage{thick, black, dashed}\addlegendentry{$O(h^4)$}\nextgroupplot[log basis x={10}, log basis y={10}, tick align=outside, tick pos=left, x grid style={darkgray176}, xlabel={Time (t)}, xlabel style={yshift=-4pt}, xmin=0.000794328234724281, xmax=0.125892541179417, xmode=log, xtick style={color=black}, xtick={1e-05,0.0001,0.001,0.01,0.1,1,10}, xticklabels={${10^{-5}}$,${10^{-4}}$,${10^{-3}}$,${10^{-2}}$,${10^{-1}}$,${10^{0}}$,${10^{1}}$}, y grid style={darkgray176}, ylabel={$\ell_2$ error}, ylabel style={yshift=8pt}, ymin=1.41949157180871e-12, ymax=3.66592986113268, ymode=log, ytick style={color=black}, ytick={1e-14,1e-12,1e-10,1e-08,1e-06,0.0001,0.01,1,100,10000}, yticklabels={${10^{-14}}$,${10^{-12}}$,${10^{-10}}$,${10^{-8}}$,${10^{-6}}$,${10^{-4}}$,${10^{-2}}$,${10^{0}}$,${10^{2}}$,${10^{4}}$}]\addplot[thick, steelblue31119180] coordinates {(0.1, 5.90995331273363e-05)(0.03125, 5.96642431767327e-07)(0.01, 6.28647280764416e-09)(0.00315457413249211, 6.39343884713583e-11)(0.001, 5.20375654071971e-12)};
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\addplot[thick, black, dashed] coordinates {(0.1, 0.01)(0.03125, 9.5367431640625e-05)(0.01, 1e-06)(0.00315457413249211, 9.90291271421585e-09)(0.001, 1e-10)};
\foreach \x in {0.1,0.09,0.08,0.07,0.06,0.05,0.04,0.03,0.02,0.01,0.009,0.008,0.007,0.006,0.005,0.004,0.003,0.002,0.001} {\addplot[gray,dotted,forget plot] coordinates {(\x, 1.419e-12)(\x, 3.66)};
}\foreach \p in {-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0} {\addplot[gray,dotted,forget plot] coordinates {(0.00079, 10^\p)(0.125, 10^\p)};
}
\end{groupplot}
\end{tikzpicture}
\end{document}