A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method

<table class="table-group" id="tab5"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td align="left"></td><td align="center"><svg height="13.45" id="M156" style="vertical-align:-2.21956pt" version="1.1" viewbox="0 0 38.325001 13.45" width="38.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
<g transform="translate(72,-61.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">D</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="15.278666" y="0">Q</tspan>
<tspan style="font-size: 12.50px; " x="24.305832" y="0">2</tspan>
</text>
</g>
</g>
</svg> model</td><td align="center"><svg height="13.45" id="M157" style="vertical-align:-2.21956pt" version="1.1" viewbox="0 0 38.325001 13.45" width="38.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
<g transform="translate(72,-61.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">D</tspan>
<tspan style="font-size: 12.50px; " x="9.0271664" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="15.278666" y="0">Q</tspan>
<tspan style="font-size: 12.50px; " x="24.305832" y="0">3</tspan>
</text>
</g>
</g>
</svg> model</td><td align="center"> Our model</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td align="left"> GRE</td><td align="center"><span style="width: 92.75px;"><svg height="10.9125" id="M158" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 92.75 10.9125" width="92.75" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-67.78,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="6.2515001" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="9.3772497" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="15.62875" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="21.880251" y="0">8</tspan>
<tspan style="font-size: 12.50px; " x="28.13175" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="34.383251" y="0">𝐸</tspan>
<tspan style="font-size: 12.50px; " x="46.08606" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="57.426277" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="63.67778" y="0">3</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 87.550003px;"><svg height="11.0375" id="M159" style="vertical-align:-0.27588pt" version="1.1" viewbox="0 0 87.550003 11.0375" width="87.550003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="6.2515001" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="9.3772497" y="0">7</tspan>
<tspan style="font-size: 12.50px; " x="15.62875" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="21.880251" y="0">9</tspan>
<tspan style="font-size: 12.50px; " x="28.13175" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="34.383251" y="0">𝐸</tspan>
<tspan style="font-size: 12.50px; " x="46.08606" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="57.426277" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="63.67778" y="0">3</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><span style="width: 87.550003px;"><svg height="11.1" id="M160" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 87.550003 11.1" width="87.550003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="6.2515001" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="9.3772497" y="0">8</tspan>
<tspan style="font-size: 12.50px; " x="15.62875" y="0">8</tspan>
<tspan style="font-size: 12.50px; " x="21.880251" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="28.13175" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="34.383251" y="0">𝐸</tspan>
<tspan style="font-size: 12.50px; " x="46.08606" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="57.426277" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="63.67778" y="0">4</tspan>
</text>
</g>
</g>
</svg></span></td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Global relative errors with different models. 

Journal of Applied Mathematics

tab5

Table 5

Table 5: A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method 