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Simple question
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Consider the equation in a cube <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center"> u<sub>t</sub>=u<sub>xx</sub>+u<sub>yy</sub>+u<sub>zz</sub></td></tr></table> </td></tr></table> I replace derivatives with differences, e.g., <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center" cellspacing="0" cellpadding="0"><tr><td nowrap="nowrap" align="center"> u<sub>xx</sub>=</td><td nowrap="nowrap" align="center"> *1 <div class="hrcomp"><hr noshade="noshade" size="1"/></div>h<sup>2</sup> </td><td nowrap="nowrap" align="center"> (u<sub>i<font face="symbol">-</font> 1,jk</sub><font face="symbol">-</font> 2u<sub>ijk</sub>+u<sub>i+1jk</sub>)</td></tr></table> </td></tr></table> and then use explicit Runge-Kutta method with automatic choice of timestep. On a grid 100 x 100 x 100 all is OK. But if I take a grid 500 x 500 x 500 the timestep becomes too small since the stiffness raises. How to overcome this? Thank. </html> |
Simple answer
Use an implicit method.
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