A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
A particle-in-Burgers model: theory and numerics - Laboratoire de ...
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Mo<strong>de</strong>l <strong>and</strong> motivation Auxiliary steps Results h = 0: coupl<strong>in</strong>g h = 0: <strong>de</strong>f<strong>in</strong>ition, uniqueness h = 0: <strong>numerics</strong>, existence The coupled problem<br />
Coupled problem: existence, uniqueness of BV solutions / existence of<br />
L ∞ solutions<br />
The above <strong>in</strong>gredients can be used <strong>in</strong> several ways:<br />
– In a fixed-po<strong>in</strong>t argument h(·) ↦→ u(·,·) ↦→ h(·)<br />
(compactness: work <strong>in</strong> C 1 ([0, T]), exploit a W 2,∞ (0, T) bound on h(·) )<br />
– In a time splitt<strong>in</strong>g algorithm (alternatively evolv<strong>in</strong>g u <strong>and</strong> h on small time<br />
<strong>in</strong>tervals):<br />
· u updated from h us<strong>in</strong>g the <strong>theory</strong> of entropy solutions for h frozen;<br />
· h updated from u us<strong>in</strong>g the above weak formulation of the ODE.<br />
– In a numerical scheme (same time splitt<strong>in</strong>g + approximation <strong>in</strong> space of the<br />
conservation law); an <strong>in</strong>terest<strong>in</strong>g possibility is the r<strong>and</strong>om-choice algorithm<br />
(Glimm ), <strong>in</strong> or<strong>de</strong>r not to adapt the space mesh<strong>in</strong>g to the <strong>particle</strong> location.<br />
Theorem (Ma<strong>in</strong> result)<br />
For all BV datum u0 <strong>and</strong> given h(0), h ′ (0), there exists a unique entropy<br />
solution to the coupled problem.<br />
For all L ∞ datum u0 <strong>and</strong> given h(0), h ′ (0), there exists an entropy solution to<br />
the coupled problem.