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 />
...Auxiliary steps to approach the full <strong>mo<strong>de</strong>l</strong><br />
Other way around, we should un<strong>de</strong>rst<strong>and</strong> how to evolve the <strong>particle</strong><br />
location given the fluid state at time t. Recall the equation (ODE) for<br />
the <strong>particle</strong>:<br />
mh ′′ (t) = λ (u(t, h(t)) − h ′ (t)).<br />
Recall that u(t,·) has a jump at x = h(t)...<br />
Difficulty 2 : un<strong>de</strong>rst<strong>and</strong> the equation <strong>in</strong> the Carathéodory sense ? In<br />
the Filippov sense ?? We will see that a nice mathematical <strong>and</strong><br />
physical <strong>in</strong>terpretation is possible:<br />
• the <strong>particle</strong> is driven by the lack of mass conservation <strong>in</strong> the<br />
equation for u ; or, equivalently, the total quantity of movement<br />
<br />
R u(t,·)+mh′ (t) is conserved.<br />
• the ODE for h can be written <strong>in</strong> a weak form that <strong>in</strong>volves the values<br />
of u(t,·) on R (which is more "robust")<br />
With these auxiliary steps well un<strong>de</strong>rstood, we can<br />
• th<strong>in</strong>k of the appropriate <strong>de</strong>f<strong>in</strong>ition of solution<br />
• use fixed-po<strong>in</strong>t arguments to guarantee existence<br />
• use time splitt<strong>in</strong>g algorithms (evolve the PDE <strong>and</strong> the ODE<br />
alternatively) for existence (constructive) <strong>and</strong> efficient <strong>numerics</strong>.