Investigating the Method of Transport by Writing it as a Boltzmann Scheme

Description

Boltzmann Schemes for the Euler equations use the fact that the Euler equations are the moments of the Boltzmann equation when the distribution function is locally Maxwellian. We will consider a group of such schemes introduced by Sanders and Prendergast (1974), and further investigated by Deshpande (1986) and Perthame (1990, 1992). These schemes have the following properties:

• They are unconditionally stable in L¹, in particular, density and pressure remain positive.
• Often, the numerical solution fulfills an entropy inequality.

The Method of Transport is a genuinely multidimensional finite volume scheme developed by M. Fey for the Euler equations. We have proved that the first order method can be written as a Boltzmann scheme, therefore it is unconditionally stable. A slightly modified version of the second order method can also be written as a Boltzmann scheme. It is stable under certain conditions on the gradients.

We will now investigate whether numerical solutions obtained by the Method of Transport fulfill a discrete version of the entropy inequality.

Contacts

Prof. Rolf Jeltsch and Susanne Zimmermann (szimmer@sam.math.ethz.ch).