Resolving phase transitions with discontinuous Galerkin methods
Eduardo Grossi, Nicolas Wink
Abstract
We demonstrate the applicability and advantages of Discontinuous Galerkin (DG) schemes in the context of the Functional Renormalization Group (fRG). We investigate the O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -model in the large N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> limit. It is shown that the flow equation for the effective potential can be cast into a conservative form. We discuss results for the Riemann problem, as well as initial conditions leading to a first and second order phase transition. In particular, we unravel the mechanism underlying first order phase transitions, based on the formation of a shock in the derivative of the effective potential.