The Fractal Burgers Equation - Theory and Numerics

Abstract

We study the Cauchy problem for the 1-D fractal Burgers equation, which is a non-linear and non-local scalar conservation law used to for instance model overdriven detonation in gases. Properties of classical solutions of this problem are studied using techniques mainly developed for the study of entropy solutions. With this approach we prove several a-priori estimates, using techniques such as Kruzkow doubling of variables. The main theoretical result of this study is a L1-type contraction estimate, where we show the contraction in time of the positive part of solutions of the fractal Burgers equation. This result is used to show several other a priori estimates, as well as the uniqueness and regularity in time of solutions. We also solve our Cauchy problem numerically, by proposing, analyzing and implementing one explicit and one implicit-explicit method, both based on finite volume methods. The methods are proved to be monotone, consistent and conservative under suitable CFL conditions. Subsequently, several a priori estimates for the numerical solutions are established. A discussion on how the numerical methods may be implemented efficiently, as well as discussions of some of the numerical results obtained conclude this study.