Discrete entropy structures in nonlinear diffusive equations

Partial differential equations from science and technology typically contain some structu-ral information reflecting inherent physical properties such as positivity, mass and energy conservation, or entropy dissipation. These properties are of major importance in the mathematical analysis for the derivation of a priori estimates which are needed, for instance, in the existence and long-time analysis. Numerical schemes should be designed in such a way that the structural features are preserved on the discrete level in order to obtain accurate and stable algorithms. Whereas consistency and stability of numerical schemes have received much attention in the literature, much less is known about structure-preserving schemes. In this project, we wish to explore the entropy structure of certain highly nonlinear diffusion equations and to derive new numerical schemes which preserve their structure. Equations considered in this project include second- order (porous-medium) equations, fourth-order (thin-film, Derrida-Lebowitz-Speer-Spohn) equations, and Maxwell-Stefan systems for multicomponent gaseous mixtures. We stress the fact that this project is about the mathematical analysis of discrete schemes and their numerical tests, but we do not seek for efficient implementations, optimal iterative techniques, or other aspects of numerical analysis. First, we design temporally higher-order schemes and spatially finite-volume schemes for nonlinear parabolic equations, which are entropy-stable or even entropy-dissipative. This is achieved by an algebraic viewpoint: The continuous entropy estimates are translated to the discrete level by using suitable discrete derivatives and summation-of-parts formulas. Convergence properties are proved and numerical experiments in one and two space dimensions allow for the comparison and evaluation of the various schemes. Second, in a geometric viewpoint, the gradient-flow structure of certain dissipative equations and systems will be exploited by proposing a semi-discretization in time of the evolution by means of the so-called minimizing movement scheme. In particular, the entropy structure of Maxwell-Stefan systems is analyzed. The numerical results from the minimizing movement scheme are compared to those from a standard finite-difference or finite- element discretization using the entropy-variable formulation. Furthermore, temporally higher-order minimizing movement schemes for gradient flows are derived and applied to the fourth-order Derrida-Lebowitz-Speer-Spohn equation.

Partial differential equations from science and technology typically contain some structural information reflecting inherent physical properties such as positivity, mass and energy conservation, or entropy dissipation. These properties are of major importance to understand the behavior of the solutions of the equations and eventually the behavior of the underlying physical system. Numerical schemes, which approximate the partial differential equations, should be designed in such a way that the structural features are preserved on the discrete level in order to obtain accurate and stable algorithms.The main aim of this project was to analyze the entropy structure of certain diffusive partial differential equations, in particular with respect to higher-order time and so-called finite-volume approximations. We have determined conditions under which these approximations preserve the entropy dissipation. Our ideas were based on a combination of tools from different mathematical fields, namely partial differential equations, ordinary differential equations, and time-continuous Markov chains. Our results may help to improve numerical predictions of scientific or technological processes from fluid dynamics, population dynamics, and quantum diffusion theory.