RT info:eu-repo/semantics/article
T1 Analysis of a scheme which preserves the dissipation and positivity of Gibbs' energy for a nonlinear parabolic equation with variable diffusion
A1 Serna-Reyes, Adán
A1 Macías Díaz, Jorge E.
A1 Reguera López, Nuria
K1 Nonlinear diffusion-reaction equation
K1 Dissipation of Gibbs' free energy
K1 Structure-preserving numerical model
K1 Stability and convergence analysis
K1 Matemáticas
K1 Mathematics
AB In this work, we design and analyze a discrete model to approximate the solutions of a parabolic partial differential equation in multiple dimensions. The mathematical model considers a nonlinear reaction term and a space-dependent diffusion coefficient. The system has a Gibbs' free energy, we establish rigorously that it is non-negative under suitable conditions, and that it is dissipated with respect to time. The discrete model proposed in this work has also a discrete form of the Gibbs' free energy. Using a fixed-point theorem, we prove the existence of solutions for the numerical model under suitable assumptions on the regularity of the component functions. We prove that the scheme preserves the positivity and the dissipation of the discrete Gibbs' free energy. We establish theoretically that the discrete model is a second-order consistent scheme. We prove the stability of the method along with its quadratic convergence. Some simulations illustrating the capability of the scheme to preserve the dissipation of Gibb's energy are presented.
PB Elsevier
SN 0168-9274
YR 2023
FD 2023-01
LK http://hdl.handle.net/10259/8079
UL http://hdl.handle.net/10259/8079
LA eng
NO The corresponding author (J.E.M.-D.) wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.
DS Repositorio Institucional de la Universidad de Burgos
RD 10-may-2024