Energy-production-rate preserving numerical approximations to network generating partial differential equations

• Published in: Computers & mathematics with applications (1987) Vol. 84; pp. 148 - 165
• Main Authors: Hong, Qi, Zhao, Jia, Wang, Qi
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 15.02.2021; Elsevier BV
• Subjects: Algorithms; Boundary conditions; Energy quadratization; Energy-dissipation-rate preserving; Finite difference method; Finite difference methods; Flux density; Gradient flow; Gradient flows; Mathematical models; Network-generating PDEs; Partial differential equations; Network-generating PDEs; Energy quadratization; Gradient flows; Energy-dissipation-rate preserving; Finite difference methods
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2020.11.014

We recast a network generating partial differential equation system into a singular limit of a dissipative gradient flow model, which not only identifies the consistent physical boundary conditions but also generates networks. We then develop a set of structure-preserving numerical algorithms for the gradient flow model. Using the energy quadratization (EQ) method, we reformulate the gradient flow system into an equivalent one with a quadratic energy density by introducing auxiliary variables. Subsequently, we devise a series of fully discrete, linear, second order, energy-production-rate preserving, finite difference algorithms to solve the EQ-reformulated PDE system subject to various compatible boundary conditions. We show that the numerical schemes are energy-production-rate preserving for any time steps. Numerical convergence tests are given to validate the accuracy of the fully discrete schemes. Several 2D numerical examples are given to demonstrate the capability of the schemes in predicting network generating phenomena with the gradient flow PDE system, especially, the original network generating PDE model.