Efficient and Structure-Preserving Time-Stepping Methods for Nonlinear Evolution Equations and Multiphysics Systems

Abstract

This dissertation is devoted to the design and analysis of efficient and structure-preserving time-stepping methods for nonlinear evolution equations and multiphysics systems. Four novel classes of methods are developed and analyzed: low regularity integrators (LRIs) for phase field models, noniterative localized exponential time differencing (ETD) methods for hyperbolic conservation laws, dynamically regularized Lagrange multiplier (DRLM) methods for the incompressible Navier-Stokes equations and the coupled Cahn-Hilliard-Navier-Stokes system, and generalized transferable neural networks (GTransNet) for the Cahn-Hilliard equation.

We first construct LRI schemes based on Duhamel's principle for the classical and conservative Allen-Cahn equations. The schemes are shown to preserve the maximum bound principle, energy stability, and mass conservation (for the conservative Allen-Cahn equation). Optimal temporal and fully discrete error estimates are rigorously derived under minimal regularity assumptions on the exact solution, requiring only continuity in time for the Allen-Cahn equation and Lipschitz continuity for its conservative counterpart. As a result, LRI schemes outperform classical exponential integrators, particularly when the interfacial parameter approaches zero.

For hyperbolic conservation laws, we combine ETD with the discontinuous Galerkin method and nonoverlapping domain decomposition to develop global and localized ETD schemes that are fully explicit and can be coupled with the total variation bounded (TVB) slope limiter to effectively capture moving shocks. Notably, the localized scheme is noniterative, allowing local problems to be solved in parallel across subdomains while preserving the large time step sizes of the global method. Rigorous temporal error analysis and mass conservation under periodic boundary conditions are established for both schemes.

We then develop the DRLM method for the incompressible Navier-Stokes equations, where a time-dependent Lagrange multiplier and a regularization parameter are introduced to reformulate the equations into an equivalent system. First- and second-order DRLM schemes, with explicit treatment of the convection term, are shown to be unconditionally energy stable with respect to the original variables. Optimal error estimates are established for the first-order scheme, with error bounds that decay as the regularization parameter increases. The DRLM framework is further extended to the Cahn-Hilliard-Navier-Stokes system, where the corresponding schemes are fully decoupled, mass-conserving, and require no stabilization term.

Finally, we propose a neural network-based framework for the Cahn-Hilliard equation by combining GTransNet for spatial approximation with stabilized backward differentiation formulas (BDF) for time discretization. A key feature of the resulting GTransNet-BDF schemes is the introduction of a mass-conserving projection that corrects output-layer weights produced by the least-squares system to enforce discrete mass conservation at negligible computational cost. In addition, the schemes are shown to be energy stable, and their mesh-free nature and predetermined hidden layers make the method applicable to complex domains and variable mobility.