Computational Methods for Multi-Scale Temporal Problems: Algorithms, Analysis, and Numerical Experiments

A major challenge in numerical simulation of most natural phenomena is the presence of disparate temporal and spatial scales. Capturing all the fine features can be computationally prohibitive. Hence, development of efficient and accurate multi-scale numerical algorithms has gained immense attention from engineers and scientists. Typically, a single numerical method cannot efficiently capture all the aforementioned features. Due to the assumptions made in construction of numerical methods and mathematical models, the range of applicability to various length and time-scales is often limited. A direction in resolving this issue is to apply different numerical methods in different regions of the computational domain. This strategy enables computation of necessary details as desired by the user. In this work, we propose numerical methodologies based on domain partitioning techniques that allow different time-steps and time-integrators in different regions of the computational domain. The first problem of interest is elastodynamics, which can pose various temporal scales in impact, contact and wave propagation problems. A monolithic (strong) coupling algorithm based on non-overlapping domain partitioning is proposed. The proposed algorithm is based on the theory of differential/algebraic equations and its numerical stability, energy conservation and accuracy is studied in detail. Following these findings, we extend this algorithm to advection-diffusion-reaction problems. The proposed algorithm proves useful especially in cases where the relative strength of the involved processes changes dramatically with respect to spatial coordinates. Numerical stability and accuracy of this method is studied and its application to fast bimolecular chemical reactions is showcased. Further on, we confine our attention to single and multiple-relaxation-time lattice Boltzmann methods for the advection-diffusion equation and study their performance in preserving the maximum principle and the non-negative constraint. Finally, a computational framework based on overlapping domain decomposition methods is proposed. This framework is designed for advection-diffusion problems and allows coupling of the finite element method and lattice Boltzmann methods with different time-steps and grid sizes. Additionally, a new method for enforcing the Dirichlet and Neumann boundary conditions on the numerical solution from the lattice Boltzmann method is proposed. This method is based on maximization of entropy and ensures non-negativity of the discrete distributions on the boundary of the domain. We study the performance of this framework through numerical experiments and showcase its application to fast and equilibrium chemical reactions.